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# Boundary regularity of a fourth order Alt-Caffarelli problem and applications to the minimization of the critical buckling load

Jimmy Lamboley, Mickaël Nahon

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Jimmy Lamboley, Mickaël Nahon. Boundary regularity of a fourth order Alt-Caffarelli problem and applications to the minimization of the critical buckling load. 2025. hal-05427874

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# Boundary regularity of a fourth order Alt-Caffarelli problem and applications to the minimization of the critical buckling load

Jimmy Lamboley, Mickaël Nahon

## Abstract

We study a higher order analogue to the Alt-Caffarelli functional that arises in several shape optimization problems, among which the minimization of the critical buckling load of a clamped plate of fixed area. We obtain several regularity results up to the boundary in two dimensions, in particular we prove the full regularity of the boundary (analytic outside angles of opening ≈ 1.43π) near any point of density less than 1 of the optimal shape. These results are based on the monotonicity formula discovered by Dipierro, Karakhanyan, and Valdinoci in [11], which we improve with a new epiperimetric inequality.

## Contents

<table><tr><td>1</td><td>Introduction</td><td>2</td></tr><tr><td>1.1</td><td>General context</td><td>2</td></tr><tr><td>1.2</td><td>Motivations</td><td>3</td></tr><tr><td>1.3</td><td>Main results</td><td>5</td></tr><tr><td>1.4</td><td>Notations and outline of the paper</td><td>9</td></tr></table>

<table><tr><td>2</td><td>Preliminary estimates</td><td>10</td></tr></table>

<table><tr><td>3</td><td>Monotonicity formula and blow-ups</td><td>16</td></tr><tr><td>3.1</td><td>Monotonicity formula of Dipierro, Karakhanyan and Valdinoci recast in exponential coordinates</td><td>16</td></tr><tr><td>3.2</td><td>Classification of 2-homogeneous minimizers</td><td>24</td></tr><tr><td>3.3</td><td>Existence of blow-ups</td><td>26</td></tr></table>

<table><tr><td>4</td><td>Epiperimetric inequality</td><td>32</td></tr><tr><td>4.1</td><td>Buckling eigenbasis</td><td>34</td></tr><tr><td>4.2</td><td>Single mode case</td><td>35</td></tr><tr><td>4.3</td><td>Removal of higher modes</td><td>39</td></tr><tr><td>4.4</td><td>General case $v \neq 0$</td><td>45</td></tr><tr><td>4.5</td><td>Estimates on biharmonic extensions</td><td>49</td></tr><tr><td>4.6</td><td>Proof of the epiperimetric inequality</td><td>51</td></tr></table>

<table><tr><td>5</td><td>Description of flat, angular and isolated boundary points.</td><td>56</td></tr><tr><td>5.1</td><td>Uniqueness and speed of convergence of blow-ups</td><td>56</td></tr><tr><td>5.2</td><td>$C^{1,\alpha}$ regularity of the boundary</td><td>59</td></tr><tr><td>5.3</td><td>Higher regularity of the boundary</td><td>65</td></tr><tr><td>5.4</td><td>Proof of the main result</td><td>67</td></tr></table>
# 6 Quasi-minimizers

68

<table>
  <tbody>
    <tr>
      <td>6.1</td>
      <td>Regularity of quasi-minimizers</td>
      <td>68</td>
    </tr>
    <tr>
      <td>6.1.1</td>
      <td>Cacciopoli inequality</td>
      <td>69</td>
    </tr>
    <tr>
      <td>6.1.2</td>
      <td>BMO estimate</td>
      <td>69</td>
    </tr>
    <tr>
      <td>6.1.3</td>
      <td>Nondegeneracy lemma</td>
      <td>70</td>
    </tr>
    <tr>
      <td>6.1.4</td>
      <td>Sequence of minimizers</td>
      <td>71</td>
    </tr>
    <tr>
      <td>6.1.5</td>
      <td>Monotonicity</td>
      <td>72</td>
    </tr>
    <tr>
      <td>6.1.6</td>
      <td>Epiperimetry</td>
      <td>74</td>
    </tr>
    <tr>
      <td>6.1.7</td>
      <td>Uniqueness and speed of convergence of blow-ups</td>
      <td>75</td>
    </tr>
    <tr>
      <td>6.1.8</td>
      <td>Boundary decomposition for quasiminimizers</td>
      <td>76</td>
    </tr>
    <tr>
      <td>6.2</td>
      <td>Application to buckling eigenvalue</td>
      <td>77</td>
    </tr>
  </tbody>
</table>

# 1 Introduction

## 1.1 General context

The goal of this paper is to study a variational formulation of a fourth order free boundary problem, which is a natural fourth order version of the classical Alt-Caffarelli free boundary problem from [1].

Let $D$ some open subset of $\mathbb{R}^2$ and let $H^2(D) = \{u \in L^2(D) : \nabla u, \nabla^2 u \in L^2(D)\}$. For $u \in H^2(D)$ we define

$$E(u; D) = \int_D (|\Delta u|^2 + \chi_{u \neq 0}), \qquad (1.1)$$

where $\chi$ designates the indicator function, meaning $\chi_{u \neq 0}(x) = \begin{cases} 1 & \text{if } u(x) \neq 0 \\ 0 & \text{if } u(x) = 0 \end{cases}$. We say $u$ is a minimizer of $E(\cdot; D)$ if for every $v \in H^2(D)$ such that $\{u \neq v\}$ is compactly contained in $D$, we have

$$E(u; D) \leq E(v; D).$$

We will denote the set of minimizers of $E(\cdot; D)$ by $\mathcal{M}(D)$.

When the $|\Delta u|^2$ term in the energy is replaced by $|\nabla u|^2$, and if one assumes $u \ge 0$ near $\partial D$, as mentioned above we obtain the classical (one-phase) Alt-Caffarelli problem and it is well-known in this context that solutions $u$ are nonnegative, locally Lipschitz in $D$ (this regularity is called the optimal regularity, as one expects the gradient to be discontinuous between a region where $u = 0$ and a region where $u > 0$), and that the free boundary $\partial\{u > 0\}$ is analytic. In higher dimension ($D \subset \mathbb{R}^d$ for some $d \ge 2$), it is known that the free boundary is fully regular for $d = 3, 4$, that singularities may appear in high dimension ($d \ge 7$), and the situation is still open in dimension $d = 5, 6$ (see for instance [36, Sec. 1.4]).

The situation with this fourth order version is more involved, and the existing literature is more scarce. In [11], the authors studied a similar problem where $\{u \neq 0\}$ is replaced by $\{u > 0\}$: as mentioned in [13], while these two cases lead to the same problem when considering the Alt-Caffarelli functional (as the energy of $u_+$ is lower than the energy of $u$), when dealing with a fourth order analogue, the two resulting problems have very different behaviour. The problem under study here is therefore closer to the one studied in [13], although the mathematical tools will have similarities with [11], particularly with the use of a monotonicity formula as explained later in this introduction.

**Remark 1.** By integration by parts, $u$ belongs to the set of minimizers $\mathcal{M}(D)$ if and only if it is a minimizer (in the same sense) of one of the following functionals:

$$u \mapsto \int_D (|\nabla^2 u|^2 + \chi_{u \neq 0}) \quad \text{or} \quad u \mapsto \int_D ((\partial_{x,x}u - \partial_{y,y}u)^2 + 4(\partial_{x,y}u)^2 + \chi_{u \neq 0}).$$
The purpose of this paper is twofold:

* On one hand, we study the regularity properties of the free boundary related to minimizers of $E(\cdot, D)$: in particular, we classify the blow-ups for this problem, highlighting the possibility of angular singularities, and we provide regularity properties near flat and angular points of the free boundary, using an approach based on a monotonicity formula (in the spirit of [11]) and an epiperimetric inequality (in the spirit of [30, 32] and subsequent works). Some questions remain open, in particular the behavior of the free boundary near points of density 1 of the support of $u$, and the optimal regularity for $u$ which is expected to be $C^{1,1}$.

We describe these results in Section 1.3 below.

* On the other hand, we extend our results to more general energies, by introducing a suitable notion of quasi-minimizer, and we show that this applies in particular to the minimization of buckling eigenvalues. This is described in Section 1.2 below.

Note that in the whole paper, we only consider the two-dimensional case (though as seen in the next paragraph, this includes several rotationally symmetric three-dimensional problems). Another natural variant would be to study minimizers of

$$ \vec{v} \in H^1(D, \mathbb{R}^d) \mapsto \int_D (|\nabla \vec{v}|^2 + \chi_{\vec{v} \neq 0}) \quad (1.2) $$

under the constraint $\nabla \cdot \vec{v} = 0$. Indeed for $d = 2$, the minimization of the functional (1.2) is equivalent (by identifying $\vec{v} = \nabla^{\perp} u$) to the minimization of the functional

$$ u \in H^2(D, \mathbb{R}) \mapsto \int_D (|\Delta u|^2 + \chi_{\nabla u \neq 0}). \quad (1.3) $$

When $\{\nabla u = 0\}$ is fully included (up to a negligible set) in a single level set of $u$, then a minimizer of (1.3) is (up to a translation) a minimizer of (1.1), however the reciprocal is not true in general.

A minimizer $\vec{v}$ of (1.2) is expected to verify Stokes' equation with an overdetermined boundary condition ($\vec{v} = 0, \|\nabla v\| = 1$) on the regular free boundary.

This is reminiscent of some vectorial versions of the one phase Alt-Caffarelli problem, see for instance [21, 7, 24, 25, 23, 6, 5], and the recent survey [35]. A notable difference is that in each of these works, some version of the maximum principle is involved to prove the regularity of the boundary, something which is not available for the Stokes system.

## 1.2 Motivations

Generalized versions of the functional $E(\cdot; D)$ arise in several settings: we give examples coming both from spectral optimization and fluid mechanics.

**Towards a Faber-Krahn inequality for the first buckling eigenvalue:** Consider the first buckling eigenvalue of a clamped plate of shape $\Omega$

$$ \Lambda_1(\Omega) = \inf_{u \in H_0^2(\Omega)} \frac{\int_{\Omega} |\Delta u|^2}{\int_{\Omega} |\nabla u|^2} \quad (1.4) $$

defined for any open set $\Omega \subset \mathbb{R}^2$ of finite area (here $H_0^2(\Omega)$ is the $H^2$-closure of compact support functions in $\Omega$, see subsection 1.4 for notations). It was conjectured by Pólya and Szegö in [34, 27] that among open sets of given area, $\Lambda_1(\Omega)$ is minimal when $\Omega$ is a disk, by analogy with Rayleigh's since solved conjectures on the principal frequency of membranes or of clamped plates. It is known by an argument of Weinberger and Willms (reproduced in [17, Prop 4.4] and [2]) that
under the assumption that there exists a minimizer $\Omega^{\text{opt}}$ that is bounded, of class $C^2$, and simply connected, then $\Omega^{\text{opt}}$ must be a disk. The regularity hypothesis is central as it implies (by a shape derivative argument) that $\Delta u$ is locally constant on $\partial\Omega$.

One of the motivations in this work is to partially weaken the regularity hypotheses of Weinberger & Willms's argument: we will prove that minimizers of $\Lambda_1(\Omega)$ under area constraint are necessarily bounded, and that they satisfy some partial regularity properties, stated in Theorem 8. In other words, we lower the hypothesis on Weinberger and Willms's argument on buckling eigenvalue as follows:

Assume $\Omega \subset \mathbb{R}^2$ is a minimizer of $|\Omega|\Lambda_1(\Omega)$ among open sets of finite area, such that $\Omega$ is simply connected and for all $p \in \partial\Omega$ we have

$$ \liminf_{r \to 0} \frac{|\mathbb{D}_{p,r} \cap \Omega|}{|\mathbb{D}_{p,r}|} < \frac{t_1}{2\pi} (\approx 0.715). \qquad (1.5) $$

Then $\Omega$ is a disk.

Here $\mathbb{D}_{p,r}$ is the disk centered at $p$ of radius $r$ and $t_1 \approx 1.43\pi$ is uniquely defined by $\tan(t_1) = t_1$ and $t_1 \in (\pi, 2\pi)$. In particular, if $\Omega$ is a minimizer and $\Omega$ is convex, then it must be a disk (this follows from the fact that $t_1 > \pi$). These results are discussed in subsection 6.2, as well as some alternative conditions on $\Omega$.

**Remark 2.** Note that the topological hypothesis of Weinberger & Willms ($\Omega$ being simply connected) may be considered as a constraint, in an attempt to prove the minimality of the disk only among simply connected sets of given area. However, proving the regularity of the boundary under this type of topological constraint goes beyond the scope of this work: as far as the authors know, this is not well-understood even for the classical Alt-Caffarelli problem.

Let us now mention some other problems that lead to similar free boundary problems and where our results may apply, though they all exhibit some specific difficulties, which is why we leave these applications for future work.

For a rotationally symmetric set $K \subset \mathbb{R}^3$, the drag of $K$ in the direction $\vec{e}_x$ is defined as

$$ \mathcal{D}(K) = \inf \left\{ \int_{\mathbb{R}^3 \setminus K} \left| \frac{\nabla \vec{v} + (\nabla \vec{v})^*}{2} \right|^2 , \vec{v} \in \vec{e}_x + H^1(\mathbb{R}^3, \mathbb{R}^3) : \nabla \cdot \vec{v} = 0, \vec{v}|_{\partial K} \equiv 0 \right\}. \quad (1.6) $$

The minimization of $D(K)$ under the constraint $|K| = 1$ has been widely studied, see for instance in [26, 4] the study of a conjectured optimal shape with two sharp ends of angular opening $\frac{2\pi}{3}$. This has also been considered in two dimensions in [31], with an explicit conjectured optimal shape for the drag with two angles of opening $\approx 0.57\pi$ (which corresponds to $2\pi - t_1$ where $t_1$ is as defined in (1.5): the reason for this precise angle is explained in the classification of 2-homogeneous minimizers below). Let us also mention the works [29, 3] for the minimization of the energy dissipated by a fluid following Stokes' equation in a pipe.

In the three-dimensional rotationally invariant case, we can define a stream function $u \in H_{\text{loc}}^2(\mathbb{R} \times \mathbb{R}_{>0}, \mathbb{R})$ by $\vec{v}(x, y, 0) = y^{-1}\nabla^{\perp}(y^{\frac{1}{2}}u)$, and it can be checked that the minimality of the drag $\mathcal{D}(K)$ under the constraint $|K| = |B|$ corresponds to the minimality of the energy

$$ u \mapsto \int_D \left( |\Delta u|^2 + \frac{3}{2}y^{-2}|\nabla u|^2 - \frac{63}{16}y^{-4}u^2 \right) $$

for any $D \in \mathbb{R} \times \mathbb{R}_{>0}$, under the constraint $2\pi \int_{\mathbb{R}^2} y\chi_{\nabla u \neq 0} = 1$: this is a variant of the energy (1.3), and we consider the study of (1.1) to be a first step towards understanding this free boundary problem.
## 1.3 Main results

Let us first remind several known results from the literature, essentially based on [11, 13] (although the reference [11] deals with a different free boundary problem, we will see that several tools developed in [11] find applications here).

We start by studying the regularity of $u \in \mathcal{M}(D)$. In [13] it is proved that any minimizer $u$ of $E(\cdot; D)$ is in $C_{\text{loc}}^{1,\alpha}(D, \mathbb{R})$ for any $\alpha \in (0, 1)$, and more precisely that

$$ \Delta u \in \text{BMO}_{\text{loc}}(D, \mathbb{R}). $$

The expected optimal regularity for $u$ is $u \in C_{\text{loc}}^{1,1}(D, \mathbb{R})$, which we will not obtain here. Let us mention the work [33] where the regularity of the optimal shape for $\Omega \mapsto \Lambda_1(\Omega)|\Omega|$ (which is essentially the same free boundary problem) is analyzed, with an argument for the $C^{1,1}$ regularity that appears erroneous to the authors.

Then we note that a function $u$ is a solution in $D$ if and only if $x \mapsto r^{-2}u(rx)$ is also a solution in $D/r$, for any $r > 0$, so the expected regularity of the boundary is linked to the study of 2-homogeneous minimizers. As will be seen in subsection 3.2, we prove that these 2-homogeneous minimizers belong to one of the four following types:

(I) Flat solutions:

$$ u = su^I \circ \text{rot}, \quad u^I(x, y) = \frac{y_+^2}{2}, \qquad (1.7) $$

where $s \in \{-1, +1\}$ and $\text{rot}$ is a rotation.

(II) Angular solutions:

$$ u = su^{II} \circ \text{rot}, \quad u^{II}(re^{i\theta}) = \frac{r^2}{4} \left( 1 - \cos(2\theta) - \frac{2}{t_1} \left( \theta - \frac{\sin(2\theta)}{2} \right) \right) \chi_{0 \le \theta \le t_1}, \qquad (1.8) $$

where $t_1 \approx 1.43\pi$ is the unique fixed point of tan in $(\pi, 2\pi)$, $s \in \{-1, +1\}$ and $\text{rot}$ is a rotation.

(III) Nodal solutions:

$$ u = \lambda u^{III} \circ \text{rot}, \quad u^{III}(x, y) = \frac{y^2}{2}, \qquad (1.9) $$

where $|\lambda| \ge 1$ and $\text{rot}$ is a rotation.

(IV) Isolated points:

$$ u(x, y) = u_{a,b,c}(x, y) = a(x^2 + y^2) + b(x^2 - y^2) + 2cxy, \qquad (1.10) $$

for some $(a, b, c) \in \mathbb{R}^3$ such that $a^2 \neq b^2 + c^2$ ($a^2 = b^2 + c^2$ corresponds to the previous case).

Note that we do not claim each of these functions are minimizers, only that all homogeneous minimizers are among these. A detailed discussion on this issue may be found in Remark 28.

A central property of elements of $\mathcal{M}(\mathbb{D}_1)$ is the monotonicity formula discovered in a different context by Dipierro, Karakhanyan, and Valdinoci in [11, Th. 1.12], which may be seen as a higher-order analogue of the monotonicity formula introduced by Weiss in [40] for the classical Alt-Caffarelli problem. We define

$$ W(u, r) = \frac{1}{r^2} \int_{\mathbb{D}_r} \{ |\Delta u|^2 + \chi_{u \neq 0} \} + D(u, r), \qquad (1.11) $$
where

$$D(u, r) = \frac{1}{r} \int_{\partial \mathbb{D}_r} \left\{ \frac{2\partial_r u \Delta u}{r} - \frac{10(\partial_r u)^2}{r^2} - \frac{4u \Delta u}{r^2} + \frac{24u \partial_r u}{r^3} + \frac{4\partial_\theta u \partial_{r,\theta} u}{r^3} - \frac{16u^2}{r^4} - \frac{6(\partial_\theta u)^2}{r^4} \right\}$$

is a “corrective term” that is well-defined for almost every $r$. Then we show (see Theorem 19) that $r \mapsto W(u, r)$ is continuous, nondecreasing, and it is constant if and only if $u$ is one of the 2-homogeneous minimizers described above. Moreover, in this case we identify

$$W(u, r) = |\{u \neq 0\} \cap \mathbb{D}_1| \in \left\{\frac{\pi}{2}, \frac{t_1}{2}, \pi\right\},$$

the value $\frac{\pi}{2}$ corresponding to type I, $\frac{t_1}{2}$ corresponding to type II, and $\pi$ corresponding to types III and IV. For a general minimizer $u$, we write $W(u, 0) \in \mathbb{R} \cup \{-\infty\}$ the limit of $W(u, r)$ as $r \to 0$ (which always exists since $r \mapsto W(u, r)$ is nondecreasing).

When $u \in \mathcal{M}(D)$, it is known from [13, Th. 2] (see also section 2 below) that $u$ and $\nabla u$ are continuous, and we define the support of $u$ as the open set

$$Spt(u) = \{u \neq 0\} \cup \{\nabla u \neq 0\}.$$

Defining the support of a function as an open set may seem non-standard, however it is necessary to capture some part of the free boundary that would otherwise be in the interior of the support. Our main result is as follows:

**Theorem 3.** Let $D$ be an open set of $\mathbb{R}^2$ and $u \in \mathcal{M}(D)$. Then $u \in C_{\text{loc}}^{1,\alpha}(D)$ for every $\alpha \in (0,1)$ and there is a partition

$$D \cap \partial \text{Spt}(u) = \mathcal{R}_u \sqcup \mathcal{A}_u \sqcup \mathcal{N}_u \sqcup \mathcal{J}_u \sqcup \mathcal{E}_u,$$

where

* $\mathcal{R}_u$ (regular points) is the regular boundary of $\text{Spt}(u)$, meaning every point for which $\text{Spt}(u)$ is (after rotation) the epigraph of an analytic function in a neighbourhood, $\mathcal{R}_u$ is relatively open in $\partial \text{Spt}(u)$, $u|_{\text{Spt}(u)}$ is analytic up to $\mathcal{R}_u$, and the trace of $\Delta u|_{\text{Spt}(u)}$ on $\mathcal{R}_u$ takes values in $\{-1, +1\}$.

* $\mathcal{A}_u$ (angular points) is a discrete set where two connected components of $\mathcal{R}_u$ (where $\Delta u|_{\mathcal{R}_u}$ takes opposite signs) join with an angular opening $t_1$.

* $\mathcal{N}_u$ (nodal points) is the set of points of $\partial \text{Spt}(u)$ such that $u$ is biharmonic in a neighbourhood. It is a disjoint union of singletons, and of real analytic curves in $\partial \text{Spt}(u)$ on which $|\Delta u| \ge 1$.

* $\mathcal{J}_u$ (junction points) is the set of points $p$, such that $r^{-2}u(p + r \cdot)$ converges as $r \to 0$ in $H_{\text{loc}}^2(\mathbb{R}^2)$ to a homogeneous solution of type III and such that for any $r > 0$, we have $|\{u = 0\} \cap \mathbb{D}_{p,r}| > 0$.

* $\mathcal{E}_u$ (explosion points) is a set of points $p \in \partial \text{Spt}(u)$ such that

$$\limsup_{r \to 0} \frac{|\{u = 0\} \cap \mathbb{D}_{p,r}|}{\pi r^2} = 0, \quad W(u(p + \cdot), 0) < \pi, \quad \|u_{p,r}\|_{H^1(\mathbb{D}_1)} \xrightarrow{r \to 0} +\infty,$$

where $u_{p,r} := r^{-2}u(p + r \cdot)$. Moreover, any accumulation point of $\frac{u_{p,r}}{\|u_{p,r}\|_{H^1(\mathbb{D}_1)}}$ in $H^1(\mathbb{D}_1)$ as $r \to 0$ is a non-zero 2-homogeneous biharmonic polynomial.

More details on the behavior of $u$ at points of $\mathcal{A}_u$ and $\mathcal{J}_u$ may be found in Theorems 6 and 7. In order to close the regularity question in this context, the two remaining open problems are:
* Whether the explosion set $\mathcal{E}_u$ is empty, which is strongly linked to the conjectured $C^{1,1}$ regularity of $u$.

* A finer understanding of the junction set $\mathcal{J}_u$. A key step here would be to first prove the blow-up of $u$ at a point of $\mathcal{J}_u$ is necessarily $\pm u^{\text{III}} \circ \text{rot}$ for some rotation rot (in other words, proving that $|\lambda| = 1$ in (1.9)).

The proof of Theorem 3 can be found in section 5, and is split into several results: Theorem 4 gives the existence and uniqueness of blow-ups at any boundary point that is not of density 1 in the support of $u$, or at points for which $W(u, 0)$ is sufficiently large, and it provides a speed of convergence to these blow-ups as well. Then Theorems 5, 6 and 7 are $\epsilon$-regularity results for minimizers that are assumed to be close to one of the homogeneous solutions: for type I and II we prove that any solution that is close enough to these is itself a smooth perturbation of a homogeneous solution, whereas for type III and IV we prove a quantified speed of convergence of the blow-up sequence.

**Theorem 4 (Existence and uniqueness of blow-ups).** There exists $\gamma \in (0, 1)$ such that the following holds: let $D \subset \mathbb{R}^2$ be an open set and $u \in \mathcal{M}(D)$. Let $p \in D \cap \partial \text{Spt}(u)$ such that one of the following hypothesis is verified:

(i) $\liminf_{r \to 0} \frac{|\text{Spt}(u) \cap \mathbb{D}_{p,r}|}{\pi r^2} < 1$.

(ii) $W(u(p + \cdot), 0) \ge \pi$.

Define the blow-up sequence $u_{p,r}(z) = \frac{u(p+rz)}{r^2}$, then there exists $u_{p,0} \in H_{\text{loc}}^2(\mathbb{R}^2)$ a 2-homogeneous minimizer of type I or II in case (i), III or IV in case (ii), such that

$$ \|u_{p,r} - u_{p,0}\|_{H^1(\mathbb{D}_1)} = \mathcal{O}_{r \to 0} (r^\gamma). $$

In particular, this implies that any point that is not of Lebesgue density 1 in $\text{Spt}(u)$ has density $\frac{1}{2}$ (corresponding to type I blow-ups) or $\frac{t_1}{2\pi}$ (corresponding to type II blow-ups).

We now state the first $\epsilon$-regularity theorem: any minimizer that is $H^1$-close to a flat solution is itself smooth.

**Theorem 5 ($\epsilon$-regularity : flat case).** There exist $\alpha, \kappa \in (0, 1)$, $\epsilon, C > 0$ with the following property: for any $u \in \mathcal{M}(\mathbb{D}_1)$ such that

$$ \|u - u^I\|_{H^1(\mathbb{D}_1)} \le \epsilon, $$

there exists an analytic function $h : (-\frac{1}{2}, \frac{1}{2}) \to (-\frac{1}{2}, \frac{1}{2})$ such that

$$ \|h\|_{C^{1,\alpha}} ((-\frac{1}{2}, \frac{1}{2})) \le C \|u - u^I\|_{H^1(\mathbb{D}_1)}^{\kappa} $$

and

$$ \mathbb{D}_{\frac{1}{2}} \cap \text{Spt}(u) = \{(x, y) \in \mathbb{D}_{\frac{1}{2}} : y > h(x)\}. $$

We now state the second $\epsilon$-regularity theorem at angular boundary points: one notable difference in the statement is that we now assume that $W(u, 0) \ge \frac{t_1}{2}$. This hypothesis is verified for instance when there exists one blow-up of type other than I at the origin, by Lemma 29.
**Theorem 6** ($\epsilon$-regularity : angular case). There exist $\nu, \mu \in (0, 1)$, $\epsilon, C > 0$ with the following property: for any $u \in \mathcal{M}(\mathbb{D}_1)$ such that

$$W(u, 0) \geq \frac{t_1}{2}, \quad \|u - u^{\mathrm{II}}\|_{H^1(\mathbb{D}_1)} \leq \epsilon,$$

then there exists a diffeomorphism $\Phi \in C^{1,\gamma}(\mathbb{D}_{\frac{1}{2}}, \mathbb{D}_{\frac{1}{2}})$ such that

$$\Phi(0) = 0, \quad D\Phi(0) = I_2, \quad \|\Phi - \mathrm{id}\|_{C^{1,\nu}(\mathbb{D}_{\frac{1}{2}})} \leq C \|u - u^{\mathrm{II}}\|_{H^1(\mathbb{D}_1)}^{\mu}$$

and

$$\mathrm{Spt}(u) \cap \mathbb{D}_{1/2} = \Phi(\mathrm{Spt}(u^{\mathrm{II}}) \cap \mathbb{D}_{1/2}),$$

where $\mathrm{Spt}(u^{\mathrm{II}})$ is the union of two half-lines meeting with the angle $t_1$.

The last $\epsilon$-regularity result concerns homogeneous solutions of type III and IV (which we do not differentiate at this stage), that we remind are of the form

$$u_{a,b,c}(x, y) = a(x^2 + y^2) + b(x^2 - y^2) + 2cxy$$

for some $(a, b, c) \in \mathbb{R}^3$. The main feature of this case (compared to Theorem 5 and 6) is that the set of homogeneous solutions of type III, IV is non-compact.

**Theorem 7** ($\epsilon$-regularity : nodal and isolated case). There exist $\epsilon, \gamma, C > 0$ such that the following holds: let $a, b, c \in \mathbb{R}$ and $u \in \mathcal{M}(\mathbb{D}_1)$ such that

$$0 \in \partial \mathrm{Spt}(u), \quad W(u, 0) \geq \pi, \quad \|u - u_{a,b,c}\|_{H^1(\mathbb{D}_1)} \leq \epsilon \|u_{a,b,c}\|_{H^1(\mathbb{D}_1)}.$$

Then there exists $a', b', c' \in \mathbb{R}$ such that

$$|a' - a| + |b' - b| + |c' - c| \leq C \|u - u_{a,b,c}\|_{H^1(\mathbb{D}_1)}^{\gamma} \|u_{a,b,c}\|_{H^1(\mathbb{D}_1)}^{1-\gamma}$$

and such that for any $r \in (0, 1]$:

$$\|r^{-2}u(r \cdot) - u_{a',b',c'}\|_{H^1(\mathbb{D}_1)} \leq C \min \left( r, \frac{\|u - u_{a,b,c}\|_{H^1(\mathbb{D}_1)}}{\|u_{a,b,c}\|_{H^1(\mathbb{D}_1)}} \right)^{\gamma} \|u_{a,b,c}\|_{H^1(\mathbb{D}_1)}.$$

Finally, we state a generalization of Theorem 3 to a notion a quasi-minimizer.

**Theorem 8.** Let $D \subset \mathbb{R}^2$ be an open set, and $u \in H^2(D, \mathbb{R})$ such that for any $v \in H^2(D, \mathbb{R})$ with $\{u \neq v\} \in D$ we have

$$\int_D (|\Delta u|^2 + \chi_{u \neq 0}) \leq \int_D (|\Delta v|^2 + \chi_{v \neq 0}) + \|v - u\|_{L^2(D)}.$$

Then $u \in C_{\mathrm{loc}}^{1,\gamma}(D)$ for every $\gamma \in (0, 1)$, and there is a partition

$$D \cap \partial \mathrm{Spt}(u) = \mathcal{R}_u \sqcup \mathcal{A}_u \sqcup \mathcal{N}_u \sqcup \mathcal{J}_u \sqcup \mathcal{E}_u,$$

where

* $\mathcal{R}_u$ is the regular boundary of $\mathrm{Spt}(u)$, meaning every point for which $\mathrm{Spt}(u)$ is the epigraph of a $C^{1,\alpha}$ function in a neighbourhood, $\mathcal{R}_u$ is relatively open in $\partial \mathrm{Spt}(u)$, and $\Delta u|_{\mathrm{Spt}(u)}$ is continuous up to $\mathcal{R}_u$ with value $\pm 1$,

* the sets $\mathcal{A}_u, \mathcal{N}_u, \mathcal{J}_u, \mathcal{E}_u$ satisfy the same conclusions as in Theorem 3.

The discussion of this notion of quasi-minimizer as well as applications to the buckling problem can be found in section 6.
## 1.4 Notations and outline of the paper

In all the paper, we make use of the following notation: we write $a \lesssim b$ when $a, b$ are two quantities such that $b > 0$ and $a \le Cb$ for some universal constant $C > 0$, i.e. some constant that does not depend on any parameter.

We denote $\mathbb{D}_r$ the centered disk of radius $r$ in $\mathbb{R}^2$, and $\mathbb{D}_{p,r} = p + \mathbb{D}_r$ for $p \in \mathbb{R}^2$. We also let $\mathbb{S}^1 = \partial\mathbb{D}_1$, that we identify with $\frac{\mathbb{R}}{2\pi\mathbb{Z}}$.

For any $k \in \mathbb{N}$, $q \in [1, +\infty[$, any smooth bounded open set $D \subset \mathbb{R}^2$, and any measurable function $u: D \to \mathbb{R}$ with $k$ derivative in the distributional sense we let by convention

$$ \|u\|_{W^{k,q}(D)} = \left( \sum_{\alpha \in \mathbb{N}^2: \alpha_1 + \alpha_2 \le k} \int_D |\partial_x^{\alpha_1} \partial_y^{\alpha_2} u|^q \right)^{\frac{1}{q}}. $$

We denote $H^k(D) = W^{k,2}(D)$, and $H_0^k(D)$ the closure of $C_c^\infty(D, \mathbb{R})$ functions with respect to the $H^k(D)$ norm.

For a function $u: D \to \mathbb{R}$, $p \in D$, $r > 0$, we let

$$ u_{p,r}(z) = \frac{u(p + rz)}{r^2}. \qquad (1.12) $$

the associated rescaling defined on $\frac{D-p}{r}$. When $p=0$ we simply write $u_r := u_{0,r}$. In particular, if $u$ is a local minimizer of $E(\cdot; D)$ (which we remind is written $u \in \mathcal{M}(D)$), then $u_{p,r} \in \mathcal{M}(\frac{D-p}{r})$.

In all the paper, when $u$ is a $C^1$ function, we denote by $\text{Spt}(u)$ the set of points $p$ where either $u(p)$ or $\nabla u(p)$ is non-zero.

The plan of the paper is as follows.

* In section 2, we prove several low regularity estimate on $u$ that will be repeatedly used later. This includes a log-lipschitz bound on $\nabla u$ depending on the $H^1$ norm of $u$ in a neighbourhood, as well as a nondegeneracy lemma that states that a minimizer cannot be “too small” near a point of the support. The methods of proofs are relatively standard with respect to the classical Alt-Caffarelli problem, based on [1], [10], [36]. The main difference is that there is no maximum principle nor notion of viscosity solutions in our setting.

* In section 3 we introduce and prove the monotonicity formula for the renormalized energy $W(u, r)$ (defined in (1.11)) and several of its consequences, namely a classification of 2-homogeneous minimizers as well as a proof of Theorem 4 about existence of blow-ups. Even though the monotonicity formula was already found in [11], we propose a new way to tackle the computation, which most importantly leads to a new interpretation of the corrective term. The proof is based on the criticality of the solution for the energy, with respect to inner variation (i.e. comparison of $E(u; \mathbb{D}_1)$ with $E(u \circ (id + t\zeta); \mathbb{D}_1)$ at $t \to 0$, for any vector field $\zeta \in C_c^\infty(\mathbb{D}_1, \mathbb{R}^2)$) similarly to [38]. The monotonicity implies in particular a maximal speed of growth of the $H^1(\mathbb{D}_r)$ norm of $u$ with respect to $r$ (given by Lemma 33). This property is central in the proof of existence of blow-ups under weak geometrical hypothesis (being a non-Lebesgue point of the support of $u$).

* In section 4 we prove an epiperimetric inequality for the renormalized energy $W$, that takes the form

$$ W(u, e^{-1}) \le (1 - \eta)W(u, 1) + \eta \frac{\Theta}{2} $$

when $u$ is sufficiently close to a blow-up of opening $\Theta \in \{\pi, t_1, 2\pi\}$. This is obtained by constructing, for almost every $r$ in the interval $[e^{-1}, 1]$, a competitor $v$ for $u$ in $\mathbb{D}_r$ such that $v-u \in H_0^2(\mathbb{D}_r)$ while the energy of $v$ is well-controlled by the boundary data $u|_{\partial\mathbb{D}_r}$, $\partial_r u|_{\partial\mathbb{D}_r}$.
* In section 5 we prove Theorems 4, 5, 6, and 7. This is obtained by iterating the epiperimet- ric inequality to deduce geometrical information on the boundary (see Proposition 52). The main difficulty compared to the classical Alt-Caffarelli problem is that blow-ups are only known to exist under some a priori condition, either on the density of the support or on a lower bound of $W(u, r)$: we obtain the $C^{1,\alpha}$ regularity of the boundary near type I blow-ups by a careful analysis of contact points of the boundary with a cone from outside the sup- port. Higher regularity is obtained by a conformal hodograph transform on the associated overdetermined Stokes equation. The combination of these results implies Theorem 3.

* In section 6, we define a suitable notion of quasi-minimizers of $E(\cdot; D)$ and prove Theorem 8. We then apply these results to the minimization of the first buckling eigenvalue (1.4). Our quasiminimality is similar to the quasiminimality condition from [24]; a central point is that a suitable quasiminimality condition must be sufficiently strong to extract information from small inner variations, which is not the case for weaker quasiminimality conditions (like the one found in [10] for classical Alt-Caffarelli problem). The reason for this is that the monotonicity formula is proved via inner variations of $u$, and it is unclear whether it can be accessed differently.

## 2 Preliminary estimates

In this section we state and prove several regularity estimates on minimizers: a Cacciopoli- type estimate (Lemma 10) to show that the $H^1$ norm of $u$ locally controls its energy, a BMO estimate (Lemma 11) on $\Delta u$ that implies the gradient of $u$ is almost Lipschitz (Lemma 12), a non-degeneracy lemma (Lemma 14) that shows a minimizer always carry some minimal threshold of energy near the boundary of the support, and finally a lemma showing that a bounded sequence of minimizers converges (after extraction) to some minimizer in a strong way. The first two results already appear in some form in [13] or [33] and our proofs in this section are not original in this regard, however we carry them here with a more explicit statement for later uses.

In this section, we will need to consider the following slightly more general version: for any $\lambda \ge 0$ we let $\mathcal{M}_\lambda(D)$ be the minimizers (in the sense given in the beginning of the introduction) of the energy

$$u \in H^2(D) \mapsto E_\lambda(u; D) := \int_D (|\Delta u|^2 + \lambda \chi_{u \neq 0}).$$

We start by a remark on our definition of minimality, showing that it is equivalent to consider compactly supported perturbations, or perturbations up to the boundary with fixed traces of $u, \nabla u$.

**Lemma 9.** Let $u \in H^2(\mathbb{D}_1)$, $\lambda \ge 0$, then $u \in \mathcal{M}_\lambda(\mathbb{D}_1)$ if and only if for every $v \in u + H_0^2(\mathbb{D}_1)$ we have $E_\lambda(u; \mathbb{D}_1) \le E_\lambda(v; \mathbb{D}_1)$.

*Proof.* Assume $u \in \mathcal{M}(\mathbb{D}_1)$, let $w \in H_0^2(\mathbb{D}_1)$, $r \in (0, 1)$, and $\eta \in C_c^\infty(\mathbb{D}_1)$ such that $\eta \equiv 1$ on $\overline{\mathbb{D}_r}$. Then $(1 - \eta)w \in H_0^2(\mathbb{D}_1 \setminus \overline{\mathbb{D}_r})$, so there exists some sequence $\chi_n \in C_c^\infty(\mathbb{D}_1 \setminus \overline{\mathbb{D}_r})$ that converges to $(1 - \eta)w$ in $H^2(\mathbb{D}_1 \setminus \overline{\mathbb{D}_r})$. Since $u \in \mathcal{M}(\mathbb{D}_1)$, we have $E_\lambda(u; \mathbb{D}_1) \le E_\lambda(u + \eta w + \chi_n; \mathbb{D}_1)$, and

$$\limsup_{n \to +\infty} E_\lambda(u + \eta w + \chi_n; \mathbb{D}_1) \le E_\lambda(u + w; \mathbb{D}_1) + \lambda \pi(1 - r^2)$$

by the $H^2$ convergence of $\chi_n$ to $(1 - \eta)w$. We conclude by taking $r$ arbitrarily close to 1. $\square$

We then prove a Cacciopoli-type estimate: in this result we only use the fact that $u$ is bi- harmonic on its support, with a competitor of the form $e^{t\phi}u$ for some smooth function $\phi$ with compact support.
**Lemma 10.** Let $r \in [\frac{1}{2}, 1)$, let $\lambda \ge 0$ and $u \in \mathcal{M}_\lambda(\mathbb{D}_1)$. Then

$$ \|u\|_{H^2(\mathbb{D}_r)} \lesssim \frac{1}{(1-r)^2} \|u\|_{H^1(\mathbb{D}_1 \setminus \mathbb{D}_r)}. $$

Note that this estimate does not depend on $\lambda$.

*Proof.* For this proof, it is more convenient to see $u$ as a local minimizer of $u \mapsto \int_{\mathbb{D}_1} (|\nabla^2 u|^2 + \lambda \chi_{u \neq 0})$ (see remark 1). Let $\eta \in \mathcal{C}_c^\infty(\mathbb{D}_1, [0, 1])$ to be fixed later. The optimality condition on $u$ gives

$$ \left. \frac{d}{dt} \right|_{t=0} \int_{\mathbb{D}_1} (|\nabla^2(e^{\eta^4 t} u)|^2 + \lambda \chi_{e^{\eta^4 t} u \neq 0}) = 0, $$

which after computations develops as $\int_{\mathbb{D}_1} \nabla^2(\eta^4 u) : \nabla^2 u = 0$, with

$$ \nabla^2(\eta^4 u) = \eta^4 \nabla^2 u + 8\eta^3 \nabla^2 \eta \otimes \nabla u + 4 (3\eta^2 \nabla^2 \eta \otimes \nabla \eta + \eta^3 \nabla^2 \eta) u. $$

The previous relation becomes

$$ \begin{aligned} \int_{\mathbb{D}_1} \eta^4 |\nabla^2 u|^2 &= - \int_{\mathbb{D}_1} \eta^2 \nabla^2 u : (8\eta \nabla \eta \otimes \nabla u + 4 (3(\nabla \eta \otimes \nabla \eta) + \eta \nabla^2 \eta) u) \\ &\le \frac{1}{2} \int_{\mathbb{D}_1} \eta^4 |\nabla^2 u|^2 + \frac{1}{2} \int_{\mathbb{D}_1} |8\eta \nabla \eta \otimes \nabla u + 4 (3(\nabla \eta \otimes \nabla \eta) + \eta \nabla^2 \eta) u|^2, \end{aligned} $$

which simplifies to

$$ \int_{\mathbb{D}_1} \eta^4 |\nabla^2 u|^2 \lesssim \int_{\mathbb{D}_1} \left( [|\nabla \eta|^4 + |\eta \nabla^2 \eta|^2] |u|^2 + \eta^2 |\nabla \eta|^2 |\nabla u|^2 \right). $$

Now, choosing an appropriate profile $\eta$ equal to 1 in $\mathbb{D}_r$, with $|\nabla^k \eta| \lesssim (1-r)^{-k}$ for $k=0, 1, 2$, we obtain $\|\nabla^2 u\|_{L^2(\mathbb{D}_r)} \lesssim (1-r)^{-2} \|u\|_{H^1(\mathbb{D}_1 \setminus \mathbb{D}_r)}$.

The bound on the full $H^1(\mathbb{D}_r)$ norm is then obtained by the general inequality

$$ \|v\|_{H^2(\mathbb{D}_r)}^2 \lesssim \|\nabla^2 v\|_{L^2(\mathbb{D}_r)}^2 + \|v\|_{L^2(\partial\mathbb{D}_r)}^2 \lesssim \|\nabla^2 v\|_{L^2(\mathbb{D}_r)}^2 + \frac{1}{1-r} \|v\|_{H^1(\mathbb{D}_1 \setminus \mathbb{D}_r)}^2, $$

for any $v \in H^2(\mathbb{D}_r) \cap H^1(\mathbb{D}_1)$, applied to $v = u$. $\square$

Next we prove an estimate on the BMO semi-norm of $\nabla^2 u$. We remind our definition of this seminorm on a smooth set $\Omega$: a function $f \in L^1_{\text{loc}}(\Omega)$ is in BMO($\Omega$) if for every disk $\mathbb{D}_{p,r} \subset \Omega$ we have

$$ \int_{\mathbb{D}_{p,r}} |f - \int_{\mathbb{D}_{p,r}} f|^2 \le M^2 $$

for some constant $M > 0$, and the best constant $M$ is denoted $[f]_{\text{BMO}(\Omega)}$. Here $f_U g := \frac{\int_U g}{|U|}$ by convention.

We recall that the BMO space embeds in every $L^p$ space, for $p \in [1, +\infty)$, by the John-Nirenberg Lemma [16, Lem. 1], but not in $L^\infty(\Omega)$.

**Lemma 11.** Let $u \in \mathcal{M}(\mathbb{D}_1)$, then

$$ [\nabla^2 u]_{\text{BMO}(\mathbb{D}_{1/2})} \lesssim 1 + \|u\|_{H^2(\mathbb{D}_1)}. $$
Proof. In this proof we will only use the following property: if $v$ is the biharmonic extension of $u$ on some open set $\Omega \subset \mathbb{D}_1$ - meaning $u - v \in H_0^2(\Omega)$ and $\Delta^2 v = 0$ in $\Omega$ - then by minimality of $u$ we have

$$ \int_{\Omega} |\Delta(u - v)|^2 = \frac{1}{|\Omega|} \int_{\Omega} (|\Delta u|^2 - |\Delta v|^2) \le 1. $$

We recall (see for instance [15, Lem. 1.41]) that for any harmonic function $h: \mathbb{D}_R \to \mathbb{R}$ and $r < R$ we have the decay property

$$ \int_{\mathbb{D}_r} |h - \int_{\mathbb{D}_r} h|^2 \lesssim \left(\frac{r}{R}\right)^2 \int_{\mathbb{D}_R} |h - \int_{\mathbb{D}_R} h|^2. \quad (2.1) $$

Let now $\mathbb{D}_{p,r} \subset \mathbb{D}_1$, $v$ the biharmonic extension of $u$ in $\mathbb{D}_{p,r}$ and $\tau \in (0, 1)$: then we have

$$ \begin{aligned} \int_{\mathbb{D}_{p,\tau r}} |\Delta u - \langle \Delta u \rangle_{p,\tau r}|^2 &\lesssim \int_{\mathbb{D}_{p,\tau r}} (|\Delta(u-v)|^2 + |\langle \Delta u \rangle_{p,\tau r} - \langle \Delta v \rangle_{p,\tau r}|^2 + |\Delta v - \langle \Delta v \rangle_{p,\tau r}|^2) \\ &\lesssim \tau^{-2} \int_{\mathbb{D}_{p,r}} |\Delta(u-v)|^2 + \tau^2 \int_{\mathbb{D}_{p,r}} |\Delta v - \langle \Delta v \rangle_{p,r}|^2 \quad \text{by (2.1)} \\ &\lesssim \tau^{-2} + \tau^2 \int_{\mathbb{D}_{p,r}} (|\Delta(u-v)|^2 + |\langle \Delta u \rangle_{p,r} - \langle \Delta v \rangle_{p,r}|^2 + |\Delta u - \langle \Delta u \rangle_{p,r}|^2) \\ &\lesssim \tau^{-2} + \tau^2 \int_{\mathbb{D}_{p,r}} |\Delta u - \langle \Delta u \rangle_{p,r}|^2. \end{aligned} $$

As a consequence, letting $g(p, r) := \int_{\mathbb{D}_{p,r}} |\Delta u - \langle \Delta u \rangle_{p,r}|^2$, we may fix a sufficiently small $\tau \in (0, 1)$ such that for some (universal) constant $C \gtrsim 1$:

$$ g(p, \tau r) \le C + \frac{1}{2} g(p, r). $$

By induction on $g(p, \tau^k r)$, we obtain that for any $x \in \mathbb{D}_{\frac{3}{4}}$ and for any $r \in (0, \frac{1}{4})$ we have

$$ g(p, \tau^k r) \le \left(1 + \frac{1}{2} + \dots + \frac{1}{2^{k-1}}\right) C + \frac{1}{2^k} g(p, r). $$

Thus, for any $r \in (0, \frac{1}{4})$, we have

$$ \int_{\mathbb{D}_{p,r}} |\Delta u - \langle \Delta u \rangle_{p,r}|^2 \lesssim 1 + \int_{\mathbb{D}_1} |\Delta u|^2, $$

Hence $[\Delta u]_{\text{BMO}(\mathbb{D}_{\frac{3}{4}})} \lesssim 1 + \|\Delta u\|_{L^2(\mathbb{D}_1)}$. By [12, Cor. 2], this implies $[\nabla^2 u]_{\text{BMO}(\mathbb{D}_{\frac{1}{2}})} \lesssim 1 + \|u\|_{H^2(\mathbb{D}_1)}$. $\square$

**Lemma 12.** Let $u \in \mathcal{M}(\mathbb{D}_1)$, then for any $p, q \in \mathbb{D}_{1/2}$:

$$ |\nabla u(p) - \nabla u(q)| \lesssim |p - q| \log \left( \frac{1}{|p - q|} \right) \left( 1 + \int_{\mathbb{D}_1} |\nabla u|^2 + u^2 \right)^{\frac{1}{2}}. $$

Proof. This is a consequence of Lemmas 10, 11, to which we apply general Orlicz space embeddings (see for instance [9, Th. 3]). $\square$

**Remark 13.** This lemma implies that when $u \in \mathcal{M}(\mathbb{D}_1)$ with $u(0) = |\nabla u(0)| = 0$, then

$$ \forall p \in \mathbb{D}_{1/2}, \quad |\nabla u(p)| + \frac{|u(p)|}{|p|} \lesssim (1 + \|u\|_{H^1(\mathbb{D}_1)}) |p| \log \left(\frac{1}{|p|}\right). $$
The non-degeneracy property is central in the classical Alt-Caffarelli problem (see [1, Lem. 3.4]): it implies that blow-ups, when they exist, are not zero. The next lemma is a non-degeneracy property for minimizers of (1.1), and our proof is similar to the proof of non-degeneracy in [10], by an induction on the energy of $u$ relying on the three following estimates:

a) The $H^2$ norm of $u$ is locally controlled by its $H^1$ norm by our Cacciopoli-type estimate.

b) The area of the support of $u$ is locally controlled by its $H^2$ norm, by minimality of $u$.

c) The $H^1$ norm of $u$ is locally controlled by a **product** of the area of its support and its $H^2$ norm. The nonlinear nature of this estimate is what makes the induction work.

**Lemma 14.** There exists $\epsilon > 0$ such that for any $u \in \mathcal{M}(\mathbb{D}_1)$, if $\|u\|_{H^1(\mathbb{D}_1)} \le \epsilon$, then $u|_{\mathbb{D}_{1/2}} = 0$.

*Proof.* It is enough to prove that for any minimizer in $\mathbb{D}_1$ and for a small enough $\epsilon$,

$$ \int_{\mathbb{D}_1} |u|^2 + |\nabla u|^2 \le \epsilon \quad \text{implies} \quad u(0) = 0, $$

because this can be applied to $u_{p,1/2}$ for every $p \in \mathbb{D}_{1/2}$ (the definition of $u_{p,r}$ is given in (1.12)).

* $\int_{\mathbb{D}_1} |\nabla u|^2 + |u|^2 \le \epsilon$ implies $E(u; \mathbb{D}_{1/2}) \lesssim \epsilon$. Indeed, by the Cacciopoli-type inequality from Lemma 10, we have

$$ \int_{\mathbb{D}_{\frac{9}{10}}} |\Delta u|^2 \le \|u\|_{H^2(\mathbb{D}_{\frac{9}{10}})}^2 \lesssim \int_{\mathbb{D}_1 \setminus \mathbb{D}_{\frac{9}{10}}} (|u|^2 + |\nabla u|^2) \lesssim \epsilon. $$

Let now $\eta \in C_c^\infty(\mathbb{D}_{\frac{9}{10}}, [0, 1])$ such that $\eta \equiv 1$ in $\mathbb{D}_{\frac{1}{2}}$ with $\|\eta\|_{C^2(\mathbb{D}_1)} \lesssim 1$, we then compare the energies of $u$ and $(1 - \eta)u$:

$$ |\mathbb{D}_{1/2} \cap \{u \neq 0\}| + \int_{\mathbb{D}_1} |\Delta u|^2 \le \int_{\mathbb{D}_1} |\Delta(u - \eta u)|^2. $$

Since $u$ is biharmonic on its support (see [13, Theorem 2]) we have

$$ \int_{\mathbb{D}_1} |\Delta(u - \eta u)|^2 = \int_{\mathbb{D}_1} |\Delta u|^2 + |\Delta(\eta u)|^2, $$

so

$$ |\mathbb{D}_{\frac{1}{2}} \cap \{u \neq 0\}| \le \int_{\mathbb{D}_1} |\Delta(\eta u)|^2 \lesssim \|u\|_{H^2(\mathbb{D}_{\frac{9}{10}})}^2 \lesssim \epsilon, $$

and thus $E(u; \mathbb{D}_{\frac{1}{2}}) \lesssim \epsilon$.

* $E(u; \mathbb{D}_1) + \int_{\mathbb{D}_1} (|\nabla u|^2 + |u|^2) \le \epsilon$ implies $\int_{\mathbb{D}_1} (|\nabla u|^2 + |u|^2) \lesssim \epsilon^{\frac{3}{2}}$.

Indeed, let $\eta$ be defined as previously, then

$$ \begin{aligned} \int_{\mathbb{D}_{\frac{1}{2}}} |\nabla u|^2 &\le \int_{\mathbb{D}_1} |\nabla(\eta u)|^2 \le |\{\eta u \ne 0\}|^{\frac{1}{2}} \left( \int_{\mathbb{D}_1} |\nabla(\eta u)|^4 \right)^{1/2} \\ &\lesssim E(u; \mathbb{D}_1)^{\frac{1}{2}} \int_{\mathbb{D}_1} |\Delta(\eta u)|^2 \text{ by the Sobolev inclusion } W_0^{2,2}(\mathbb{D}_1) \hookrightarrow W_0^{1,4}(\mathbb{D}_1) \\ &\lesssim E(u; \mathbb{D}_1)^{\frac{1}{2}} \left( E(u; \mathbb{D}_1) + \int_{\mathbb{D}_1} (|\nabla u|^2 + |u|^2) \right) \\ &\lesssim \epsilon^{\frac{3}{2}} \end{aligned} $$
and similarly

$$ \int_{\mathbb{D}_{1/2}} u^2 \leq \int_{\mathbb{D}_1} (\eta u)^2 \leq |\{\eta u \neq 0\}|^{1/2} \left( \int_{\mathbb{D}_1} (\eta u)^4 \right)^{1/2} \lesssim \epsilon^{3/2}. $$

We now proceed by induction. We let $u_r(x) := \frac{u(rx)}{r^2}$. Then for some constant $C_1 \gtrsim 1$ we have (by the first point above):

$$ E(u_{1/2}; \mathbb{D}_1) + \|u_{1/2}\|_{H^1(\mathbb{D}_1)}^2 \leq C_1 \|u\|_{H^1(\mathbb{D}_1)}^2. $$

By the second point applied to $u_{1/2}$, for some constant $C_2 \gtrsim 1$ we have

$$ \|u_{1/4}\|_{H^1(\mathbb{D}_1)}^2 \leq C_2 (C_1 \|u\|_{H^1(\mathbb{D}_1)}^2)^{3/2}. $$

In particular suppose that $\|u\|_{H^1(\mathbb{D}_1)}^2 \leq \frac{1}{4}C_1^{-3}C_2^{-2}$, then we obtain by induction

$$ \forall k \in \mathbb{N}, \quad \|u_{4-k}\|_{H^1(\mathbb{D}_1)}^2 \leq 2^{-k} \|u\|_{H^1(\mathbb{D}_1)}^2. $$

This implies $u(0) = 0$.

<figure><img src="image_4.png" /></figure>

Finally, we prove a result about sequences of minimizers.

**Lemma 15.** Let $(\lambda^{(n)})_{n \in \mathbb{N}^*}$ be a sequence of nonnegative numbers converging to some limit $\lambda \geq 0$, and $(u^{(n)})_{n \in \mathbb{N}^*}$ a sequence of $H^2(\mathbb{D}_1)$ such that for any $n \in \mathbb{N}^*$, $u^{(n)} \in \mathcal{M}_{\lambda^{(n)}}(\mathbb{D}_1)$. Suppose moreover that

$$ \sup_{n \in \mathbb{N}^*} \|u^{(n)}\|_{H^1(\mathbb{D}_1)} < \infty. $$

Then there exists some extraction $(n_i)_{i \in \mathbb{N}^*}$, and some $u \in \mathcal{M}_{\lambda}(\mathbb{D}_1)$ such that

$$ \begin{gathered} u^{(n_i)} \xrightarrow{i \to +\infty} u \text{ in } H_{\text{loc}}^2(\mathbb{D}_1), \\ \lambda^{(n_i)} \chi_{u^{(n_i)} \neq 0} \xrightarrow{i \to +\infty} \lambda \chi_{u \neq 0} \text{ in } L^1(\mathbb{D}_1). \end{gathered} $$

*Proof.* The proof is similar to its equivalent in the Alt-Caffarelli problem and is fairly standard (see for instance [36, Lem. 6.3]). Since $(u^{(n)})_n$ is bounded in $H^1(\mathbb{D}_1)$, by the Cacciopoli inequality from Lemma 10 it is bounded in $H^2(\mathbb{D}_r)$ for every $r \in (0, 1)$. Up to extraction we may suppose that for every $r \in (0, 1)$, $(u^{(n)})_{n \in \mathbb{N}^*}$ converges strongly in $H^1(\mathbb{D}_r)$, weakly in $H^2(\mathbb{D}_r)$, as well as almost everywhere, to some limit $u$. By the weak $H^2$ convergence we have

$$ \int_{\mathbb{D}_1} |\Delta u|^2 \leq \liminf_{n \to +\infty} \int_{\mathbb{D}_1} |\Delta u^{(n)}|^2. $$

Since $u^{(n)}$ converges almost everywhere to $u$, then by Fatou lemma:

$$ \int_{\mathbb{D}_1} \chi_{u \neq 0} \leq \int_{\mathbb{D}_1} \liminf_{n \to \infty} \chi_{u^{(n)} \neq 0} \leq \liminf_{n \to \infty} \int_{\mathbb{D}_1} \chi_{u^{(n)} \neq 0}. $$

Thus, $E_{\lambda}(u; \mathbb{D}_1) \leq \liminf_{n \to \infty} E_{\lambda^{(n)}}(u^{(n)}; \mathbb{D}_1)$. Let now $v$ be a competitor for $u$, meaning $v \in H^2(\mathbb{D}_1)$ and $\{u \neq v\} \subset \mathbb{D}_s$ for some $s < 1$. Let $r \in (s, 1)$, and $\zeta \in C_c^{\infty}(\mathbb{D}_1)$ a function such that $\zeta = 1$ in $\mathbb{D}_s$, $\zeta = 0$ in $\mathbb{D}_1 \setminus \mathbb{D}_r$. Then

$$ v^{(n)} := (1 - \zeta)u^{(n)} + \zeta v $$
is a valid competitor for $u^{(n)}$. By our assumption on $u^{(n)}$ we have

$$E_{\lambda^{(n)}}(v^{(n)}; \mathbb{D}_1) - E_{\lambda^{(n)}}(u^{(n)}; \mathbb{D}_1) \geq 0.$$

We first compute

$$\begin{aligned}
\int_{\mathbb{D}_1} (|\Delta v^{(n)}|^2 - |\Delta u^{(n)}|^2) &= \int_{\mathbb{D}_1} \left( |(1-\zeta)\Delta u^{(n)} + \zeta\Delta v + 2\nabla(v-u^{(n)}) \cdot \nabla\zeta + (v-u^{(n)})\Delta\zeta|^2 - |\Delta u^{(n)}|^2 \right) \\
&= \int_{\mathbb{D}_1} \left( \left( (1-\zeta)^2 - 1 \right) |\Delta u^{(n)}|^2 + \zeta^2|\Delta v|^2 + 2\zeta(1-\zeta)\Delta u \Delta v \right) + o_n(1) \\
&= \int_{\mathbb{D}_1} \left( |\Delta v|^2 - |\Delta u|^2 + \left( (1-\zeta)^2 - 1 \right) (|\Delta u^{(n)}|^2 - |\Delta u|^2) \right) + o_n(1)
\end{aligned}$$

where the $o_n(1)$ term is due to the weak $H^2(\mathbb{D}_1)$ (and strong $H^1(\mathbb{D}_1)$) convergence $v^{(n)} \to v$, which is equal to $u$ on the support of $\nabla\eta, \Delta\eta$. Next,

$$\begin{aligned}
\lambda^{(n)} \int_{\mathbb{D}_1} (\chi_{v^{(n)} \neq 0} - \chi_{u^{(n)} \neq 0}) &\geq \lambda^{(n)} \int_{\mathbb{D}_1} (\chi_{\mathbb{D}_s} (\chi_{v \neq 0} - \chi_{u^{(n)} \neq 0}) - \chi_{\mathbb{D}_r \setminus \mathbb{D}_s}) \\
&= \lambda^{(n)} \int_{\mathbb{D}_1} ((\chi_{v \neq 0} - \chi_{u \neq 0}) + \chi_{\mathbb{D}_s} (\chi_{u \neq 0} - \chi_{u^{(n)} \neq 0}) - \chi_{\mathbb{D}_r \setminus \mathbb{D}_s}),
\end{aligned}$$

so we may rearrange $E(u^{(n)}; \mathbb{D}_1) \leq E(v^{(n)}; \mathbb{D}_1)$ into

$$E_{\lambda^{(n)}}(v; \mathbb{D}_1) - E_{\lambda^{(n)}}(u; \mathbb{D}_1) \geq \int_{\mathbb{D}_1} \left( (1 - (1 - \eta)^2) (|\Delta u^{(n)}|^2 - |\Delta u|^2) + \lambda^{(n)} \chi_{\mathbb{D}_s} (\chi_{u^{(n)} \neq 0} - \chi_{u \neq 0}) \right) + o_n(1) - 2\pi\lambda^{(n)}(r - s).$$

Taking $n \to +\infty$, we get

$$\begin{aligned}
E_{\lambda}(v; \mathbb{D}_1) - E_{\lambda}(u; \mathbb{D}_1) &\geq \liminf_{n \to \infty} \int_{\mathbb{D}_1} \left( (1 - (1 - \eta)^2) (|\Delta u^{(n)}|^2 - |\Delta u|^2) + \lambda^{(n)} \chi_{\mathbb{D}_s} (\chi_{u^{(n)} \neq 0} - \chi_{u \neq 0}) \right) \\
&\qquad - 2\pi\lambda(r - s) \\
&\geq - 2\pi\lambda(r - s). \tag{2.2}
\end{aligned}$$

Taking $r \searrow s$, we obtain $E_{\lambda}(v, \mathbb{D}_1) \geq E_{\lambda}(u, \mathbb{D}_1)$ so $u \in \mathcal{M}_{\lambda}(\mathbb{D}_1)$. Moreover, the convergence is strong in $H_{\text{loc}}^2$: indeed taking $v = u$, inequality (2.2) gives

$$2\pi\lambda(r-s) + \int_{\mathbb{D}_1} (1 - (1 - \eta)^2) |\Delta u|^2 \geq \liminf_{n \to \infty} \int_{\mathbb{D}_1} (1 - (1 - \eta)^2) |\Delta u^{(n)}|^2,$$

which implies

$$2\pi\lambda(r-s) + \int_{\mathbb{D}_r} |\Delta u|^2 \geq \liminf_{n \to \infty} \int_{\mathbb{D}_s} |\Delta u^{(n)}|^2.$$

By taking $r$ arbitrarily close to $s$, we obtain $\int_{\mathbb{D}_s} |\Delta u|^2 \geq \liminf_{n \to \infty} \int_{\mathbb{D}_s} |\Delta u^{(n)}|^2$ for every $s$, so $u^{(n)}$ converges strongly (in $H^2(\mathbb{D}_s)$) to $u$. Similarly, for the convergence of the area term, inequality (2.2) with $v = u$ gives

$$\int_{\mathbb{D}_s} \lambda \chi_{u \neq 0} \geq \liminf_{n \to \infty} \int_{\mathbb{D}_s} \lambda^{(n)} \chi_{u^{(n)} \neq 0},$$

for every $s$ (and the opposite inequality is a consequence of Fatou's lemma). By dominated convergence, $\lambda^{(n)}\chi_{u^{(n)} \neq 0} \underset{n \to +\infty}{\longrightarrow} \lambda\chi_{u \neq 0}$ in $L^1(\mathbb{D}_1)$. $\square$
## 3 Monotonicity formula and blow-ups

Monotonicity formulas are roughly speaking a way to express the notion that, as we zoom in on a solution, the solution can only look simpler and simpler. For the Alt-Caffarelli problem, i.e. for minimizers of

$$u \mapsto \int_{\mathbb{D}_1} (|\nabla u|^2 + \chi_{u>0}),$$

Weiss found in [38, 40] that the quantity

$$r \mapsto \frac{1}{r^2} \int_{\mathbb{D}_r} (|\nabla u|^2 + \chi_{u>0}) - \frac{1}{r^3} \int_{\partial \mathbb{D}_r} u^2$$

is always non-decreasing, and it is constant if and only if $u$ is 1-homogeneous. From this monotonicity alone one can deduce that blow-ups are always 1-homogeneous, and that for any solution $u$, the $H^1(\mathbb{D}_r)$ norm is controlled by the $L^2(\partial \mathbb{D}_r)$ norm (with appropriate scaling) for small $r$. This monotonicity formula was later refined in several settings (see [39], [32], and also the monograph [36]) to prove the convergence to a unique blow-up by an epiperimetric inequality (which originates from [30] with a similar estimate for the perimeter of a minimal surface, and later in [39] in the setting of obstacle problems). Our interest in these techniques is due to the fact that even for classical Alt-Caffarelli problem, these epiperimetric methods use no maximum principle or notion of viscosity solutions, something which is not available in our setting.

In [11, Th. 1.12], the authors found a similar monotone quantity for higher order problem (note that their functional is not exactly the same as ours, however it is applied to points where the solution has homogeneity 2). We start by stating again the monotonicity of this quantity in our setting, with a proof that is made slightly shorter by an exponential change of variable.

The monotonicity formula of Weiss for the classical Alt-Caffarelli problem may be obtained in several ways. The first is by comparing the energy of the solution $u$ with its 1-homogeneous extension $|x|u(\frac{\tau x}{|x|})$ in $\mathbb{D}_r$. The second is by using that $u$ is critical for the energy with respect to small deformation $u(x + tf(|x|)x)$ for a smooth function $f$ that approximates $\chi_{[0,r]}$. In our setting, the 2-homogeneous extension is not always a valid competitor (since it may create a jump in the gradient), and we do not obtain the monotonicity by the comparison with a single competitor as in the classical Alt-Caffarelli problem: however we are able to adapt the second method to our setting.

### 3.1 Monotonicity formula of Dipierro, Karakhanyan and Valdinoci recast in exponential coordinates

In this paragraph, we deal with the monotonicity formula. We start choosing a specific set of coordinates that is convenient to express the energy. We deduce in this set of coordinates a monotonicity formula (Theorem 18) and then express it again in the usual coordinates (Theorem 19).

We will denote $C_T = (T, +\infty) \times \mathbb{S}^1$ the half-infinite cylinder, such that $(t, \theta) \in C_T \mapsto e^{-t+i\theta} \in \mathbb{D}_{e^{-T}\setminus\{0\}}$ is a bijective anticonformal map. We denote by $H_{\text{lin}}^2(C_T)$ the set of functions $v \in H_{\text{loc}}^2(C_T)$ such that for some constant $C > 0$ (that may depend on $v$), we have

$$\forall t \ge T, \quad \|v\|_{H^2(C_t \setminus C_{t+1})} \le C(t - T + 1). \qquad (3.1)$$

For any $\tau \in \mathbb{R}$, $v \in H_{\text{lin}}^2(C_0)$, we denote

$$\mathcal{G}(v, \tau) = \int_{C_{\tau}} e^{-2(t-\tau)} \left\{ (\partial_{t,t}v)^2 + 2(\partial_{t,\theta}v)^2 + (\partial_{\theta,\theta}v)^2 - 4(\partial_{\theta}v)^2 + \chi_{v \neq 0} \right\} \qquad (3.2)$$
**Lemma 16.** Let $u \in H^2(\mathbb{D}_1)$ such that $v \in H_{\text{loc}}^2(C_0)$ defined by

$$v(t, \theta) = e^{2t} u(e^{-t+i\theta})$$

is in $H_{\text{lin}}^2(C_0)$. Then for every $\tau \ge 0$, we have

$$\frac{1}{(e^{-\tau})^2} \int_{\mathbb{D}_{e^{-\tau}}} \{ |\Delta u|^2 + \chi_{u \neq 0} \} = \mathcal{G}(v, \tau) + 2 \int_{\partial C_{\tau}} \{ 2 (\partial_t v)^2 - 4v \partial_t v + 4v^2 - (\partial_{\theta} v)^2 + \partial_{t,\theta} v \partial_{\theta} v \} .$$

*Proof.* We express $u$ in terms of $v$ with

$$u(re^{i\theta}) = r^2 v(-\log(r), \theta).$$

Then

$$\Delta u(re^{i\theta}) = (\partial_{\theta,\theta} v + \partial_{t,t} v - 4\partial_t v + 4v)(-\log(r), \theta),$$

so we may rewrite the energy of $u$ in the disk $\mathbb{D}_{e^{-\tau}}$ as

$$\int_{\mathbb{D}_{e^{-\tau}}} \{ |\Delta u|^2 + \chi_{u \neq 0} \} = \int_{C_{\tau}} e^{-2t} \{ |\partial_{\theta,\theta} v + \partial_{t,t} v - 4\partial_t v + 4v|^2 + \chi_{v \neq 0} \}. \quad (3.3)$$

We develop this expression:

$$\int_{C_{\tau}} e^{-2(t-\tau)} \{ |\partial_{t,t} v - 4\partial_t v + 4v + \partial_{\theta,\theta} v|^2 + \chi_{v \neq 0} \} \\ = \int_{C_{\tau}} e^{-2(t-\tau)} \{ (\partial_{t,t} v)^2 + 16 (\partial_t v)^2 + 16v^2 + (\partial_{\theta,\theta} v)^2 + \chi_{v \neq 0} \} \\ + \int_{C_{\tau}} e^{-2(t-\tau)} \{ -8\partial_{t,t} v \partial_t v + 8\partial_{t,t} v v + 2\partial_{t,t} v \partial_{\theta,\theta} v - 32\partial_t v v - 8\partial_t v \partial_{\theta,\theta} v + 8v \partial_{\theta,\theta} v \} .$$

By integration by parts we have:

$$\begin{align*}
\int_{C_{\tau}} e^{-2(t-\tau)} (-8\partial_{t,t} v \partial_t v) &= \int_{C_{\tau}} e^{-2(t-\tau)} (-8 (\partial_t v)^2) + \int_{\partial C_{\tau}} 4 (\partial_t v)^2 \\
\int_{C_{\tau}} e^{-2(t-\tau)} 8\partial_{t,t} v v &= \int_{C_{\tau}} e^{-2(t-\tau)} 8(-\partial_t v + 2v) \partial_t v + \int_{\partial C_{\tau}} -8v \partial_t v \\
&= \int_{C_{\tau}} e^{-2(t-\tau)} (-8 (\partial_t v)^2 + 16v^2) + \int_{\partial C_{\tau}} (-8v \partial_t v - 8v^2) \\
\int_{C_{\tau}} e^{-2(t-\tau)} (-32\partial_t v v) &= \int_{C_{\tau}} e^{-2(t-\tau)} (-32v^2) + \int_{\partial C_{\tau}} 16v^2 \\
\int_{C_{\tau}} e^{-2(t-\tau)} (-8\partial_t v \partial_{\theta,\theta} v) &= \int_{C_{\tau}} e^{-2(t-\tau)} 8\partial_{t,\theta} v \partial_{\theta} v \\
&= \int_{C_{\tau}} e^{-2(t-\tau)} 8 (\partial_{\theta} v)^2 + \int_{\partial C_{\tau}} -4 (\partial_{\theta} v)^2.
\end{align*}$$

In each integration by parts we used the growth estimate (3.1) so that there is no boundary term at $t = +\infty$. Finally, for almost every $\tau \ge 0$ we have

$$\int_{C_{\tau}} e^{-2(t-\tau)} 2\partial_{t,t} v \partial_{\theta,\theta} v = \int_{C_{\tau}} e^{-2(t-\tau)} (2 (\partial_{t,\theta} v)^2 - 4 (\partial_{\theta} v)^2) + \int_{\partial C_{\tau}} (2 (\partial_{\theta} v)^2 + 2\partial_{t,\theta} v \partial_{\theta} v).$$

Summing all these contributions, we indeed get for almost every $\tau \ge 0$:
$$
\begin{align}
& \int_{C_{\tau}} e^{-2(t-\tau)} \left\{ |\partial_{t,t} v + 4\partial_t v + 4v + \partial_{\theta,\theta} v|^2 + \chi_{v \neq 0} \right\} \nonumber \\
&= \int_{C_{\tau}} e^{-2(t-\tau)} \left\{ (\partial_{t,t} v)^2 + 2 (\partial_{t,\theta} v)^2 + (\partial_{\theta,\theta} v)^2 - 4 (\partial_{\theta} v)^2 + \chi_{v \neq 0} \right\} \tag{3.4} \\
&\quad + 2 \int_{\partial C_{\tau}} \left\{ 2 (\partial_t v)^2 - 4v \partial_t v + 4v^2 - (\partial_{\theta} v)^2 + \partial_{t,\theta} v \partial_{\theta} v \right\} d\theta. \nonumber
\end{align}
$$

<figure><img src="image_5.png" /></figure>

**Definition 17.** A function $v \in H_{\text{loc}}^2(C_\tau)$ is called a minimizer of $\mathcal{G}(\cdot, \tau)$ if $v \in H_{\text{lin}}^2(C_\tau)$ and if letting

$$u(re^{i\theta}) = r^2 v(-\log(r), \theta),$$

we have $u \in \mathcal{M}(\mathbb{D}_{e^{-\tau}})$.

In particular, for any $w \in H_{\text{lin}}^2(C_\tau)$ such that $\{v \neq w\} \subset C_{\tau+\epsilon}$ (for some $\epsilon > 0$), we have (by Lemma 16):

$$\mathcal{G}(v, \tau) \leq \mathcal{G}(w, \tau).$$

The reason we do not define minimizers of $\mathcal{G}(\cdot, \tau)$ only by this condition, is because this does not take into account competitors for $u$ for which either $u(0)$ or $\nabla u(0)$ do not vanish. Indeed, $\mathcal{G}(\cdot, \tau)$ is a priori only defined for functions that grow sufficiently slow as $t \to +\infty$.

We now define a “corrected” energy, for any $v \in H_{\text{lin}}^2(C_\tau)$:

$$W(v, \tau) = \mathcal{G}(v, \tau) + 2 \int_{\partial C_{\tau}} \partial_t v (\partial_t v - \partial_{t,t} v), \quad (3.5)$$

where the corrective term is chosen so that $W'$ has a sign. Due to the last term, $W(v, \tau)$ is a priori defined for almost every $\tau$ only. The next result implies in fact that $W(v, \tau)$ is continuous in $\tau$ when $v$ is a minimizer, so it is well-defined for every $\tau > 0$.

**Theorem 18.** Let $v$ be a minimizer of $\mathcal{G}(\cdot; 0)$, then $\tau \mapsto W(v, \tau)$ belongs to $W_{\text{loc}}^{1,1}(\mathbb{R}_+)$, and its weak derivative (defined for almost every $\tau \ge 0$) is

$$W'(v, \tau) = -4 \int_{\partial C_{\tau}} \{(\partial_{t,t} v)^2 + (\partial_{t,\theta} v)^2\}.$$

Here $W'(v, \tau)$ designates the derivative of $W(v, \tau)$ along the variable $\tau$, which exists in $L_{\text{loc}}^1(\mathbb{R}_+)$ in the weak sense since $W(v, \cdot)$ is absolutely continuous. In particular, $W(v, \tau)$ is constant in $\tau$ if and only if $\partial_t v(t, \theta)$ is constant.

*Proof.* We write $\|\cdot\|$ (resp. $\langle\cdot, \cdot\rangle$) the $L^2(\mathbb{S}^1)$ norm (resp. the $L^2(\mathbb{S}^1)$ scalar product). It is sufficient to prove the following: for any $\varphi \in C_c^\infty(\mathbb{R}_{>0}, \mathbb{R})$, we have

$$\int_{0}^{+\infty} \varphi'(t) W(v, t) dt = 4 \int_{0}^{+\infty} \varphi(t) \left\{ \|\partial_{t,t} v(t)\|^2 + \|\partial_{t,\theta} v(t)\|^2 \right\} dt. \quad (3.6)$$

Let now $f \in C^\infty(\mathbb{R}_+, \mathbb{R})$, such that $f = 0$ in a neighbourhood of 0, and $f, f', f''$ are bounded in $L^\infty(\mathbb{R}_+)$ (note that $f$ does not have compact support in general). For any small enough $\epsilon$ (positive or negative), the function $t \mapsto t + \epsilon f(t)$ is a diffeomorphism of $\mathbb{R}_+$, with inverse $\psi_\epsilon$. Let

$$v_\epsilon(t, \theta) = v(\psi_\epsilon(t), \theta).$$
Then $v_{\epsilon} \in H_{\text{lin}}^2(C_0)$ is a valid competitor for $v$. We compute

$$
\begin{align*}
\partial_{\theta} v_{\epsilon}(t, \theta) &= \partial_{\theta} v(\psi_{\epsilon}(t), \theta), \quad \partial_{\theta, \theta} v_{\epsilon}(t, \theta) = \partial_{\theta, \theta} v(\psi_{\epsilon}(t), \theta) \\
\partial_t v_{\epsilon}(t, \theta) &= (1 - \epsilon f'(t) + o(\epsilon)) \partial_t v(\psi_{\epsilon}(t), \theta), \\
\partial_{t, \theta} v_{\epsilon}(t, \theta) &= (1 - \epsilon f'(t) + o(\epsilon)) \partial_{t, \theta} v(\psi_{\epsilon}(t), \theta), \\
\partial_{t, t} v_{\epsilon}(t, \theta) &= (1 - 2\epsilon f'(t) + o(\epsilon)) \partial_{t, t} v(\psi_{\epsilon}(t), \theta) - (\epsilon f''(t) + o(\epsilon)) \partial_t v(\psi_{\epsilon}(t), \theta),
\end{align*}
$$

so

$$
\mathcal{G}(v_{\epsilon}, 0) = \mathcal{G}(v, 0) + o(\epsilon) - \epsilon \int_{C_0} e^{-2t} \{4f'(\partial_{t,t}v)^2 + 2f''\partial_t v \partial_{t,t} v + 4f'(\partial_{t,\theta}v)^2\} \\
\qquad + \epsilon \int_{C_0} e^{-2t}(f' - 2f) \{(\partial_{t,t}v)^2 + 2(\partial_{t,\theta}v)^2 + (\partial_{\theta,\theta}v)^2 - 4(\partial_{\theta}v)^2 + \chi_{v \neq 0}\}.
$$

Taking the derivative at $\epsilon = 0$ above, we have for any such $f$ that

$$
\begin{align*}
\int_{C_0} e^{-2t} (f' - 2f) \left\{ (\partial_{t,t}v)^2 + 2(\partial_{t,\theta}v)^2 + (\partial_{\theta,\theta}v)^2 - 4(\partial_{\theta}v)^2 + \chi_{v \neq 0} \right\} - 2e^{-2t} f'' \partial_t v \partial_{t,t} v \\
&= 4 \int_{C_0} e^{-2t} f' \left\{ (\partial_{t,t}v)^2 + (\partial_{t,\theta}v)^2 \right\}.
\end{align*}
$$

Let $\varphi \in C_c^\infty(\mathbb{R}_{>0}, \mathbb{R})$, and $f(t) = \int_0^t e^{2s}\varphi(s)ds$. The function $f$ verifies the previous hypothesis (in particular it is constant for any large enough $t$, meaning it is bounded), and $f'(t) = e^{2t}\varphi(t)$, $f''(t) = e^{2t}(\varphi'(t) + 2\varphi(t))$, so the previous relation becomes

$$
\int_{C_0} \left( \varphi(t) - 2e^{-2t} \int_0^t e^{2s} \varphi(s) ds \right) \left\{ (\partial_{t,t} v)^2 + 2(\partial_{t,\theta} v)^2 + (\partial_{\theta,\theta} v)^2 - 4(\partial_{\theta} v)^2 + \chi_{v \neq 0} \right\} \\
- 2 \int_{C_0} (\varphi'(t) + 2\varphi(t)) \partial_t v \partial_{t,t} v = 4 \int_{C_0} \varphi(t) \left\{ (\partial_{t,t} v)^2 + (\partial_{t,\theta} v)^2 \right\}.
$$

On one hand,

$$
-2 \int_{C_0} (\varphi'(t) + 2\varphi(t)) \partial_t v \partial_{t,t} v = 2 \int_{0}^{+\infty} \varphi'(t) \langle \partial_t v(t), \partial_t v(t) - \partial_{t,t} v(t) \rangle dt,
$$

and on the other hand,

$$
\begin{align*}
& \int_{C_0} \left( \varphi(t) - 2e^{-2t} \int_0^t e^{2s} \varphi(s) ds \right) \left\{ (\partial_{t,t} v)^2 + 2(\partial_{t,\theta} v)^2 + (\partial_{\theta,\theta} v)^2 - 4(\partial_{\theta} v)^2 + \chi_{v \neq 0} \right\} \\
&= \int_{C_0} \left( e^{2t} \varphi(t) - 2 \int_0^t e^{2s} \varphi(s) ds \right) e^{-2t} \left\{ (\partial_{t,t} v)^2 + 2(\partial_{t,\theta} v)^2 + (\partial_{\theta,\theta} v)^2 - 4(\partial_{\theta} v)^2 + \chi_{v \neq 0} \right\} \\
&= \int_0^{+\infty} \left( e^{2t} \varphi(t) - 2 \int_0^t e^{2s} \varphi(s) ds \right) \frac{d}{dt} \left\{ -e^{-2t} \mathcal{G}(v,t) \right\} dt \\
&= \int_0^{+\infty} \varphi'(t) \mathcal{G}(v,t) dt,
\end{align*}
$$

by integration by parts (we remind that $\varphi$ has compact support). As a consequence, we obtain
that for any $\varphi \in C_c^\infty(\mathbb{R}_{>0}, \mathbb{R})$, we have

$$
\int_{0}^{+\infty} \varphi'(t) \{ \mathcal{G}(v, t) + 2 \langle \partial_t v(t), \partial_t v(t) - \partial_{t,t} v(t) \rangle \} dt = 4 \int_{0}^{+\infty} \varphi(t) \{ \| \partial_{t,t} v(t) \|^2 + \| \partial_{t,\theta} v(t) \|^2 \} dt.
$$

By the definition of $\mathcal{W}$ in (3.5), we have obtained that $t \mapsto \mathcal{W}(v,t)$ admits a weak derivative, which is of the form given in (3.6). $\square$
We may now state this theorem in usual coordinates (both will be useful). Suppose $u \in \mathcal{M}(\mathbb{D}_1)$ such that $u(0) = |\nabla u(0)| = 0$, and we remind that for any $r \in (0, 1)$ we let

$$u_r(x) = \frac{u(rx)}{r^2}$$

the natural 2-homogeneous rescaling of $u$, defined in $\mathbb{D}_{1/r}$. To avoid notational ambiguity, we always denote $\partial_r$ the radial derivative of a function, and $\frac{d}{dr}$ the derivative with respect to the parameter $r$, in other words

$$\forall x \in \mathbb{S}^1, \frac{d}{dr}u_r(x) = \frac{1}{r}(\partial_r u_r(x) - 2u_r(x)).$$

We denote

$$N(u, r) = \int_{\mathbb{S}^1} \left( r^2 \left| \frac{d}{dr} u_r \right|^2 + |\partial_{\theta} u_r|^2 + |u_r|^2 \right). \quad (3.7)$$

In particular $N(u, r)$ verifies

$$\|u_r\|_{L^2(\mathbb{S}^1)}^2 + \|\nabla u_r\|_{L^2(\mathbb{S}^1)}^2 \lesssim N(u, r) \lesssim \|u_r\|_{L^2(\mathbb{S}^1)}^2 + \|\nabla u_r\|_{L^2(\mathbb{S}^1)}^2,$$

$r \mapsto N(u, r)$ is absolutely continuous, and its (weak) derivative defined almost everywhere is

$$\begin{aligned} N'(u, r) &= \frac{2}{r} \int_{\mathbb{S}^1} ((\partial_r u_r - 2u_r)(\partial_{r,r} u_r - 3\partial_r u_r + 4u_r) + (\partial_{r,\theta} u_r - 2\partial_\theta u_r)\partial_\theta u_r + (\partial_r u_r - 2u_r)u_r) \\ &= \frac{2}{r} \int_{\mathbb{S}^1} \left( \partial_r u_r \partial_{r,r} u_r - 2u_r \partial_{r,r} u_r + \partial_\theta u_r \partial_{r,\theta} u_r - 3(\partial_r u_r)^2 + 11u_r \partial_r u_r - 2(\partial_\theta u_r)^2 - 10u_r^2 \right). \end{aligned} \quad (3.8)$$

We will also denote

$$R(u, r) := 2 \int_{\mathbb{S}^1} \left\{ (\partial_{\theta} u_r)^2 - (\partial_r u_r)^2 + 2u_r^2 - u_r \partial_r u_r \right\}. \quad (3.9)$$

**Theorem 19.** Let $u \in \mathcal{M}(\mathbb{D}_1)$ such that $u(0) = |\nabla u(0)| = 0$, define

$$W(u, r) := \frac{1}{r^2} E(u; \mathbb{D}_r) + rN'(u, r) + R(u, r). \quad (3.10)$$

Then $r \mapsto W(u, r)$ is non-decreasing, and it is constant in $r$ if and only if $u$ is 2-homogeneous and in this case

$$W(u, r) = W(u, 1) = |\{u \neq 0\} \cap \mathbb{D}_1|.$$

Moreover, if we denote $v(t, \theta) = r^{-2}u(re^{i\theta})$ where $r = e^{-t}$, then $v$ is a minimizer of $\mathcal{G}(\cdot; 0)$ and for any $t \ge 0$:

$$\mathcal{W}(v, t) = W(u, r).$$

Although the corrective term looks different, once everything is developed we find that the normalized energy of (3.10) is equal to the one defined in [11, Th. 1.12] (up to a scalar factor), also recalled in (1.11).

*Proof.* By Lemma 16, we have

$$\mathcal{G}(v, \tau) = e^{2\tau} E(u; \mathbb{D}_{e^{-\tau}}) - 2 \int_{\partial \mathcal{C}_{\tau}} \left\{ 2(\partial_t v)^2 - 4v \partial_t v + 4v^2 - (\partial_{\theta} v)^2 + \partial_{t,\theta} v \partial_{\theta} v \right\} d\theta.$$
We replace $v(t, \theta) = e^{2t}u(e^{-t+i\theta})$, so

$$ \partial_t v(t, \theta) = 2e^{2t}u(e^{-t+i\theta}) - e^t \partial_r u(e^{-t+i\theta}), $$

$$ \partial_{t,\theta} v(t, \theta) = 2e^{2t}u_{\theta}(e^{-t+i\theta}) - e^t \partial_{r,\theta} u(e^{-t+i\theta}), $$

$$ \partial_{t,t} v(t, \theta) = 4e^{2t}u(e^{-t+i\theta}) - 3e^t \partial_r u(e^{-t+i\theta}) + \partial_{r,r} u(e^{-t+i\theta}), $$

$$ \partial_{\theta} v(t, \theta) = e^{2t} \partial_{\theta} u(e^{-t+i\theta}). $$

Denoting $r = e^{-\tau}$, the boundary integral term above becomes

$$ \begin{aligned} -2 \int_{\partial C_{\tau}} \{2 (\partial_t v)^2 - 4v \partial_t v + 4v^2 - (\partial_{\theta} v)^2 + \partial_{t,\theta} v \partial_{\theta} v\} d\theta & \\ = -\frac{2}{r} \int_{\partial \mathbb{D}_r} \left( 2 \left( \frac{2u}{r^2} - \frac{\partial_r u}{r} \right)^2 - 4 \frac{u}{r^2} \left( \frac{2u}{r^2} - \frac{\partial_r u}{r} \right) + 4 \frac{u^2}{r^4} - \frac{(\partial_{\theta} u)^2}{r^4} + \frac{\partial_{\theta} u}{r^2} \left( \frac{2\partial_{\theta} u}{r^2} - \frac{\partial_{r,\theta} u}{r} \right) \right) d\theta & \\ = -\frac{2}{r} \int_{\partial \mathbb{D}_r} \left( 8 \frac{u^2}{r^4} + 2 \frac{(\partial_r u)^2}{r^2} - 8 \frac{u \partial_r u}{r^3} - 8 \frac{u^2}{r^4} + 4 \frac{u \partial_r u}{r^3} + 4 \frac{u^2}{r^4} + \frac{(\partial_{\theta} u)^2}{r^4} - \frac{\partial_{\theta} u \partial_{r,\theta} u}{r^3} \right) d\theta & \\ = -\frac{2}{r} \int_{\partial \mathbb{D}_r} \left( 4 \frac{u^2}{r^4} + 2 \frac{(\partial_r u)^2}{r^2} - 4 \frac{u \partial_r u}{r^3} + \frac{(\partial_{\theta} u)^2}{r^4} - \frac{\partial_{\theta} u \partial_{r,\theta} u}{r^3} \right) d\theta & \end{aligned} $$

We obtain

$$ \mathcal{G}(v, \tau) = \frac{1}{r^2} E(u; \mathbb{D}_r) - \frac{2}{r} \int_{\partial \mathbb{D}_r} \left( 4 \frac{u^2}{r^4} + 2 \frac{(\partial_r u)^2}{r^2} - 4 \frac{u \partial_r u}{r^3} + \frac{(\partial_{\theta} u)^2}{r^4} - \frac{\partial_{\theta} u \partial_{r,\theta} u}{r^3} \right). $$

Then from the definition of $\mathcal{W}(v, \tau)$ (3.5),

$$ \begin{aligned} \mathcal{W}(v, \tau) &= \mathcal{G}(v, \tau) + 2 \int_{\partial C_{\tau}} \partial_t v (\partial_t v - \partial_{t,t} v) \\ &= \mathcal{G}(v, \tau) + \frac{2}{r} \int_{\partial \mathbb{D}_r} \left( \frac{\partial_r u}{r} - 2 \frac{u}{r^2} \right) \left( \partial_{r,r} u - 2 \frac{\partial_r u}{r} + 2 \frac{u}{r^2} \right) \\ &= \mathcal{G}(v, \tau) + \frac{2}{r} \int_{\partial \mathbb{D}_r} \left( \frac{\partial_r u \partial_{r,r} u}{r} - 2 \frac{u \partial_{r,r} u}{r^2} - 2 \frac{(\partial_r u)^2}{r^2} + 6 \frac{u \partial_r u}{r^3} - 4 \frac{u^2}{r^4} \right). \end{aligned} $$

So replacing the expression of $\mathcal{G}(v, \tau)$ found above we obtain

$$ \mathcal{W}(v, \tau) = \frac{1}{r^2} E(u; \mathbb{D}_r) + \frac{2}{r} \int_{\partial \mathbb{D}_r} \left( \frac{\partial_r u \partial_{r,r} u}{r} - 2 \frac{u \partial_{r,r} u}{r^2} + \frac{\partial_{\theta} u \partial_{r,\theta} u}{r^3} - 4 \frac{(\partial_r u)^2}{r^2} + 10 \frac{u \partial_r u}{r^3} - \frac{(\partial_{\theta} u)^2}{r^4} - 8 \frac{u^2}{r^4} \right). \quad (3.11) $$

This is the same expression as in (1.11). Moreover, the boundary integral term on the right-hand side is exactly the sum of $rN'(u, r)$ (see (3.8)) and $R(u, r)$ (see (3.9)).

Suppose now that $r \mapsto W(u, r)$ is constant. Then $t \mapsto \mathcal{W}(v, t)$ is constant, so by Theorem 18, $\partial_{t,\theta} v$ and $\partial_{t,t} v$ are identically zero. Thus for some constant $c \in \mathbb{R}$, we have

$$ v(t, \theta) = v(0, \theta) + ct. $$

We denote $g(\theta) = v(0, \theta)$, then $u$ may be rewritten as

$$ u(re^{i\theta}) = cr^2 \log(r) + r^2 g(\theta). $$

Assume by contradiction $c$ is non-zero. Since $u \in C^1(\mathbb{D}_1, \mathbb{R})$ (by Lemma 12), in particular $g$ is bounded, so there is some sufficiently small radius $\bar{r}$ such that $u \neq 0$ in $\bar{\mathbb{D}}_r \setminus \{0\}$. Let $h$ be the biharmonic extension of $u$ in $\bar{\mathbb{D}}_r$. Since $u$ has full support in $\bar{\mathbb{D}}_r$ we have by minimality of $u$

$$ \int_{\bar{\mathbb{D}}_r} |\Delta(u - h)|^2 = \int_{\bar{\mathbb{D}}_r} \left( |\Delta u| - |\Delta h|^2 \right) \le 0, $$
so $u$ is biharmonic in $\mathbb{D}_{\bar{r}}$. This is in contradiction with the fact that $c \neq 0$.
As a consequence, $u$ is of the form

$$u(re^{i\theta}) = r^2 g(\theta),$$

meaning it is 2-homogeneous. Reciprocally if $u$ is of this form, then we see that

$$W(u, r) = W(v, -\log(r)) = W(v, 0) = W(u, 1)$$

for every $r < 1$, so $W(u, \cdot)$ is constant.

<figure><img src="image_6.png" /></figure>

Although the boundary term in the definition of $W$ (3.10) is not continuous with respect to $H^2$, due to its structure the energy $W$ still verifies a type of Cacciopoli inequality:

**Lemma 20.** Let $u \in \mathcal{M}(\mathbb{D}_1)$ such that $u(0) = |\nabla u(0)| = 0$, then

$$|W(u, \frac{1}{2})| \lesssim \|u\|_{H^1(\mathbb{D}_1)}^2.$$

*Proof.* If $\text{Spt}(u) \cap \mathbb{D}_{\frac{1}{2}} = \emptyset$ there is nothing to prove: without loss of generality we assume this is not the case. By the non-degeneracy Lemma 14 this implies $\|u\|_{H^1(\mathbb{D}_1)} \gtrsim 1$.

Let $\varphi \in C_c^\infty([\frac{1}{2}, \frac{3}{4}], \mathbb{R}_+)$ be such that $\int_0^1 \varphi(s)ds = 1$ and $|\varphi'| \lesssim 1$, then by the monotonicity of $r \mapsto W(u, r)$:

$$
\begin{aligned}
W\left(u, \frac{1}{2}\right) &\le \int_{1/2}^{3/4} \varphi(r)W(u, r)dr \\
&= \int_{1/2}^{3/4} \varphi(r) \left( \frac{1}{r^2}E(u; \mathbb{D}_r) + rN'(u, r) + R(u, r) \right) dr \\
&= \int_{1/2}^{3/4} \left( \frac{\varphi(r)}{r^2}E(u; \mathbb{D}_r) - (r\varphi(r))'N(u, r) + \varphi(r)R(u, r) \right) dr \\
&\lesssim 1 + \|u\|_{H^2(\mathbb{D}_{\frac{3}{4}})}^2 \\
&\lesssim 1 + \|u\|_{H^1(\mathbb{D}_1)}^2 \quad \text{by lemma 10} \\
&\lesssim \|u\|_{H^1(\mathbb{D}_1)}^2 \quad \text{since } \|u\|_{H^1(\mathbb{D}_1)} \gtrsim 1.
\end{aligned}
$$

Similarly, choosing $\varphi \in C_c^\infty([\frac{1}{4}, \frac{1}{2}], \mathbb{R}_+)$ such that $\int_0^1 \varphi(s)ds = 1$ and $|\varphi'| \lesssim 1$ we get the opposite estimate.

The monotonicity formula extends to minimizers of $E_0(u, D) := \int_D |\Delta u|^2$ i.e. biharmonic functions. We denote

$$W_0(u, r) := \frac{1}{r^2} E_0(u; \mathbb{D}_r) + rN'(u, r) + R(u, r). \qquad (3.12)$$

Let $u \in H^2(\mathbb{D}_1)$ be a biharmonic function: by using its Goursat decomposition ($h_1 + (x^2 + y^2)h_2$ for $h_1, h_2$ harmonic in $\mathbb{D}_1$), $u$ can be decomposed as

$$u(re^{i\theta}) = \sum_{n \in \mathbb{Z}} (a_n r^{|n|+2} + b_n r^{|n|}) e^{in\theta} \qquad (3.13)$$

for some complex coefficients $(a_n)_{n \in \mathbb{Z}}$, $(b_n)_{n \in \mathbb{Z}}$, such that the sum converges absolutely in $\mathbb{D}_1$.
**Lemma 21.** Let $u \in H^2(\mathbb{D}_1)$ be a biharmonic function, and let $(a_n)_{n \in \mathbb{Z}}$, $(b_n)_{n \in \mathbb{Z}}$ be the coefficients defined in the decomposition (3.13), then for any $r \in (0, 1)$ we have

$$W_0(u, r) = 8\pi \sum_{n \in \mathbb{Z}} \left\{ |n|^3 r^{2|n|} |a_n|^2 + 2|n|^2 (|n|-2)r^{2|n|-2} \Re(\bar{a}_n b_n) + (|n|-2)(|n|^2 - 2|n| + 2)r^{2|n|-4} |b_n|^2 \right\}.$$

*Proof.* For this computation, we will directly use definition (1.11).

Let us first notice that for any $r < 1$, $W_0(u, r) = W_0(u_r, 1)$, and the coefficients associated to the Goursat decomposition of $u_r$ are $(r^{|n|}a_n, r^{|n|-2}b_n)_{n \in \mathbb{Z}}$: we lose no generality in proving the result at $r = 1$ only, for a function $u$ that is biharmonic in a neighbourhood of $\mathbb{D}_1$.

For a function $f: \mathbb{D}_1 \to \mathbb{C}$ we define

$$c_n[f] = \frac{1}{2\pi} \int_{\partial \mathbb{D}_1} e^{-in\theta} f(e^{i\theta}) d\theta$$

the Fourier coefficients of $f$. Then we compute

$$
\begin{align*}
c_n[u] &= a_n + b_n, \\
c_n[\partial_r u] &= (|n| + 2)a_n + |n|b_n, \\
c_n[\partial_{\theta, \theta} u] &= -|n|^2(a_n + b_n), \\
c_n[\Delta u] &= 4(|n| + 1)a_n.
\end{align*}
$$

$$\text{We deduce } E_0(u; \mathbb{D}_1) = \int_{\mathbb{D}_1} \left| \sum_{n \in \mathbb{Z}} 4(|n| + 1) a_n r^{|n|} e^{in\theta} \right|^2 = 8\pi \sum_{n \in \mathbb{Z}} (2|n| + 2) |a_n|^2, \text{ and}$$

$$
\begin{align*}
N'(u, 1) + R(u, 1) &= \int_{\partial \mathbb{D}_1} \left\{ 2\partial_r u \Delta u - 10(\partial_r u)^2 - 4u \Delta u + 24u \partial_r u + 4\partial_\theta u \partial_{r,\theta} u - 16u^2 - 6(\partial_\theta u)^2 \right\} \\
&= 2\pi \sum_{n \in \mathbb{Z}} \left\{ 2((|n|+2)a_n + |n|b_n)4(|n|+1)\bar{a}_n - 10(|n|+2)a_n + |n|b_n|^2 - 16(|n|+1)a_n(\bar{a}_n + \bar{b}_n) \right\} \\
&\quad + 2\pi \sum_{n \in \mathbb{Z}} \left\{ (24 + 4|n|^2)((|n|+2)a_n + |n|b_n)(\bar{a}_n + \bar{b}_n) - (16 + 6|n|^2)|a_n + b_n|^2 \right\} \\
&= 8\pi \sum_{n \in \mathbb{Z}} \left\{ (|n|^3 - 2|n| - 2)|a_n|^2 + 2|n|^2(|n| - 2)\Re(\bar{a}_n b_n) + |n|(|n| - 2)(|n|^2 - 2|n| + 2)|b_n|^2 \right\}.
\end{align*}
$$

Then $W_0(u, 1) = E_0(u; \mathbb{D}_1) + N'(u, 1) + R(u, 1)$ simplifies to the expression of the lemma with $r = 1$. $\square$

**Remark 22.** Let $u \in H^2(\mathbb{D}_1)$ be a biharmonic function, and we write $(a_n, b_n)_{n \in \mathbb{Z}}$ the coefficient in the Goursat decomposition (3.13) of $u$. Then for any $r \in (0, 1)$ we have

$$W_0'(u, r) = \frac{16\pi}{r} \sum_{n \in \mathbb{Z}} |n|^4 r^{2|n|} |a_n|^2 + 2|n|^2 (|n| - 1)(|n| - 2) r^{2|n|-2} \Re(\bar{a}_n b_n) \\ + \frac{16\pi}{r} \sum_{n \in \mathbb{Z}} (|n| - 2)^2 (|n|^2 - 2|n| + 2) r^{2|n|-4} |b_n|^2.$$

The discriminant of the n-th term (seen as a Hermitian form of $r^{|n|}a_n, r^{|n|-1}b_n$) is $-4|n|^4(|n|-2)^2$: for $n \notin \{-2, 0, 2\}$, the n-th term is positive definite, and for $n \in \{-2, 0, 2\}$, it is positive semi-definite. We find again the monotonicity of $W_0$ in this special case. Another observation is that

$$\frac{1}{8\pi} W_0(u, r) = -4r^{-4}|b_0|^2 - r^{-2}(|b_1|^2 + |b_{-1}|^2) + \mathcal{O}_{r \to 0}(1).$$

In particular, $W_0(u, 0)$ is finite if and only if $u(0) = |\nabla u(0)| = 0$. This implies that if $v \in \mathcal{M}(\mathbb{D}_1)$ and $0 \in \text{Spt}(v)$, then necessarily $W(v, r) \to -\infty$ as $r \to 0$.
## 3.2 Classification of 2-homogeneous minimizers

From the monotonicity formula of Theorem 19, we expect that limits of $r \mapsto u_r := r^{-2}u(r\cdot)$ as $r \to 0$, when they exist with a strong enough notion of convergence, should be 2-homogeneous. For this reason we classify every 2-homogeneous minimizers; we will see there are only four types. We start with a lemma on biharmonic functions in angular sectors. For any $\omega \in (0, 2\pi]$ we denote by

$$S_{\omega} := \{re^{i\theta} : 0 < r < 1, 0 < \theta < \omega\}$$

the angular sector. As mentioned previously, we will denote by $t_1$ the unique fixed point of $t \in (\pi, 2\pi) \mapsto \tan(t)$.

**Lemma 23.** Let $\omega \in (0, 2\pi]$, and $u \in H^2(S_\omega) \setminus \{0\}$ such that $u = \partial_\theta u = 0$ on $\partial S_\omega \setminus \partial D_1$ and $\Delta^2 u = 0$ on $S_\omega$. Suppose $u$ is of the form $u(re^{i\theta}) = r^2b(\theta)$ (for some $b \in H_0^2([0, \omega])$). Then either

(i) $\omega = \pi$ and $b$ is a multiple of

$$b^I(\theta) = \frac{1}{2} \sin(\theta)_+^2. \qquad (3.14)$$

(ii) $\omega = t_1$ and $b$ is a multiple of

$$b^{II}(\theta) = \frac{1}{4} \left( 1 - \cos(2\theta) - \frac{2}{t_1} \left( \theta - \frac{\sin(2\theta)}{2} \right) \right) \chi_{0 \le \theta \le t_1}. \qquad (3.15)$$

(iii) $\omega = 2\pi$ and $b$ is a multiple of

$$b^{III}(\theta) = \frac{1}{2} \sin(\theta)^2. \qquad (3.16)$$

*Proof.* The equation verified by $u$ rewrites as

$$\begin{cases} \partial_{\theta}^4 b + 4\partial_{\theta}^2 b = 0 & \text{in } (0, \omega), \\ b(0) = b'(0) = b(\omega) = b'(\omega) = 0, \end{cases}$$

so $b$ must be a linear combination of the solutions of the homogeneous differential equation, that is of $\cos(2\theta)$, $\sin(2\theta)$, $1$, $\theta$. Denote $\chi(\theta) = \theta - \frac{\sin(2\theta)}{2}$, the conditions $b(0) = b'(0) = 0$ implies that $b$ must be a linear combination of $\chi(\theta)$ and $\chi'(\theta) = 1 - \cos(2\theta)$, so for some $(\alpha, \beta) \in \mathbb{R}^2 \setminus \{(0, 0)\}$:

$$\begin{cases} \alpha\chi(\omega) + \beta\chi'(\omega) = 0 \\ \alpha\chi'(\omega) + \beta\chi''(\omega) = 0. \end{cases}$$

There is a nonzero solution if and only if $\chi''(\omega)\chi(\omega) = \chi'(\omega)^2$, meaning

$$\sin(\omega)(\omega \cos(\omega) - \sin(\omega)) = 0.$$

The only solutions in $(0, 2\pi]$ are $\omega = \pi, t_1, 2\pi$, and in each case the space of solutions has dimension 1 and is generated by the function

$$\theta \mapsto \chi(\omega)\chi'(\theta) - \chi'(\omega)\chi(\theta),$$

which gives $b^I, b^{II}, b^{III}$ (up to some multiplicative constant).

**Lemma 24.** Let $u \in \mathcal{M}(D_1)$ such that $\partial\mathrm{Spt}(u) = \{(x, y) \in D_1 : y = 0\}$.

* If $\mathrm{Spt}(u)$ is a half-disk, then the trace of $\Delta u|_{\mathrm{Spt}(u)}$ on $\partial\mathrm{Spt}(u)$ is +1 or -1.

* If $\mathrm{Spt}(u)$ is two half-disks, then $u$ is biharmonic in $D_1$ and $|\Delta u| \ge 1$ on $\partial\mathrm{Spt}(u)$.
Proof. If $u \in \mathcal{M}(\mathbb{D}_1)$, then for any $\phi = (\phi^x, \phi^y) \in \mathcal{C}_c^\infty(\mathbb{D}_1, \mathbb{R}^2)$ we have

$$ \left. \frac{d}{dt} \right|_{t=0^+} E \left( u \circ (\mathrm{Id} + t\phi)^{-1} ; \mathbb{D}_1 \right) \geq 0, $$

which is equivalent to the following inner variation relation (see [11, Lem. 4.4.] for a detailed computation):

$$ \forall \phi \in \mathcal{C}_c^\infty(\mathbb{D}_1, \mathbb{R}^2), \quad \int_{\mathbb{D}_1} (|\Delta u|^2 + \chi_{u \neq 0}) \nabla \cdot \phi \geq 2 \int_{\mathbb{D}_1} \Delta u (2\nabla^2 u : \nabla \phi + \nabla u \cdot \Delta \phi) \quad (3.17) $$

and when we replace $\phi$ with $-\phi$, we have an equality above. Suppose that $\mathrm{Spt}(u) = \{(x, y) \in \mathbb{D}_1 : y > 0\}$, then $u$ is smooth up to the boundary (since it is biharmonic with Dirichlet condition for $u$ and $\nabla u$ along a smooth boundary), so

$$ u_r \xrightarrow{r \to 0} \frac{\Delta^+ u(0)}{2} y_+^2 $$

in every $\mathcal{C}^k(\mathbb{D}_1 \cap \{y > 0\})$ space (here $\Delta^+$ designate the trace of $\Delta u|_{\{y>0\}}$ on $\{y = 0\}$). Replacing $u$ with $u_r$ in the inner variation relation (3.17) gives, as $r \to 0$, and for any $\phi \in \mathcal{C}_c^\infty(\mathbb{D}_1, \mathbb{R}^2)$,

$$ \int_{\mathbb{D}_1 \cap \{y>0\}} (|\Delta^+ u(0)|^2 + 1) \nabla \cdot \phi = 2 \int_{\mathbb{D}_1 \cap \{y>0\}} \Delta^+ u(0) (2\Delta^+ u(0)e_y \otimes e_y : \nabla \phi + \Delta^+ u(0)(ye_y) \cdot \Delta \phi), $$

which after integration by parts gives

$$ ((\Delta^+ u(0))^2 - 1) \int_{\mathbb{D}_1 \cap \{y=0\}} \phi^y = 0. $$

And so $(\Delta^+ u)^2 = 1$ on $\{(x, y) \in \mathbb{D}_1 : y = 0\}$.

Suppose now that $\mathrm{Spt}(u) = \{(x, y) \in \mathbb{D}_1 : y \neq 0\}$. Let $\phi = (\phi^x, \phi^y) \in \mathcal{C}_c^\infty(\mathbb{D}_1, \mathbb{R}^2)$ such that $\phi^y(x, 0) \geq 0$ for any $x$, and let

$$ u^t = \begin{cases} u \circ (\mathrm{Id} + t\phi)^{-1} & \text{in } (\mathrm{Id} + t\phi)(\{y > 0\}) \\ u & \text{in } \{y < 0\} \\ 0 & \text{elsewhere.} \end{cases} $$

Then $u^t$ is a valid competitor for $u$ (note that this would not be true when replacing $\phi$ with $-\phi$ due to overlapping of $(\mathrm{Id} + t\phi)(\{y > 0\})$ and $\{y < 0\}$): by the inner variation (3.17), this gives $|\Delta^+ u(0)| \geq 1$. Finally, let $v$ be the biharmonic extension of $u$ in $\mathbb{D}_1$: then by minimality we have $E(u; \mathbb{D}_1) \leq E(v; \mathbb{D}_1)$ and since $u$ has full support in $\mathbb{D}_1$, then $\int_{\mathbb{D}_1} |\Delta u|^2 \leq \int_{\mathbb{D}_1} |\Delta v|^2$, so $u$ must be equal to its own biharmonic extension. $\square$

**Proposition 25.** Let $u \in \mathcal{M}(\mathbb{D}_1) \setminus \{0\}$, suppose that $u$ is 2-homogeneous. Then $u$ belongs to one of the four types of functions described in (1.7) (1.8), (1.9), (1.10).

Proof. Since $u \in \mathcal{M}(\mathbb{D}_1)$, by Lemma 12, $\nabla u$ is continuous. Therefore, $\mathrm{Spt}(u)(= \{u \neq 0\} \cup \{\nabla u \neq 0\})$ is an open cone, and on each connected component, $u$ must be a biharmonic function with Dirichlet boundary conditions. By Lemma 23, each angular sector (that is not the full disk $\mathbb{D}_1$) must have an opening $\pi, t_1$ or $2\pi$. This reduces to five different cases, up to rotation:

* $u = cu^1$ for some $c \in \mathbb{R}$, and Lemma 24 gives $c = \pm 1$.

* $u = cu^{II}$ for some $c \in \mathbb{R}$, and Lemma 24 gives $c = \pm 1$.
• $u = cu^{\text{III}}$ for some $c \in \mathbb{R}$, and Lemma 24 gives $|c| \ge 1$.

• $u(x, y) = c^+u^1(x, y)\chi_{y>0} + c^-u^1(x, -y)\chi_{y<0}$ for some $c^+, c^- \in \mathbb{R}$, and Lemma 24 gives $c^+ = c^-$, so $u = c^+u^{\text{III}}$, and this is already covered in the previous case.

• $\partial\text{Spt}(u)$ is reduced to a point. In this case, $u$ is a general 2-homogeneous biharmonic function: using it Goursat decomposition, we may write $u = (x^2 + y^2)v + w$ where $v, w$ are harmonic and respectively 0-homogeneous (meaning it is constant) and 2-homogeneous (meaning it is a linear combination of $xy$ and $x^2 - y^2$), so it is written as in (1.10). Note that when $a^2 = b^2 + c^2$, the solution is in fact of type III.

<figure><img src="image_7.png" /></figure>

**Definition 26.** We let $\mathcal{M}_{\text{hom}}$ be the set of non-zero 2-homogeneous functions $u$ in $\mathcal{M}(\mathbb{D}_1)$. We have a splitting

$$ \mathcal{M}_{\text{hom}} = \mathcal{M}_{\text{hom}}^{\pi} \sqcup \mathcal{M}_{\text{hom}}^{t_1} \sqcup \mathcal{M}_{\text{hom}}^{2\pi}, $$

such that $u \in \mathcal{M}_{\text{hom}}^{\Theta}$ if and only if $u \in \mathcal{M}_{\text{hom}}$ and $\Theta = 2|\text{Spt}(u) \cap \mathbb{D}_1|$. Similarly, we write $\mathcal{B}_{\text{hom}}$ the set of non-zero minimizers of $\mathcal{G}(\cdot, 0)$ that are invariant with respect to the $t$ variable, and we have a splitting

$$ \mathcal{B}_{\text{hom}} = \mathcal{B}_{\text{hom}}^{\pi} \sqcup \mathcal{B}_{\text{hom}}^{t_1} \sqcup \mathcal{B}_{\text{hom}}^{2\pi} $$

such that $u \in \mathcal{M}_{\text{hom}}^{\Theta}$ if and only if $u(re^{i\theta}) = r^2b(\theta)$ where $b$ (seen as a function of $t, \theta$ that is constant with respect to $t$) belongs to $\mathcal{B}_{\text{hom}}^{\Theta}$.

**Lemma 27.** There exists $d > 0$ such that for any $u, v \in \mathcal{M}_{\text{hom}}$ with $|\text{Spt}(u) \cap \mathbb{D}_1| \ne |\text{Spt}(v) \cap \mathbb{D}_1|$, we have $\|u - v\|_{H^1(\mathbb{D}_1)} \ge d$.

*Proof.* Suppose $|\text{Spt}(u) \cap \mathbb{D}_1| < |\text{Spt}(v) \cap \mathbb{D}_1|$. Since the supports of $u$ and $v$ have an opening $\pi, t_1$ (for $u$), $t_1, 2\pi$ (for $v$), there exists a sector $S \subset \mathbb{D}_1$ of opening at least $\frac{t_1-\pi}{2}$ such that $u = 0$ on $S$ and $S \subset \text{Spt}(v)$. In particular there exists $p \in \partial\mathbb{D}_{\frac{1}{2}}$ such that $\mathbb{D}_{p, \frac{1}{10}} \subset S$: by the nondegeneracy Lemma 14 applied to $v$ on this disk we get

$$ \|u - v\|_{H^1(\mathbb{D}_1)} \ge \|v\|_{H^1(\mathbb{D}_{p, \frac{1}{10}})} \gtrsim 1. $$

<figure><img src="image_8.png" /></figure>

**Remark 28.** We will show below that functions of type I described in (1.7) are in fact minimizers, see remark 32. We conjecture (but do not prove) that every functions of type II and III described in (1.8), (1.9) also belong to $\mathcal{M}(\mathbb{D}_1)$.

On the contrary, by the non-degeneracy Lemma 14, we know that the function $u_{a,b,c}$ described in (1.10) cannot be a minimizer when the parameters $a, b, c$ are too small. In fact, in the work in preparation [14], H.-C. Grunau and M. Müller prove that for any $(a, b, c) \in \mathbb{R}^3$, the function $u_{a,b,c}$ belong to $\mathcal{M}(\mathbb{D}_1)$ when $|a| \ge \frac{1}{4}$, which includes the case of type III minimizers. The situation for $u^{\text{III}}$ and $u_{a,b,c}$ with $|a| < \frac{1}{4}$ and $(b, c) \ne (0, 0)$ remains open.

### 3.3 Existence of blow-ups

The first use of the monotonicity formula of Theorem 19 is to show that any blow-up is necessarily 2-homogeneous.

**Lemma 29.** Let $u \in \mathcal{M}(\mathbb{D}_1)$, assume $0 \in \partial\text{Spt}(u)$, and let $r_1 \ge r_2 \ge r_3 \ge \dots \to 0$ such that

$$ \sup_{n \in \mathbb{N}^*} \|u_{r_n}\|_{H^1(\mathbb{D}_1)} < \infty. $$
Then there exists some extraction $(n_i)_{i \ge 1}$, and some 2-homogeneous minimizer $v$ as characterized in Lemma 25 such that

$$u_{r_{n_i}} \underset{i \to +\infty}{\longrightarrow} v \text{ in } H_{\text{loc}}^2(\mathbb{D}_1),$$

and $W(v, r) = W(u, 0)$ for any $r \in (0, 1)$.

*Proof.* Since $u_{r_n}$ is bounded in $H^1(\mathbb{D}_1)$, in particular $u(0) = |\nabla u(0)| = 0$. The sequence $(u_{r_n})_n$ satisfies the hypothesis of Lemma 15, so after extraction we may suppose $u_{r_n}$ converges in $H_{\text{loc}}^2(\mathbb{D}_1)$ to some limit $v$. Let $\phi \in C_c^\infty([1/2, 1[, \mathbb{R}_+)$ be such that $\int_0^1 \phi(s)ds = 1$, and

$$W_{\phi}(v, s) := \int_{0}^{1} \phi(\rho)W(v, s\rho)ds.$$

Then $W_\phi(v, s)$ may be rewritten as

$$W_{\phi}(v, s) = \int_{0}^{1} \phi(\rho) \left( \frac{E(v; \mathbb{D}_{\rho s})}{(\rho s)^2} + \rho s N'(v, \rho s) + R(v, \rho s) \right) d\rho \\ = \int_{0}^{1} \left\{ \phi(\rho) \left( \frac{E(v; \mathbb{D}_{\rho s})}{(\rho s)^2} - N(v, \rho s) + R(v, \rho s) \right) - \rho \phi'(\rho) N(v, \rho s) \right\} d\rho.$$

Since $u_{r_n}$ converges to $v$ in $H_{\text{loc}}^2(\mathbb{D}_1)$ and $\chi_{u_{r_n} \ne 0}$ converges in $L^1(\mathbb{D}_1)$ to $\chi_{u \ne 0}$ (by Lemma 15), then for fixed $s$:

$$W_{\phi}(v, s) = \lim_{n \to +\infty} W_{\phi}(u_{r_n}, s) = \lim_{n \to +\infty} W_{\phi}(u, s r_n),$$

and we see the expression on the right-hand side is independent of $s$ (by the monotonicity of $W$), so $W_\phi(v, \cdot)$ is constant. Since $W(v, s/2) \le W_\phi(v, s) \le W(v, s)$ for any $s \in (0, 1)$, this implies $W(v, \cdot)$ is constant.

By Theorem 19, $v$ is therefore 2-homogeneous. Moreover, $v$ cannot be zero, since otherwise $u_{r_n}$ would be zero in $\mathbb{D}_{1/2}$ for some large enough $n$ by the nondegeneracy Lemma 14, which would contradict the fact that $0 \in \partial \text{Spt}(u)$. As a consequence, by Lemma 25, $v$ is equal to one of the four types of solutions described in (1.7), (1.8), (1.9), (1.10). $\square$

Our goal in the rest of the section is to prove the two following results, on the existence of blow-ups under some a priori information on the solution.

**Proposition 30.** Let $u \in \mathcal{M}(\mathbb{D}_1)$ such that

$$\limsup_{r \to 0} \frac{|\{u=0\} \cap \mathbb{D}_r|}{|\mathbb{D}_r|} > 0.$$

Then there exists a blow-up of type I or II at the origin.

When there is no such density estimate of $\{u=0\}$, we do not know yet whether a blow-up exists. However, we can prove that there is a nonzero “renormalized” blow-up in the following sense.

**Proposition 31.** Let $u \in \mathcal{M}(\mathbb{D}_1)$ such that $u(0) = |\nabla u(0)| = 0$ and $(r_n)_{n \in \mathbb{N}^*}$ a sequence converging to 0 such that

$$\lim_{r \to 0} \|u_{r_n}\|_{H^1(\mathbb{D}_1)} = +\infty.$$

Then there exists some subsequence $r_{n_k}$, and some nonzero 2-homogeneous biharmonic function $v$, such that

$$\frac{u_{r_{n_k}}}{\|u_{r_{n_k}}\|_{H^1(\mathbb{D}_1)}} \underset{k \to +\infty}{\longrightarrow} v \text{ in } H_{\text{loc}}^2(\mathbb{R}^2).$$
**Remark 32.** We may now prove that the function defined in (1.7) belongs to $\mathcal{M}(\mathbb{D}_1)$ by a blow-up argument. Indeed, for every small $\epsilon > 0$ consider $u^\epsilon \in \mathcal{M}(\mathbb{D}_1)$ verifying $(u^\epsilon, \partial_r u^\epsilon) = (\epsilon, 0)$ on $\partial\mathbb{D}_1$. Such minimizer exists by standard arguments. Let $\eta \in C_c^\infty(\mathbb{R}, \mathbb{R})$ with $\eta \equiv 1$ near 0, then

$$E(u^\epsilon; \mathbb{D}_1) \le E\left(\epsilon\eta\left(\frac{1-|\cdot|}{\sqrt{\epsilon}}\right); \mathbb{D}_1\right) = \mathcal{O}_{\epsilon\to 0}(\sqrt{\epsilon}).$$

So by the nondegeneracy Lemma 14, $u^\epsilon = 0$ in $\mathbb{D}_1$ for a sufficiently small $\epsilon > 0$ that we fix. Let $r > 0$ be the largest radius such that $u^\epsilon = 0$ in $\mathbb{D}_r$, and $p \in \partial\mathbb{D}_r \cap \partial\text{Spt}(u^\epsilon)$. Since $\text{Spt}(u^\epsilon)$ has density at most $\frac{1}{2}$ at $p$, then by Proposition 30, $u^\epsilon(p+\cdot)$ admits a blow-up of type I, which must then be a minimizer.

The proofs of Proposition 30, 31 rely on the structure of $W(u, r)$. We remind that by Theorem 19, $W(u, r)$ is decomposed as

$$W(u, r) := \frac{1}{r^2} E(u; \mathbb{D}_r) + rN'(u, r) + R(u, r),$$

where $N, R$ are defined in (3.7), (3.9). Since $N(u, r) \gtrsim \|u_r\|_{L^2(\partial\mathbb{D}_1)}^2 + \|\nabla u_r\|_{L^2(\partial\mathbb{D}_1)}^2 \gtrsim R(u, r)$, there exists some universal constant $\kappa > 0$ (which could be made explicit, although we will not need it) such that for any function $u \in H^2(\mathbb{D}_1)$, $r \in (0, 1)$:

$$|R(u, r)| \le \kappa N(u, r). \qquad (3.18)$$

The next lemma uses this decomposition of $W(u, r)$ to ensure that for any $u \in \mathcal{M}(\mathbb{D}_1)$, $u_r$ cannot be too small compared to $u$ for $r$ close to 1, in a uniform way.

**Lemma 33.** Let $u \in \mathcal{M}(\mathbb{D}_1)$ such that $u(0) = |\nabla u(0)| = 0$, $r \in (0, 1)$, $\kappa$ the constant from (3.18). Then

$$N(u, r) \ge r^\kappa N(u, 1) - \frac{1 - r^\kappa}{\kappa} W(u, 1).$$

A scale-invariant statement of this lemma is that for any $s < r$, $u \in \mathcal{M}(\mathbb{D}_r)$ such that $u(0) = |\nabla u(0)| = 0$:

$$N(u, s) \ge \left(\frac{s}{r}\right)^\kappa N(u, r) - \frac{1 - \left(\frac{s}{r}\right)^\kappa}{\kappa} W(u, r).$$

This is obtained by applying Lemma 33 to $u_r$ with a radius $\frac{s}{r}$.

*Proof of Lemma 33.* We start from $W(u, r) \le W(u, 1)$, (given by the monotonicity from Theorem 19), and we decompose $W(u, r)$ as:

$$\frac{1}{r^2} E(u; \mathbb{D}_r) + rN'(u, r) + R(u, r) \le W(u, 1).$$

Using the estimate $|R| \le \kappa N$ (see (3.18)), and the fact that the energy $E$ is nonnegative, we get

$$rN'(u, r) - \kappa N(u, r) \le W(u, 1)$$

which may be rewritten as

$$\frac{d}{dr} \frac{N(u, r)}{r^\kappa} \le \frac{W(u, 1)}{r^{\kappa+1}}.$$

We integrate this from $r(< 1)$ to 1:

$$N(u, 1) - \frac{N(u, r)}{r^\kappa} \le \left(\frac{1}{r^\kappa} - 1\right) \frac{W(u, 1)}{\kappa},$$

which is the result.

<figure><img src="image_9.png" /></figure>
**Lemma 34.** There exists $\sigma > 0$ such that the following holds: let $(u^{(n)})_n$ be a sequence in $\mathcal{M}(\mathbb{D}_1)$ such that

$$u^{(n)}(0) = |\nabla u^{(n)}(0)| = 0, \quad N(u^{(n)}, 1) \xrightarrow{n \to +\infty} +\infty, \quad W(u^{(n)}, 1) \le \sigma N(u^{(n)}, 1).$$

Then there exists a subsequence $(n_k)_{k \ge 1}$ and a nonzero biharmonic function $v : \mathbb{D}_1 \to \mathbb{R}$ such that

$$\frac{u^{(n_k)}}{\|u^{(n_k)}\|_{H^1(\mathbb{D}_1)}} \xrightarrow{k \to +\infty} v \text{ in } H_{\text{loc}}^2(\mathbb{D}_1).$$

*Proof.* Let $(u^{(n)})$ be such a sequence, for a value $\sigma > 0$ that will be fixed sufficiently small. By Lemma 33 we have, for all $r \in [\frac{1}{2}, 1]$, $N(u^{(n)}, r) \ge 2^{-\kappa} N(u^{(n)}, 1) - \frac{1-2^{-\kappa}}{\kappa} W(u^{(n)}, 1) \ge 2^{-\kappa-1} N(u^{(n)}, 1)$ for a small enough $\sigma$.

Let $M_n = 2 \int_{\frac{1}{2}}^1 N(u^{(n)}, r)dr$. Then $M_n \ge 2^{-\kappa-1} N(u^{(n)}, 1) \to +\infty$ and by the Cacciopoli inequality of Lemma 10 we have

$$\|u^{(n)}\|_{H^1(\mathbb{D}_1)}^2 \lesssim M_n \lesssim \|u^{(n)}\|_{H^1(\mathbb{D}_1)}^2.$$

It is enough to prove that $M_n^{-1/2}u^{(n)}$ converges to some non-zero biharmonic function. Since this sequence is bounded in $H^1(\mathbb{D}_1)$, and $M_n^{-1/2}u^{(n)} \in \mathcal{M}_{M_n^{-1}}(\mathbb{D}_1)$, by Lemma 15 it admits (after extraction of a subsequence $n_k$) some limit $v$ in the $H_{\text{loc}}^2(\mathbb{D}_1)$ sense such that $v \in \mathcal{M}_0(\mathbb{D}_1)$ (meaning $v$ is biharmonic).

We now prove that $v$ is not identically zero. There exists some $r_n \in [\frac{1}{2}, 1]$ such that $M_n = N(u^{(n)}, r_n)$, and by application of Lemma 33 we have, for all $r \in [\frac{1}{4}, \frac{1}{2}]$ ($\subset [\frac{r_n}{4}, r_n]$),

$$N(u^{(n)}, r) \ge 4^{-\kappa} M_n - \frac{1-4^{-\kappa}}{\kappa} W(u^{(n)}, 1) \ge 4^{-\kappa-1} M_n \text{ for a small enough } \sigma.$$

In particular, $\|M_n^{-1/2}u^{(n)}\|_{H^1(\mathbb{D}_{1/2})}^2$ is bounded from below by a positive constant. Since the convergence of $M_{n_k}^{-1/2}u^{(n_k)}$ to $v$ is strong in $H^1(\mathbb{D}_{1/2})$, necessarily $v$ is non-zero. $\square$

The next lemma is central for the existence of blow-ups: it states that at points where $Spt(u)^c$ has a positive density, $W(u, r)$ controls $N(u, r)$ from above. Since $r \mapsto W(u, r)$ is nondecreasing, this will imply a uniform bound of $N(u, r)$ as $r \to 0$.

**Lemma 35.** Let $\delta \in (0, 1)$, there exists $C_\delta > 0$ such that for any $u \in \mathcal{M}(\mathbb{D}_1)$ with $u(0) = |\nabla u(0)| = 0$ and

$$|\{u \ne 0\} \cap \mathbb{D}_1| \le (1-\delta)|\mathbb{D}_1|,$$

we have

$$N(u, 1) \le C_\delta(1 + W(u, 1)_+).$$

Note that this lemma fails without the hypothesis on the support: we expect $u : (x, y) \mapsto \lambda y^2$ to be a solution in $\mathcal{M}(\mathbb{D}_1)$ for any $\lambda \ge 1$, but $W(u, 1) = \pi$, whereas $N(u, 1) = c\lambda^2$ for some $c > 0$, so the estimate cannot hold for large enough $\lambda$.

*Proof.* Suppose that for some sequence of minimizers $u^{(n)}$ verifying this density estimate we have $N(u^{(n)}, 1) \ge n(1 + W(u^{(n)}, 1)_+)$. In particular Lemma 34 applies for large enough $n$: up to the extraction of a subsequence, $\frac{u^{(n)}}{\|u^{(n)}\|_{H^1(\mathbb{D}_1)}}$ converges in $H_{\text{loc}}^2(\mathbb{D}_1)$ to a non-zero biharmonic function $v$.

By Fatou's lemma we have $|\{v = 0\}| \ge \limsup_{n \to +\infty} |\{u^{(n)} = 0\}| \ge \delta$, which is a contradiction since, for instance, any biharmonic function is analytic, so its support must have full measure. $\square$
We now deduce the proofs of Proposition 30 and 31.

Proof of Proposition 30. Let $r_n \to 0$ be a sequence such that for some $\delta \in (0, \frac{1}{2})$, we have

$$ \frac{|\{u=0\} \cap \mathbb{D}_{r_n}|}{|\mathbb{D}_{r_n}|} > 2\delta. $$

Let $\tau = \sqrt{1-\delta}$, such that for any $r \in (\tau r_n, r_n)$, we have

$$ \frac{|\{u=0\} \cap \mathbb{D}_r|}{|\mathbb{D}_r|} > \delta. $$

Then for every such $n$, $r \in (\tau r_n, r_n)$, we have by Lemma 35

$$ N(u, r) \le C_{\delta}(1 + W(u, r)_{+}) \le C_{\delta}(1 + W(u, 1)_{+}), $$

where $C_\delta$ is the constant from Lemma 35. We integrate this in $r \in (\tau r_n, r_n)$:

$$ \|u_{r_n}\|_{H^1(\mathbb{D}_1 \setminus \mathbb{D}_\tau)}^2 \lesssim \int_{\tau}^{1} N(u, s r_n) ds \lesssim \delta C_\delta(1 + W(u, 1)_+), $$

and by the Cacciopoli inequality of Lemma 10, $\|u_{r_n}\|_{H^1(\mathbb{D}_\tau)}$ is also bounded independently of $n$. So we can apply Lemma 29 to the sequence $u_{r_n}$, and there exists some subsequence $u_{r_{n_k}}$ that converges in $H_{\text{loc}}^2(\mathbb{D}_1)$ to some $v \in \mathcal{M}_{\text{hom}}$. Since the density estimate is still verified at the limit, necessarily $v \in \mathcal{M}_{\text{hom}}^I \sqcup \mathcal{M}_{\text{hom}}^{II}$. $\square$

Proof of Proposition 31. By the Cacciopoli inequality of Lemma 10,

$$ \|u_{r_n}\|_{H^1(\mathbb{D}_1)}^2 \lesssim \int_{r_n/2}^{r_n} N(u, s) \frac{ds}{s} \lesssim \|u_{r_n}\|_{H^1(\mathbb{D}_1)}^2, $$

so there exists some $s_n \in [\frac{r_n}{2}, r_n]$ such that

$$ \|u_{r_n}\|_{H^1(\mathbb{D}_1)}^2 \lesssim N(u, s_n) \lesssim \|u_{r_n}\|_{H^1(\mathbb{D}_1)}^2 \xrightarrow{n \to +\infty} +\infty. $$

Since $W(u_{s_n}, 1) = W(u, s_n) \le W(u, 1)$, Lemma 34 applies to $u_{s_n}$ for large enough $n$: up to extracting a subsequence, we may suppose that $u_{s_n}/\|u_{s_n}\|_{H^1(\mathbb{D}_1)}$ converges in $H_{\text{loc}}^2(\mathbb{D}_1)$ to some nonzero biharmonic function $v$.

Let $R > 2$: $u_{s_n}$ is well-defined in $\mathbb{D}_R$ for large enough $n$, and we now prove that up to extracting another subsequence, $u_{s_n}$ converges to $v$ in $H_{\text{loc}}^2(\mathbb{D}_R)$. By Lemma 33 applied to $u$ with the radii $s_n, s_n r$ for $r \in [R/2, R]$, we have

$$ N(u, s_n) \ge R^{-\kappa} N(u, s_n r) - \frac{1 - R^{-\kappa}}{\kappa} W(u, s_n r)_+ . $$

By integrating this estimate in $r \in [\frac{R}{2}, R]$ we get

$$ \|u_{s_n}\|_{H^1(\mathbb{D}_R \setminus \mathbb{D}_{\frac{R}{2}})}^2 \lesssim R^{\kappa} (N(u, s_n) + W(u, 1)_+). $$

Thus $u_{s_n}$ is locally bounded in $H^1(\mathbb{D}_R)$ for any $R > 0$: up to extracting a subsequence, we obtain that $u_{s_n}/\|u_{s_n}\|_{H^1(\mathbb{D}_1)}$ converges in $H_{\text{loc}}^2(\mathbb{R}^2)$ to $v$.

Next we prove that $v$ is a sum of $k$-homogeneous biharmonic functions for $k \in \{0, 1, 2\}$. Since $v$ is biharmonic, it admits a decomposition (as in (3.13)):

$$ v(re^{i\theta}) = \sum_{n \in \mathbb{Z}} (a_n r^{|n|+2} + b_n r^{|n|}) e^{in\theta}, $$
for some complex coefficients $(a_n)_{n \in \mathbb{Z}}$, $(b_n)_{n \in \mathbb{Z}}$. Proving $v$ is a sum of $k$-homogeneous function for $k \le 2$ is thus equivalent to proving that every coefficient other than $a_0, b_0, b_{\pm 1}, b_{\pm 2}$ vanishes.
For any $r > 0$, we have

$$
\begin{align*}
W_0(v, r) &= \lim_{n \to +\infty} W_0 \left( \frac{u_{s_n}}{\|u_{s_n}\|_{H^1(\mathbb{D}_1)}}, r \right) = \lim_{n \to +\infty} \frac{W_0(u_{s_n}, r)}{\|u_{s_n}\|_{H^1(\mathbb{D}_1)}^2} \\
&\le \limsup_{n \to +\infty} \frac{W(u, s_n r)}{\|u_{s_n}\|_{H^1(\mathbb{D}_1)}^2} \le \limsup_{n \to +\infty} \frac{W(u, 1)}{\|u_{s_n}\|_{H^1(\mathbb{D}_1)}^2} \\
&= 0 \text{ since } \|u_{s_n}\|_{H^1(\mathbb{D}_1)} \to +\infty.
\end{align*}
$$

By Lemma 21 we have the explicit expression

$$
W_0(v, r) = 8\pi \sum_{n \in \mathbb{Z}} H_n (r^{|n|} a_n, r^{|n|-2} b_n),
$$

where $H_n$ is the Hermitian form

$$
H_n(z, w) = |n|^3|z|^2 + 2|n|^2(|n| - 2)\Re(\bar{z}w) + (|n| - 2)(|n|^2 - 2|n| + 2)|w|^2.
$$

Note that $H_n$ is positive definite for $|n| \ge 3$; using this, we get

$$
\begin{align*}
\forall r > 0, \quad 0 \ge \frac{W_0(v, r)}{8\pi} &\ge \sum_{|n| \le 2} H_n (r^{|n|} a_n, r^{|n|-2} b_n) \\
&= -4r^{-4} |b_0|^2 + 2r^2 (|a_1|^2 + |a_{-1}|^2) - 2 (\Re(\bar{a}_1 b_1) + \Re(\bar{a}_{-1} b_{-1})) \\
&\quad - r^{-2} (|b_1|^2 + |b_{-1}|^2) + 8r^4 (|a_2|^2 + |a_{-2}|^2),
\end{align*}
$$

so $a_{\pm 2} = a_{\pm 1} = 0$, and

$$
\sum_{|n| \le 2} H_n (r^{|n|} a_n, r^{|n|-2} b_n) = -4r^{-4} |b_0|^2 - r^{-2} (|b_1|^2 + |b_{-1}|^2).
$$

Now, for any $N \in \mathbb{Z}$ such that $|N| \ge 3$ we have

$$
0 \ge \frac{W_0(v, r)}{8\pi} \ge H_N (r^{|N|} a_N, r^{|N|-2} b_N) - 4r^{-4} |b_0|^2 - r^{-2} (|b_1|^2 + |b_{-1}|^2).
$$

If $a_N \neq 0$, the highest degree term of the right-hand side is $|N|^3|a_N|^{2r^2|N|}$, which is a contradiction
as $r \to +\infty$. So necessarily $a_N = 0$, and the previous inequality becomes

$$
0 \ge (|N| - 2)(|N|^2 - 2|N| + 2) |b_N|^2 r^{2(|N|-2)} - 4r^{-4} |b_0|^2 - r^{-2} (|b_1|^2 + |b_{-1}|^2).
$$

Since $|N| \ge 3$, then necessarily $b_N = 0$ (otherwise the inequality above does not hold as $r \to +\infty$).
This ends the proof that $v$ is a sum of $k$-homogeneous biharmonic polynomials for $k \le 2$.

Finally, we prove that $v$ is 2-homogeneous. Indeed, for any $n \in \mathbb{N}^*$, for any $p \in \mathbb{D}_{\frac{1}{2}}$, by Lemma 12 applied to $u_{r_n}$ we have

$$
|u_{r_n}(p)| \leq C|p|^2 \log \left( \frac{1}{|p|} \right) \|u_{r_n}\|_{H^1(\mathbb{D}_1)},
$$

for some universal constant $C > 0$. As a consequence, we have for any $p \in \mathbb{D}_{\frac{1}{2}}$:

$$
|v(p)| \leq C|p|^2 \log \left( \frac{1}{|p|} \right).
$$

Since $v$ is already known to be a polynomial of degree 2, this implies that $v$ is 2-homogeneous. $\square$
# 4 Epiperimetric inequality

Our goal is now to prove a quantitative version of the monotonicity formula of Theorems 18 and 19. In the classical Alt-Caffarelli problem (see for instance [32], or [36, Lem. 12.14.]), this is based mainly on two observations: first, the derivative of the normalized energy

$$W^{\text{AC}}(u, r) = \frac{1}{r^2} \int_{\mathbb{D}_r} (|\nabla u|^2 + \chi_{u>0}) - \frac{1}{r^3} \int_{\partial \mathbb{D}_r} u^2,$$

with respect to $r$ is bounded from below by $\frac{W^{\text{AC}}(u^r, r)-W^{\text{AC}}(u, r)}{r}$, where $u^r$ is the 1-homogeneous extension of $u$ in $\mathbb{D}_r$. The second observation is that for some universal constant $\eta > 0$, we have

$$W^{\text{AC}}(u, r) < (1 - \eta)W^{\text{AC}}(u^r, r) + \eta \frac{\pi}{2}.$$

Here $\frac{\pi}{2}$ is the energy of homogeneous minimizers. This leads to a differential inequality of the form

$$\frac{d}{dr} \left( W^{\text{AC}}(u, r) - \frac{\pi}{2} \right) \geq \frac{\eta}{(1-\eta)r} \left( W^{\text{AC}}(u, r) - \frac{\pi}{2} \right).$$

When $W^{\text{AC}}(u, 0) \geq \frac{\pi}{2}$, this integrates to an explicit rate of convergence

$$\forall r \in (0, 1), W^{\text{AC}}(u, r) - \frac{\pi}{2} \leq r^{\frac{\eta}{1-\eta}} \left( W^{\text{AC}}(u, 1) - \frac{\pi}{2} \right).$$

See [36, Lem. 12.14] for a more extensive discussion on the epiperimetric inequality and on how to deduce geometric information from this decay.

Neither of these observations is known in our setting: the (2-)homogeneous extension of $u$ in $\mathbb{D}_r$ is generally not a valid competitor for $u$ (since the radial derivatives may not match on each side of $\partial \mathbb{D}_r$, so the 2-homogeneous extension does not belong to $H^2$), and it is unclear whether the derivative of $W(u, r)$ is bounded from below by $\frac{W(u^r, r)-W(u, r)}{r}$, where $u^r$ would be some other “natural” competitor for $u$ in $\mathbb{D}_r$.

Instead of proving a differential inequality on $W(u, r)$, we prove directly the growth rate estimate

$$W(u, e^{-1}) - \frac{\Theta}{2} \leq (1 - \eta) \left( W(u, 1) - \frac{\Theta}{2} \right),$$

for some universal constant $\eta \in (0, 1)$, when $u$ is sufficiently close to some homogeneous solution of opening $\Theta \in \{\pi, t_1, 2\pi\}$. Here $e^{-1}$ does not play any particular role and could be replaced by any constant strictly lower than 1.

We also obtain a control of the variation $\|u_r - u_s\|_{H^1(\mathbb{D}_1)}$ at different scales $0 < s < r < 1$ by $\sqrt{W(u, 1) - \frac{\Theta}{2}}$. This will be enough to obtain the uniqueness and rate of convergence of blow-ups.

**Theorem 36.** Let $\Theta \in \{\pi, t_1, 2\pi\}$. There exist constants $c_1 > 0$, $\eta \in (0, 1)$ such that the following holds: let $u \in \mathcal{M}(\mathbb{D}_1)$, $\bar{u} \in \mathcal{M}_{\text{hom}}^{\Theta}$ if $\Theta \in \{\pi, t_1\}$ (resp. $\bar{u} \in \text{Span}(x^2 + y^2, x^2 - y^2, xy)$ if $\Theta = 2\pi$) such that

$$\|u - \bar{u}\|_{H^1(\mathbb{D}_1 \setminus \mathbb{D}_{e-1})} \leq c_1 \|\bar{u}\|_{H^1(\mathbb{D}_1)}, \quad \text{and} \quad W(u, 0) \geq \frac{\Theta}{2}.$$

Then

$$W(u, e^{-1}) - \frac{\Theta}{2} \leq (1 - \eta) \left( W(u, 1) - \frac{\Theta}{2} \right),$$

and

$$\|u - u_{e^{-1}}\|_{H^1(\mathbb{D}_1 \setminus \mathbb{D}_{e-1})} \leq C_1 \sqrt{W(u, 1) - \frac{\Theta}{2}}.$$
Similarly to Section 3, it is more convenient to first prove the epiperimetric inequality in exponential coordinates. In all that follows, a function of $\theta$ will also be seen, by an abuse of notation, as a function of $(t, \theta)$ that is constant with respect to $t$.

**Theorem 37.** Let $\Theta \in \{\pi, t_1, 2\pi\}$. There exist constants $c_0 > 0$, $\eta \in (0, 1)$, such that the following holds: let $u \in H_{\text{lin}}^2(C_0)$ be a minimizer of $\mathcal{G}(\cdot, 0)$, assume that there exists $b \in \mathcal{B}_{\text{hom}}^{\Theta}$ if $\Theta \in \{\pi, t_1\}$ (resp. $b \in \text{Span}(1, \cos(2\theta), \sin(2\theta))$ if $\Theta = 2\pi$) such that

$$ \|u - b\|_{H^1(C_0 \setminus C_1)} \leq c_0 \|b\|_{H^1(\mathbb{S}^1)} \quad \text{and} \quad \mathcal{W}(u, 1) \geq \frac{\Theta}{2}. \qquad (4.1) $$

Then

$$ \mathcal{W}(u, 1) \leq \mathcal{W}(u, 0) - \eta \left( \left( \mathcal{W}(u, 0) - \frac{\Theta}{2} \right) + \|\partial_t u\|_{H^1(C_0 \setminus C_1)}^2 \right). $$

Theorem 36 will be obtained as a direct consequence of Theorem 37. A central point in the statement of Theorem 37 is that the monotonicity formula of Theorem 18 only gives a control

$$ \|\partial_{t,t} u\|_{L^2(C_0 \setminus C_1)}^2 + \|\partial_{t,\theta} u\|_{L^2(C_0 \setminus C_1)}^2 \leq \frac{\mathcal{W}(u, 0) - \mathcal{W}(u, 1)}{4}, $$

and it is unclear whether $\|\partial_t u\|_{L^2(C_0 \setminus C_1)}^2$ is also controlled by the right-hand side, which explains the presence of the term $\|\partial_t u\|_{H^1(C_0 \setminus C_1)}^2$ in the conclusion of Theorem 37). When $u$ vanishes on a sufficiently large subset of $C_0 \setminus C_1$ we may apply a Poincaré inequality to $\partial_t u$, whereas the case where the support of $u$ is close to $C_0 \setminus C_1$ will be handled separately.

The proof of Theorem 37 relies on a comparison of the solution $u$ with an appropriate competitor. The construction of the competitor is summarized in the two lemmas below, and their proof will occupy subsections 4.2 to 4.5.

We recall that $\text{Spt}(u)$ designates every point where either $u$ or $\nabla u$ does not vanish, and we restrict it to $\mathbb{S}^1$ for functions only depending on $\theta$.

**Lemma 38.** Let $\Theta \in \{\pi, t_1, 2\pi\}$. There exist $c, \epsilon, C > 0$ such that for any $u \in H^2(\mathbb{S}^1)$, $v \in H^1(\mathbb{S}^1)$, with $\{v \neq 0\} \subset \text{Spt}(u)$ and

$$ \left\{ \begin{array}{ll} (c, \Theta - c) \subset \text{Spt}(u) \subset (-c, \Theta + c) & \text{when } \Theta \in \{\pi, t_1\}, \\ (c, \pi - c) \cup (\pi + c, 2\pi - c) \subset \text{Spt}(u) \subset (0, 2\pi) & \text{when } \Theta = 2\pi, \end{array} \right. \qquad (4.2) $$

there exists $U \in H_{\text{lin}}^2(C_0)$ such that $U(0, \theta) = u(\theta)$, $\partial_t U(0, \theta) = v(\theta)$ and

$$ \mathcal{G}(U, 0) \leq \mathcal{G}(u, 0) - \epsilon \left( \mathcal{G}(u, 0) - \frac{\Theta}{2} \right)_+ + C \int_{\mathbb{S}^1} (\partial_{\theta} v)^2. \qquad (4.3) $$

Here $\mathcal{G}(u, 0) = \frac{1}{2} \int_{\mathbb{S}^1} ((\partial_{\theta,\theta} u)^2 - 4(\partial_{\theta} u)^2 + \chi_{\{u \neq 0\}})$, since $(t, \theta) \mapsto u(\theta)$ is constant with respect to $t$; in a way we are comparing the energy of $U(t, \theta)$ with the energy of the homogeneous extension $(t, \theta) \mapsto u(\theta)$, and the last term takes into account the effect of the boundary condition $\partial_t U(0, \theta) = v(\theta)$ on the estimate.

The support condition (4.2), and more particularly the case disjunction on the value of $\Theta$, is based on the following observation (see the proof of Theorem 37): assume $u \in H_{\text{lin}}^2(C_0)$ is a minimizer of $\mathcal{G}(\cdot, 0)$, such that $\frac{\|u-b\|_{H^1(C_0 \setminus C_1)}}{\|b\|_{H^1(\mathbb{S}^1)}}$ is small for some $b \in \mathcal{B}_{\text{hom}}^{\Theta}$ with $\Theta \in \{\pi, t_1, 2\pi\}$:

* if $\Theta \in \{\pi, t_1\}$, then up to a translation the support of $b$ is $(0, \Theta)$, and we may prove that the support of $u(t, \cdot)$ for $t \in (\frac{1}{4}, \frac{3}{4})$ verifies the support assumption (4.2).
• if $\Theta = 2\pi$, then up to a translation the support of $b$ is either $\mathbb{S}^1$ or $(0, \pi) \cup (\pi, 2\pi)$; when $u$ is sufficiently close to $b$, the support of $u(t, \cdot)$ contains $(c, \pi - c) \cup (\pi + c, 2\pi - c)$. Lemma 38 treats this case under the additional assumption that $\text{Spt}(u(t, \cdot))$ is not $\mathbb{S}^1$. The case where $\text{Spt}(u) = \mathbb{S}^1$ is purposefully set aside, and is treated separately by the following lemma.

**Lemma 39.** Let $v \in H_{\text{lin}}^2(C_0)$ be a nonzero minimizer of $\mathcal{G}(\cdot, 0)$ such that $|\text{Spt}(v(0, \cdot))| = 2\pi$, then

$$ \mathcal{G}(v, 0) \le \mathcal{G}(v(0, \cdot), 0) - \frac{1}{10} (\mathcal{G}(v(0, \cdot), 0) - \pi)_+ + 6 \|\partial_{t,\theta} v(0, \cdot)\|^2 - 2 \|\partial_t v(0, \cdot)\|^2. $$

The proof of Lemma 38 is broken down into several steps, where we construct competitors for increasingly complex boundary conditions $(u, v)$:

• In subsection 4.2, we treat the case where $v = 0$, and $u$ is a single one-dimensional buckling mode (see subsection 4.1 for an introduction of this basis).

• In subsection 4.3, we suppose $v = 0$, for a general $u$.

• In subsection 4.4, we remove the $v = 0$ hypothesis and prove the full estimate of (4.3).

Lemma 39 is obtained by a direct comparison with the biharmonic extension in the disk model, in subsection 4.5.

Finally, in subsection 4.6 we deduce the epiperimetric inequality from Lemmas 38 and 39.

## 4.1 Buckling eigenbasis

We will denote by $\| \cdot \|$ (resp. $\langle \cdot, \cdot \rangle$) the $L^2(\mathbb{S}^1)$ norm (resp. scalar product). Let $(b_n)_{n \ge 1}$ be an eigenbasis associated to the buckling operator in the segment $[0, \pi]$: in other words it is an orthonormal basis of $H_0^1([0, \pi])$ for the scalar product $\langle \partial_\theta b_n, \partial_\theta b_m \rangle = \delta_{n,m}$, such that $b_n \in H_0^2([0, \pi])$ and $\partial_\theta^4 b_n = -\mu_n \partial_\theta^2 b_n$ where $\mu_n$ is the $n$-th buckling eigenvalue. The eigenvalues $(\mu_n)_{n \ge 1}$ can be defined as the $n$-th positive root of the function

$$ \mu \mapsto \sin\left(\frac{\pi\sqrt{\mu}}{2}\right) \left(\pi\sqrt{\mu}\cos\left(\frac{\pi\sqrt{\mu}}{2}\right) - 2\sin\left(\frac{\pi\sqrt{\mu}}{2}\right)\right). \quad (4.4) $$

In particular $\mu_n = (n+1)^2$ for odd $n$, and $n^2 < \mu_n < (n+2)^2$ for even $n$. The first few values are

$$ \mu_1 = 4, \mu_2 = \left(\frac{2t_1}{\pi}\right)^2 \approx 8.183, \mu_3 = 16, \mu_4 \approx 24.187, \dots $$

and $b_1(\theta) = \sqrt{\frac{2}{\pi}} \sin(\theta)_+^2$. For any $\omega > 0$, we denote

$$ b_{n,\omega}(\theta) = \sqrt{\frac{\omega}{\pi}} b_n \left(\frac{\pi}{\omega}\theta\right), \quad \mu_{n,\omega} = \frac{\pi^2}{\omega^2} \mu_n \quad (4.5) $$

Such that $b_{n,\omega}$ is a $\langle \partial_\theta^*, \partial_\theta^* \rangle$-orthonormal basis of $H_0^1([0, \omega])$, such that $b_{n,\omega} \in H_0^2([0, \omega])$ and $\partial_\theta^4 b_{n,\omega} = -\mu_{n,\omega} \partial_\theta^2 b_{n,\omega}$. A central point is that $(b_{n,\omega})$ is **not** orthogonal for the $L^2([0, \omega])$ scalar product ($\langle b_1, b_3 \rangle \ne 0$ for instance, since both do not change sign). We will use the following estimate.

**Lemma 40.** For any $\omega \in (0, 2\pi]$, for any $n \ge 1$, we have

$$ \|b_{n,\omega}\|^2 \lesssim \frac{1}{\mu_{n,\omega}}. $$
*Proof.* It is enough to prove this for $\omega = \pi$, by change of variable.
When $n$ is odd, we have $\mu_n = (n+1)^2$ and

$$b_n(\theta) = \sqrt{\frac{2}{\pi} \frac{1 - \cos((n+1)\theta)}{n+1}}$$

And so $\|b_n\|^2 \lesssim \frac{1}{(n+1)^2} = \frac{1}{\mu_n}$.

When $n$ is even, the computation is simpler on the interval $[-\frac{\pi}{2}, +\frac{\pi}{2}]$. We know that $\mu_n \in (n^2, (n+2)^2)$ and $\theta \mapsto b_n(\theta + \frac{\pi}{2})$ is a scalar multiple of the function

$$f_n(\theta) = \sin(\sqrt{\mu_n}\theta) - \beta_n\theta$$

where $\beta_n \in \mathbb{R}$ is chosen such that $f_n(\pm\frac{\pi}{2}) = f'_n(\pm\frac{\pi}{2}) = 0$, meaning

$$\sin\left(\frac{\pi\sqrt{\mu_n}}{2}\right) = \beta_n \frac{\pi}{2}, \quad \sqrt{\mu_n}\cos\left(\frac{\pi\sqrt{\mu_n}}{2}\right) = \beta_n. \qquad (4.6)$$

The first condition gives $|\beta_n| \lesssim 1$, so $|f_n| \lesssim 1$. Then, using these two estimates,

$$\|b_n\|^2 = \frac{\|f_n\|^2}{\|\partial_\theta f_n\|^2} \lesssim \frac{1}{\int_{-\pi/2}^{+\pi/2} (\sqrt{\mu_n}\sin(\sqrt{\mu_n}\theta) - \beta_n)^2 d\theta} \lesssim \frac{1}{\mu_n}.$$

<figure><img src="image_10.png" /></figure>

From the expression (4.5), we see that for $\omega \in (0, 2\pi]$, $\mu_{n,\omega} = 4$ is only verified in three cases:

(I) $\omega = \pi$, $n = 1$. Then we have $\mu_{1,\pi} = 4$, $\mu_{2,\pi} \approx 8.187$.

(II) $\omega = t_1$, $n = 2$. Then we have $\mu_{1,t_1} \approx 1.955$, $\mu_{2,t_1} = 4$, $\mu_{3,t_1} \approx 7.821$.

(III) $\omega = 2\pi$, $n = 3$. Then we have $\mu_{2,2\pi} \approx 2.046$, $\mu_{3,2\pi} = 4$, $\mu_{4,2\pi} \approx 6.047$.

For $\Theta \in \{\pi, t_1, 2\pi\}$, we will denote

$$i(\Theta) = \begin{cases} 1 & \text{if } \Theta = \pi, \\ 2 & \text{if } \Theta = t_1, \\ 3 & \text{if } \Theta = 2\pi, \end{cases}$$

and the numerical values above give us the following estimate: for any $\omega \in (0, 2\pi]$ such that $|\omega - \Theta| \le 0.1\Theta$, $n \in \mathbb{N}^*$, we have

$$n < i(\Theta) \Rightarrow \mu_{n,\omega} \le 3, \quad n > i(\Theta) \Rightarrow \mu_{n,\omega} \ge 5. \qquad (4.7)$$

Similarly, if $\omega \le \frac{\pi}{2}$, then $\mu_{n,\omega} \ge 5$ for any $n \in \mathbb{N}^*$.

## 4.2 Single mode case

We start by proving Lemma 38 under the simplest boundary conditions: $v = 0$, and $u$ is a single buckling mode on a segment with eigenvalue close to 4, meaning the first buckling mode (for angle close to $\pi$), or the second buckling mode (for angles close to $t_1$), or the third buckling mode (for angles close to $2\pi$). The proof in each case is exactly the same, so we prove all three at the same time.
**Lemma 41.** Let $\Theta \in \{\pi, t_1, 2\pi\}$. There exist $c, \epsilon > 0$ such that for any $a \in \mathbb{R}^*$, $\omega \in [\Theta - c, \Theta + c] \cap (0, 2\pi)$, if one denotes

$$u(\theta) = ab_{i(\Theta),\omega}(\theta),$$

then there exists $U \in H_{\text{lin}}^2(C_0)$ such that $U(0, \theta) = u(\theta)$, $\partial_t U(0, \theta) = 0$ and

$$\mathcal{G}(U, 0) \le \mathcal{G}(u, 0) - \epsilon \left( \mathcal{G}(u, 0) - \frac{\Theta}{2} \right)_+ . \qquad (4.8)$$

*Proof.* We denote in this proof

$$\mathcal{G}_{\text{norm}}(U, 0) = \mathcal{G}(U, 0) - \frac{\Theta}{2}$$

so that our goal is to find some competitor $U$ such that $\mathcal{G}_{\text{norm}}(U, 0) \le \mathcal{G}_{\text{norm}}(u, 0) - \epsilon \mathcal{G}_{\text{norm}}(u, 0)_+$. When $\mathcal{G}_{\text{norm}}(u, 0) \le 0$, this is obvious by taking $U(t, \theta) = u(\theta)$ for every $t \ge 0$: we therefore suppose in what follows that $\mathcal{G}_{\text{norm}}(u, 0) > 0$. We let $\epsilon > 0$ to be fixed sufficiently small later, and

$$\chi_{\epsilon}(t) = 1 - 9\epsilon(1 - (1 + t)e^{-t}),$$

which is positive in $\mathbb{R}_+$ (for small enough $\epsilon$), decreasing, with $\chi_{\epsilon}(0) = 1$, $\chi'_{\epsilon}(0) = 0$, $\chi_{\epsilon}(t) \xrightarrow{t \to +\infty} 1 - 9\epsilon$ and

$$\int_{0}^{+\infty} e^{-2t} (\chi_{\epsilon}(t) - (1 - \epsilon)) dt = 0. \qquad (4.9)$$

We define $\delta_0 \in \mathbb{R}$ by

$$\omega = (1 + \delta_0)\Theta,$$

and for any $\delta$ we write

$$\phi_{\delta}(\theta) = b_{i(\Theta),\Theta} \left( \frac{\theta}{1 + \delta} \right)$$

so that $u(\theta) = a\sqrt{1 + \delta_0\phi_{\delta_0}(\theta)}$. We define our competitor $U = U(t, \theta)$ by

$$U(t, \theta) = a\sqrt{1 + \delta_0\phi_{\delta_0\chi_{\epsilon}(t)}(\theta)}.$$

First observe that $U(0, \theta) = u(\theta)$, $\partial_t U(0, \theta) = 0$ (since $\chi_{\epsilon}(0) = 1$, $\chi'_{\epsilon}(0) = 0$) and $U \in H_{\text{lin}}^2(C_0)$ (since $\chi_{\epsilon} \to 1 - 9\epsilon$, and $\chi'_{\epsilon}, \chi''_{\epsilon} \to 0$ at $+\infty$). Then

$$\mathcal{G}_{\text{norm}}(U, 0) = \int_{0}^{+\infty} e^{-2t} \left\{ \|\partial_{t,t} U\|^2 + 2\|\partial_{t,\theta} U\|^2 \right\} dt \\ + \int_{0}^{+\infty} e^{-2t} \left\{ \|\partial_{\theta,\theta} U\|^2 - 4\|\partial_{\theta} U\|^2 + \Theta \delta_0 \chi_{\epsilon} \right\} dt,$$

so

$$\begin{aligned} \mathcal{G}_{\text{norm}}(U, 0) - (1 - \epsilon)\mathcal{G}_{\text{norm}}(u, 0) = & a^2(1 + \delta_0) \int_0^{+\infty} e^{-2t} \left\{ \|\partial_{t,t}\phi_{\delta_0\chi_\epsilon}\|^2 + 2\|\partial_{t,\theta}\phi_{\delta_0\chi_\epsilon}\|^2 \right\} dt \\ & + (1 - a^2(1 + \delta_0))\Theta\delta_0 \int_0^{+\infty} e^{-2t} \{\chi_\epsilon - (1 - \epsilon)\} dt \\ & + a^2(1 + \delta_0) \int_0^{+\infty} e^{-2t} \left\{ \|\partial_{\theta,\theta}\phi_{\delta_0\chi_\epsilon}\|^2 - 4\|\partial_\theta\phi_{\delta_0\chi_\epsilon}\|^2 + \Theta\delta_0\chi_\epsilon \right\} dt \\ & - a^2(1 + \delta_0)(1 - \epsilon) \int_0^{+\infty} e^{-2t} \left\{ \|\partial_{\theta,\theta}\phi_{\delta_0}\|^2 - 4\|\partial_\theta\phi_{\delta_0}\|^2 + \Theta\delta_0 \right\} dt. \end{aligned} \qquad (4.10)$$
Since $|\chi'_\epsilon| + |\chi''_\epsilon| \lesssim \epsilon$, the first line of (4.10) is bounded by

$$A_1(1 + \delta_0)a^2\epsilon^2\delta_0^2$$

for some universal constant $A_1 > 0$. The second line is zero by our normalization condition (4.9). Now we estimate the sum of the third and fourth lines, which is equal to $a^2(1 + \delta_0)F(\delta_0)$, where

$$F_{\epsilon}(\delta) := \int_{0}^{+\infty} e^{-2t} (f(\delta\chi_{\epsilon}) - (1-\epsilon)f(\delta)) dt,$$

$$f(\delta) := \|\partial_{\theta,\theta}\phi_{\delta}\|^2 - 4\|\partial_{\theta}\phi_{\delta}\|^2 + \Theta\delta.$$

We first compute $f(\delta)$ explicitly:

$$f(\delta) = \Theta\delta + \int_{0}^{\Theta(1+\delta)} \left\{ \frac{1}{(1+\delta)^4} \partial_{\theta,\theta} b_{i(\Theta),\Theta} \left( \frac{\theta}{1+\delta} \right)^2 - \frac{4}{(1+\delta)^2} \partial_{\theta} b_{i(\Theta),\Theta} \left( \frac{\theta}{1+\delta} \right)^2 \right\} d\theta \\ = \Theta\delta + 4 \left( \frac{1}{(1+\delta)^3} - \frac{1}{1+\delta} \right).$$

In particular $f(0) = 0$, $f''(0) = 40$, and there exists $C > 0$ such that for any $\delta \in [-1/2, 1/2]$, we have

$$|f(\delta) - f'(0)\delta - 20\delta^2| \le C|\delta|^3.$$

Then

$$F_{\epsilon}(0) = \int_{0}^{+\infty} e^{-2t} (f(0) - (1-\epsilon)f(0)) dt = 0 \quad \text{since } f(0) = 0$$

$$F'_{\epsilon}(0) = \int_{0}^{+\infty} e^{-2t} (\chi_{\epsilon}(t)f'(0) - (1-\epsilon)f'(0)) dt = 0 \quad \text{by (4.9)}$$

$$\begin{aligned} F''_{\epsilon}(0) &= \int_{0}^{+\infty} e^{-2t} \left( \chi_{\epsilon}(t)^2 f''(0) - (1-\epsilon)f''(0) \right) dt \\ &= f''(0) \int_{0}^{+\infty} e^{-2t} \left( (1-9(1-(1+t)e^{-t})\epsilon)\chi_{\epsilon}(t) - (1-\epsilon) \right) dt \\ &= 40 \int_{0}^{+\infty} e^{-2t} \left( -9(1-(1+t)e^{-t})\epsilon)\chi_{\epsilon}(t) \right) dt \quad \text{by (4.9)} \\ &= -20 \left( 1 - \frac{45}{16}\epsilon \right) \epsilon. \end{aligned}$$

Moreover, for any $\delta \in [-1/2, 1/2]$ we can estimate uniformly

$$\begin{aligned} |F_{\epsilon}(\delta) + 10\left(1 - \frac{45}{16}\epsilon\right)\epsilon\delta^2| &= |F_{\epsilon}(\delta) - F_{\epsilon}(0) - F'_{\epsilon}(0)\delta - \frac{1}{2}F''_{\epsilon}(0)\delta^2| \\ &\le \int_{0}^{+\infty} e^{-2t} \left|f(\delta\chi_{\epsilon}) - f(0) - f'(0)\delta\chi_{\epsilon} - \frac{1}{2}f''(0)(\delta\chi_{\epsilon})^2\right| dt \\ &\quad + (1-\epsilon) \int_{0}^{+\infty} e^{-2t} \left|f(\delta) - f(0) - f'(0)\delta - \frac{1}{2}f''(0)\delta^2\right| dt \\ &\le A_2|\delta|^3, \end{aligned}$$

for some $A_2 > 0$ that is independent of $\epsilon$: in particular, for $\epsilon \le \frac{8}{45}$ we obtain

$$F_{\epsilon}(\delta) \le -5\epsilon\delta^2 + A_2|\delta|^3.$$
Going back to (4.10), we may now bound

$$
\begin{align*}
\mathcal{G}_{\text{norm}}(U, 0) - (1 - \epsilon)\mathcal{G}_{\text{norm}}(u, 0) &= a^2(1 + \delta_0) \int_0^{+\infty} e^{-2t} \left\{ \|\partial_{t,t}\phi_{\delta_0\chi_\epsilon}\|^2 + 2\|\partial_{t,\theta}\phi_{\delta_0\chi_\epsilon}\|^2 \right\} dt \\
&\qquad + a^2(1 + \delta_0)F(\delta_0) \\
&\leq (1 + \delta_0)a^2 \left( A_1\epsilon^2\delta_0^2 - 5\epsilon\delta_0^2 + A_2|\delta_0|^3 \right).
\end{align*}
$$

Thus we may fix a small enough $\epsilon \le \min(\frac{8}{45}, \frac{1}{A_1})$ such that for any $\delta_0$ verifying $|\delta_0| \le \frac{\epsilon}{A_2}$, the
right-hand side above is negative so

$$
\mathcal{G}_{\text{norm}}(U, 0) \leq (1 - \epsilon)\mathcal{G}_{\text{norm}}(u, 0).
$$

<figure><img src="image_11.png" /></figure>

In the case $\Theta = 2\pi$, there is an additional possibility we have to consider here: the support of $u$ is the union of two disjoint segments of length close to $\pi$, and $u$ is a first buckling eigenmode on each connected component of the support.

**Lemma 42.** There exist $c, \epsilon > 0$ such that for any $a_1, a_2 \in \mathbb{R}^*$, $\omega_1, \omega_2 \in [\pi - c, \pi + c]$ with $\omega_1 + \omega_2 \le 2\pi$, for any $\beta \in [0, 2\pi - \omega_1 - \omega_2]$, if one denotes

$$
u(\theta) = \begin{cases} a_1 b_{1,\omega_1}(\theta) & \text{if } \theta \in [0, \omega_1] \\ a_2 b_{1,\omega_2}(\theta - \omega_1 - \beta) & \text{if } \theta \in [\omega_1 + \beta, \omega_1 + \beta + \omega_2] \end{cases},
$$

then there exists $U \in H_{\text{lin}}^2(C_0)$ such that $U(0, \theta) = u(\theta)$, $\partial_t U(0, \theta) = 0$ and

$$
\mathcal{G}(U, 0) \leq \mathcal{G}(u, 0) - \epsilon (\mathcal{G}(u, 0) - \pi)_+ \quad (4.11)
$$

*Proof.* Again the case $\mathcal{G}(u, 0) \le \pi$ is direct by choosing $U(t, \theta) = u(\theta)$, so we assume $\mathcal{G}(u, 0) > \pi$.

Without loss of generality (up to switching the roles of the two sectors) we suppose that
$\beta \ge \frac{2\pi-\omega_1-\omega_2}{2}$. Let $U_1, U_2$ be the competitors built in Lemma 41 for $a_1b_{1,\omega_1}$, $a_2b_{1,\omega_2}$ respectively.
We let $\epsilon$ be the constant from Lemma 41, so that $\text{Spt}(U_i(t, \cdot)) = [0, \pi + (\omega_i - \pi)\chi_\epsilon(t)]$ where we
remind $\chi_\epsilon(t) = 1 - 9\epsilon(1 - (1+t)e^{-t})$ takes values between $1 - 9\epsilon$ and 1. We define

$$
U(t, \theta) = U_1(t, \theta) + U_2(t, \omega_1 + \omega_2 + \beta - \theta).
$$

We claim the support of the two terms on the right-hand side are disjoint: indeed for any $t \ge 0$, we have

$$
\begin{align*}
\operatorname{Spt}(U_1(t, \cdot)) &= [0, \pi + (\omega_1 - \pi)\chi_\epsilon(t)], \\
\operatorname{Spt}(U_2(t, \omega_1 + \omega_2 + \beta - \cdot)) &= [\omega_1 + \omega_2 + \beta - \pi - (\omega_2 - \pi)\chi_\epsilon(t), \omega_1 + \omega_2 + \beta]
\end{align*}
$$

and the condition

$$
\pi + (\omega_1 - \pi)\chi_\epsilon(t) \le \omega_1 + \omega_2 + \beta - \pi - (\omega_2 - \pi)\chi_\epsilon(t)
$$

reduces to

$$
(2\pi - \omega_1 - \omega_2)\chi_{\epsilon}(t) \geq 2\pi - \omega_1 - \omega_2 - \beta.
$$

By our hypothesis on $\beta$, the right-hand side is smaller than $\frac{2\pi-\omega_1-\omega_2}{2}$, while the left-hand side is
larger than $(1-9\epsilon)(2\pi-\omega_1-\omega_2)$: since $\epsilon$ may be chosen smaller than $\frac{1}{18}$, this condition is always
verified. Thus, we have by additivity

$$
\begin{align*}
\mathcal{G}(U, 0) &= \mathcal{G}(U_1, 0) + \mathcal{G}(U_2, 0) \\
&\le \mathcal{G}(a_1 b_{1,\omega_1}, 0) + \mathcal{G}(a_2 b_{1,\omega_2}, 0) - \epsilon \left( \mathcal{G}(a_1 b_{1,\omega_1}, 0) - \frac{\pi}{2} \right)_+ - \epsilon \left( \mathcal{G}(a_2 b_{1,\omega_2}, 0) - \frac{\pi}{2} \right)_+ \\
&\le \mathcal{G}(u, 0) - \epsilon \left( \mathcal{G}(u, 0) - \pi \right)_+ \text{ by the general inequality } x_+ + y_+ \ge (x+y)_+, \forall x, y \in \mathbb{R}.
\end{align*}
$$

This concludes the proof.

<figure><img src="image_12.png" /></figure>
## 4.3 Removal of higher modes

We now prove Lemma 38 in the case where $v = 0$, and $u$ is a general function such that Spt($u$) verifies the hypothesis of Lemma 38. We reduce the problem to the single mode case to obtain the estimate (4.3).

We first prove some useful computations. Let $u \in H_0^2([0, \omega])$, written in the eigenbasis $(b_{n,\omega})$ as

$$u(\theta) = \sum_{n \ge 1} u_n b_{n,\omega}(\theta)$$

where $(u_n)_{n \ge 1}$ is uniquely defined, and $\sum_{n \ge 1} \mu_{n,\omega} |u_n|^2 = \|\partial_{\theta,\theta} u\|_{L^2([0,\omega])}^2 < \infty$. For any uniformly bounded sequence of functions $f_n \in C^2(\mathbb{R}_+, \mathbb{R})$ with $f_n(0) = 1, f'_n(0) = 0$, we let

$$u^{(f_n)_{n \ge 1}}(t, \theta) = \sum_{n \ge 1} u_n f_n(t) b_{n,\omega}(\theta).$$

In particular, $u^{(f_n)} \in H_{\text{lin}}^2(C_0)$ verifies $u^{(f_n)_{n \ge 1}}(0, \theta) = u(\theta)$, $\partial_t|_{t=0} u^{f,N}(t, \theta) = 0$.

**Lemma 43.** There exists some universal constant $C_0 > 0$ such that the following holds: for any $\omega \in (0, 2\pi]$, $u \in H_0^2([0, \omega])$, $(f_n)_{n \ge 1}$ a sequence of uniformly bounded functions in $C^2(\mathbb{R}_+, \mathbb{R})$ such that $f_n(0) = 1, f'_n(0) = 0$, and $u^{(f_n)_{n \ge 1}}$ defined as above, we have

$$\mathcal{G}(u^{(f_n)_{n \ge 1}}, 0) \le \omega \int_0^{+\infty} e^{-2t} \chi_{\exists n: f_n(t) \ne 0} dt + \sum_{n \ge 1} Q_{\mu_{n,\omega}-4}(f_n) u_n^2, \quad (4.12)$$

where

$$Q_{\lambda}(f) := \int_{0}^{+\infty} e^{-2t} \{C_{0}f''(t)^{2} + 2f'(t)^{2} + \lambda f(t)^{2}\} dt. \quad (4.13)$$

Moreover, when $f_n \equiv 1$ for every $n$ and $|\{u \ne 0\}| = \omega$, then (4.12) is an equality, and in this case we have

$$\mathcal{G}(u^{(1)_{n \ge 1}}, 0) = \mathcal{G}(u, 0) = \frac{\omega}{2} + \sum_{n \ge 1} \frac{\mu_{n,\omega} - 4}{2} u_n^2.$$

*Proof.* We compute directly

$$\begin{aligned} \mathcal{G}(u^{(f_n)_{n \ge 1}}, 0) = & \int_0^{+\infty} e^{-2t} \left( \int_{\mathbb{S}^1} \chi_{u^{(f_n)_{n \ge 1}} \ne 0} dt + \int_0^{+\infty} e^{-2t} \left\{ \left\| \sum_{n \ge 1} f_n''(t) u_n b_{n,\omega} \right\|^2 + 2 \left\| \sum_{n \ge 1} f_n'(t) u_n \partial_\theta b_{n,\omega} \right\|^2 \right\} dt \right. \\ & \left. + \int_0^{+\infty} e^{-2t} \left( \left\| \sum_{n \ge 1} f_n(t) u_n \partial_{\theta,\theta} b_{n,\omega} \right\|^2 - 4 \left\| \sum_{n \ge 1} f_n(t) u_n \partial_\theta b_{n,\omega} \right\|^2 \right) dt \right. \\ & \le \omega \int_0^{+\infty} e^{-2t} \chi_{\exists n: f_n(t) \ne 0} dt + \sum_{n,m \ge 1} \int_0^{+\infty} e^{-2t} f_n''(t) f_m''(t) dt \langle b_{n,\omega}, b_{m,\omega} \rangle u_n u_m \\ & + 2 \sum_{n \ge 1} \int_0^{+\infty} e^{-2t} f_n'(t)^2 dt u_n^2 + \sum_{n \ge 1} (\mu_{n,\omega} - 4) \int_0^{+\infty} e^{-2t} f_n(t)^2 dt u_n^2 \end{aligned}$$

Here we recall the orthogonality properties $\langle \partial_\theta b_{n,\omega}, \partial_\theta b_{m,\theta} \rangle = \delta_{n,m}$, $\langle \partial_{\theta,\theta} b_{n,\omega}, \partial_{\theta,\theta} b_{m,\theta} \rangle = \mu_{n,\omega} \delta_{n,m}$, but $\langle b_{n,\omega}, b_{m,\omega} \rangle$ is generally not zero; by Cauchy-Schwarz inequality, we have

$$\begin{aligned} \sum_{n,m \ge 1} \left( \int_0^{+\infty} e^{-2t} f_n''(t) f_m''(t) dt \right) \langle b_{n,\omega}, b_{m,\omega} \rangle u_n u_m & \le \left( \sum_{n \ge 1} \left( \int_0^{+\infty} e^{-2t} f_n''(t)^2 dt \right)^{\frac{1}{2}} \|b_{n,\omega}\| \|u_n\| \right)^2 \\ & \le \left( \sum_{n \ge 1} \|b_{n,\omega}\|^2 \right) \left( \sum_{n \ge 1} \int_0^{+\infty} e^{-2t} f_n''(t)^2 dt \, u_n^2 \right). \end{aligned}$$
Denote $C_0 = \sup_{\omega \in (0,2\pi]} (\sum_{n \ge 1} \|b_{n,\omega}\|^2)$, $C_0$ is finite since by Lemma 40, $\|b_{n,\omega}\|^2 \lesssim \mu_{n,\omega}^{-1} = \frac{\omega^2}{\pi^2} \mu_n^{-1}$ which is summable (we recall that $\mu_n^{-1} = O(\frac{1}{n^2})$). Then we identify

$$ \mathcal{G}(u^{(f_n)_{n \ge 1}}, 0) \le \omega \int_0^{+\infty} e^{-2t} \chi_{\exists n: f_n(t) \ne 0} dt + \sum_{n \ge 1} Q_{\mu_{n,\omega}-4}(f_n) u_n^2 $$

with an equality in the chain of inequality above when every $f_n$ is constant and the measure of the support of $u$ is $\omega$. In this case $Q_\lambda(1) = \frac{\lambda}{2}$, which concludes the proof. □

The constant function $f \equiv 1$ is clearly not a minimizer of the integral $Q_\lambda(f)$ for $\lambda > 0$, since it does not verify the associated Euler-Lagrange equation. It is however non-trivial whether there is a function with **compact support** and quantifiably lower energy $Q_\lambda$ for every $\lambda \ge 1$: this is shown in the following Lemma.

**Lemma 44.** There exist $\eta > 0$, $f \in C_c^2([0, +\infty[, [0, 1])$ verifying $f(0) = 1$, $f'(0) = 0$, such that for any $\lambda \ge 1$ we have

$$ Q_{\lambda}(f) \le (1 - \eta)Q_{\lambda}(1), $$

where $Q_\lambda$ is defined in (4.13).

Note that the condition $\lambda \ge 1$ could be replaced by any condition $\lambda \ge \lambda_0$ for some $\lambda_0 > 0$.

*Proof.* It is sufficient to prove the result for $\lambda = 1$, and it implies the result for any $\lambda \ge 1$. Indeed, suppose $f$ verifies the inequality for the value $\lambda = 1$, since $f \le 1$ and $f$ is not identically 1, then for some constant $\eta' \in (0, 1)$ we have

$$ \int_0^{+\infty} e^{-2t} f(t)^2 dt \le \frac{1 - \eta'}{2} $$

so for any $\lambda \ge 1$,

$$ \begin{aligned} Q_{\lambda}(f) &= Q_{1}(f) + (\lambda - 1) \int_{0}^{+\infty} e^{-2t} f(t)^{2} dt \\ &\le (1 - \eta) \frac{1}{2} + (1 - \eta') \frac{\lambda - 1}{2} \\ &\le (1 - \eta \wedge \eta') \frac{\lambda}{2}. \end{aligned} $$

Now we prove the result for $\lambda = 1$. Let $\epsilon \in (0, 1)$, $T > 1$ to be fixed later, $\chi \in C^\infty(\mathbb{R}, [0, 1])$ equal to 1 on $]-\infty, -1[$, $0$ on $[0, +\infty[$, and

$$ f_{\epsilon}(t) = (1 + \epsilon t)e^{-\epsilon t}, \quad f_{\epsilon,T}(t) = \chi(t - T)f_{\epsilon}(t). $$

We prove that for any sufficiently small $\epsilon$ we have the two estimates:

$$ (i) \quad Q_1(f_\epsilon) \le \left(1 - \frac{1}{4}\epsilon^2\right) Q_1(1), \quad (ii) \quad Q_1(f_{\epsilon,T}) \le \left(1 + C e^{-2T}\right) Q_1(f_\epsilon), \quad (4.14) $$

for some universal constants $c, C > 0$ (that are independent of $T, \epsilon$). The result is then obtained by fixing a sufficiently small $\epsilon$ such that the estimates hold, and then choosing $T$ large enough depending on $\epsilon$ such that $Q_1(f_{\epsilon,T}) \le \left(1 - \frac{1}{8}\epsilon^2\right) Q_1(1)$.
(i) We compute $f'_{\epsilon}(t) = -\epsilon^2 t e^{-\epsilon t}$, $f''_{\epsilon}(t) = -\epsilon^2(1-\epsilon t)e^{-\epsilon t}$. So for some constant $C > 0$ we may bound the first two terms of $Q_1(f_{\epsilon})$ (see (4.13)) by $C\epsilon^4$. Then the last term of (4.13) is estimated by

$$ \int_{0}^{+\infty} e^{-2t} f_{\epsilon}(t)^2 dt = \int_{0}^{+\infty} e^{-2(1+\epsilon)t} (1+\epsilon t)^2 dt = \frac{1}{2} \left(1 - \frac{\epsilon^2}{2} + \mathcal{O}(\epsilon^3)\right). $$

This proves (i) for sufficiently small $\epsilon$ (we remind that $Q_1(1) = \frac{1}{2}$).

(ii) We compute directly, using the (rough) estimate $f_{\epsilon}, f'_{\epsilon}, f''_{\epsilon} \lesssim 1$ and $Q(f_{\epsilon}) \gtrsim 1$ (for any $\epsilon \in (0, 1)$):

$$ \begin{aligned} Q_1(f_{\epsilon,T}) - Q_1(f_{\epsilon}) &\le \int_{T-1}^{T} e^{-2t} (C_0(f_{\epsilon,T}^{\prime\prime})^2 + 2(f_{\epsilon,T}^{\prime})^2 + (f_{\epsilon,T})^2) dt \\ &\le C e^{-2T} Q_1(f_{\epsilon}), \end{aligned} $$

for some constant $C > 0$. So we get the estimate (ii).

This concludes the proof.

<figure><img src="image_13.png" /></figure>

We now prove an analogue of Lemma 44 for **negative** values of $\lambda$.

**Lemma 45.** There exists $f \in C_c^2([0, +\infty[, \mathbb{R})$ verifying $f(0) = 1$, $f'(0) = 0$, such that for any $\lambda \le -1$ we have

$$ Q_{\lambda}(f) \le Q_{\lambda}(1), $$

where $Q_{\lambda}$ is defined in (4.13).

Note that in this case $Q_{\lambda}(1) = \frac{\lambda}{2} < 0$ since $\lambda < 0$, in particular we have $Q_{\lambda}(1) \le (1-\eta)Q_{\lambda}(1)$ for any $\eta > 0$.

*Proof.* Define, as in the previous proof, $\chi \in C^{\infty}(\mathbb{R}, [0, 1])$ equal to 1 on $]-\infty, -1]$, 0 on $[0, +\infty[$. We let

$$ f_{\epsilon}(t) = \epsilon t + e^{-\epsilon t}, $$

for some $\epsilon \in (0, 1)$ to be fixed. We have $f_{\epsilon}(0) = 1$, $f'_{\epsilon}(0) = 0$, and moreover $f'_{\epsilon}(t) = \epsilon(1 - e^{-\epsilon t})$, $f''_{\epsilon}(t) = \epsilon^2 e^{-\epsilon t}$. Thus

$$ \begin{aligned} \int_{0}^{+\infty} e^{-2t} f_{\epsilon}''(t)^2 dt &\lesssim \epsilon^4, \qquad \int_{0}^{+\infty} e^{-2t} f_{\epsilon}'(t)^2 dt \lesssim \epsilon^3 \\ \int_{0}^{+\infty} e^{-2t} f_{\epsilon}(t)^2 dt &= \frac{1}{2} + \frac{\epsilon^2}{4} + \mathcal{O}(\epsilon^3) \ge \frac{1}{2} \left(1 + \frac{\epsilon^2}{3}\right) \quad \text{for small } \epsilon. \end{aligned} $$

We now fix $\epsilon \in (0, 1)$ small enough such that the last inequality holds. Let

$$ f_{\epsilon,T}(t) = \chi(t-T)f_{\epsilon}(t) $$

for some $T > 1$ to be fixed. With the rough estimate $|f_{\epsilon}(t)| + |f'_{\epsilon}(t)| + |f''_{\epsilon}(t)| \lesssim 1 + t$ (for small $\epsilon$), we have for any $\lambda \le -1$:

$$ \begin{aligned} Q_{\lambda}(f_{\epsilon,T}) &= Q_{\lambda}(f_{\epsilon}) + (Q_{\lambda}(f_{\epsilon,T}) - Q_{\lambda}(f_{\epsilon})) \\ &\le \frac{\lambda}{2} \left(1 + \frac{\epsilon^2}{3}\right) + \int_{T-1}^{+\infty} e^{-2t} \left(C_0 f_{\epsilon,T}^{\prime\prime 2} + 2f_{\epsilon,T}^{\prime 2} - \lambda f_{\epsilon}^2\right) dt \\ &\le \frac{\lambda}{2} \left(1 + \frac{\epsilon^2}{3} - CT^2 e^{-2T}\right) \end{aligned} $$

for some universal constant $C > 0$. Thus for a large enough $T$ we have $Q_{\lambda}(f_{\epsilon,T}) \le \frac{\lambda}{2} = Q_{\lambda}(1)$. $\square$

<figure><img src="image_14.png" /></figure>
We now use Lemma 44 and 45, to prove the Lemma 38 in the case where $v = 0$. We prove in fact a slightly stronger estimate, with an additional control on $\partial_{\theta}u$ that will be useful in the next section.

We first introduce some notations: consider $u \in H^2(\mathbb{S}^1)$ such that $\mathrm{Spt}(u) \neq \mathbb{S}^1$. Its support is a countable union of disjoint intervals $(\theta_k, \theta_k + \omega_k)_{k \in \mathbb{N}}$ of respective lengths

$$ \omega_0 \geq \omega_1 \geq \omega_2 \geq \dots \to 0. $$

We let $u^k(\theta) = u(\theta_k + \theta)\chi_{0<\theta<\omega_k}$ such that $u^k \in H_0^2([0, \omega_k])$ is decomposed as

$$ u^k = \sum_{n \ge 1} u_n^k b_{n, \omega_k} \qquad (4.15) $$

for some sequence of coefficients $(u_n^k)_{n \ge 1, k \ge 0}$.

**Lemma 46.** Let $\Theta \in \{\pi, t_1, 2\pi\}$. There exist $c, \epsilon, a > 0$ such that for any $u \in H^2(\mathbb{S}^1)$ that verifies

$$ \mathrm{Spt}(u) \neq \mathbb{S}^1, \ [c, \Theta - c] \subset \mathrm{Spt}(u) \subset [-c, \Theta + c], $$

there exists $U \in H_{\mathrm{lin}}^2(\mathcal{C}_0)$ such that $U(0, \theta) = u(\theta)$, $\partial_t U(0, \theta) = 0$ and

$$ \mathcal{G}(U, 0) \le \mathcal{G}(u, 0) - \epsilon \left( \mathcal{G}(u, 0) - \frac{\Theta}{2} \right)_+ - a \left( \left\| \partial_{\theta, \theta} \left( u^0 - \sum_{n=1}^{i(\Theta)} u_n^0 b_{n, \omega_0} \right) \right\|^2 + \sum_{k \ge 1} \| \partial_{\theta, \theta} u^k \|^2 \right) \quad (4.16) $$

where $u^k$ is defined from the decomposition of $u$ in (4.15).

*Proof.* Let $\omega = |\mathrm{Spt}(u)|$. Let $(\theta_k, \omega_k, u^k, u_n^k)$ be defined from the decomposition of $u$ as in (4.15). By our hypothesis we have $|\omega_0 - \Theta| \le 2c$ and $|\omega_1| \le 2c$. We suppose $c$ is sufficiently small so that $|\omega_0 - \Theta| < 0.1\Theta$ and $\omega_1 \le \frac{\pi}{2}$ (see equation (4.7))

Let now $g$ be the function constructed from Lemma 44, and $h$ the function constructed from Lemma 45. Denote $T > 0$ the smallest value such that $g = h = 0$ on $[T, +\infty[$. We now define, for any $n \ge 1, k \ge 0$,

$$ f_n^k(t) = \begin{cases} g(t) & \text{if } \mu_{n,\omega_k} \ge 5 \text{ i.e. } k \ge 1 \text{ or } n > i(\Theta), \\ h(t) & \text{if } \mu_{n,\omega_k} \le 3 \text{ i.e. } k = 0 \text{ and } n < i(\Theta), \\ 1 & \text{if } |\mu_{n,\omega_k} - 4| < 1 \text{ i.e. } k = 0 \text{ and } n = i(\Theta). \end{cases} $$

We define the competitor

$$ \tilde{U}(t, \theta) = \sum_{k \ge 0} \sum_{n \ge 1} f_n^k(t) u_n^k b_{n, \omega_k}(\theta - \theta_k). $$

By the definition of $(f_n^k)$, we have for any $t \ge T$:

$$ \tilde{U}(t, \theta) = u_{i(\Theta)}^0 b_{i(\Theta), \omega_0}(\theta - \theta_0). $$

Denote $\eta > 0$ the constant from Lemma 44, $\epsilon > 0$ the constant from Lemma 41, and $\tilde{U}(t, \theta)$ the competitor built in Lemma 41 from the initial data $u_{i(\Theta)}^0 b_{i(\Theta), \omega_0}$, so that

$$ \mathcal{G}(\tilde{U}, 0) \le \mathcal{G}(u_{i(\Theta)}^0 b_{i(\Theta), \omega_0}, 0) - \epsilon \left( \mathcal{G}(u_{i(\Theta)}^0 b_{i(\Theta), \omega_0}, 0) - \frac{\Theta}{2} \right)_+ . $$

Let then

$$ U(t, \theta) = \begin{cases} \tilde{U}(t, \theta) & \text{if } t < T \\ \tilde{U}(t - T, \theta + \theta_0) & \text{if } t \ge T \end{cases} $$
For any $k \ge 1$, since every function $(f_n^k)_{n \ge 1}$ vanishes on $[T, +\infty[$, then

$$ \int_0^{+\infty} e^{-2t} \chi_{\exists n: f_n^k \neq 0} dt \le \frac{1 - e^{-2T}}{2}. $$

We compute:

$$
\begin{aligned}
\mathcal{G}(U, 0) &= \mathcal{G}(\tilde{U}, 0) + (\mathcal{G}(U, 0) - \mathcal{G}(\tilde{U}, 0)) \\
&\le \mathcal{G}(u_{i(\Theta)}^0 b_{i(\Theta),\omega_0}(0) + (1-\eta) \sum_{n>i(\Theta)} Q_{\mu_{n,\omega_0}-4}(1) |u_n^0|^2 + \sum_{n<i(\Theta)} Q_{\mu_{n,\omega_0}-4}(1) |u_n^0|^2 \\
&\qquad + \sum_{k \ge 1} \left( \frac{1-e^{-2T}}{2} \omega_k + (1-\eta) \sum_{n \ge 1} Q_{\mu_{n,\omega_k}-4}(1) |u_n^k|^2 \right) \\
&\qquad + e^{-2T} (\mathcal{G}(\tilde{U}, 0) - \mathcal{G}(u_{i(\Theta)}^0 b_{i(\Theta),\omega_0}(0)) \text{ by Lemmas 43, 44, 45} \\
&\le \mathcal{G}(u_{i(\Theta)}^0 b_{i(\Theta),\omega_0}(0) - e^{-2T} \epsilon \left( \mathcal{G}(u_{i(\Theta)}^0 b_{i(\Theta),\omega_0}(0) - \frac{\Theta}{2} \right)_+ \\
&\qquad + (1-\eta) \sum_{n>i(\Theta)} Q_{\mu_{n,\omega_0}-4}(1) |u_n^0|^2 + \sum_{n<i(\Theta)} Q_{\mu_{n,\omega_0}-4}(1) |u_n^0|^2 \\
&\qquad + \sum_{k \ge 1} \left( \frac{1-e^{-2T}}{2} \omega_k + (1-\eta) \sum_{n \ge 1} Q_{\mu_{n,\omega_k}-4}(1) |u_n^k|^2 \right) \text{ by Lemma 41.}
\end{aligned}
$$

Denote $\bar{\epsilon} = \min(e^{-2T}\epsilon, e^{-2T}, \frac{\eta}{2})$. We remind that when $n < i(\Theta)$, $Q_{\mu_{n,\omega_0}-4}(1) < 0$ so $Q_{\mu_{n,\omega_0}-4}(1) < (1-\bar{\epsilon})Q_{\mu_{n,\omega_0}-4}(1)$. We remind also that for any $x, y \in \mathbb{R}$, we have $x_+ + y_+ \ge (x+y)_+$, which we will use repeatedly to gather each term. We identify

$$ \mathcal{G}(u, 0) = \mathcal{G}(u_{i(\Theta)}^0 b_{i(\Theta),\omega_0}(0) + \sum_{n \neq i(\Theta)} Q_{\mu_{n,\omega_0}-4}(1) |u_n^0|^2 + \sum_{k \ge 1} \left( \frac{1}{2}\omega_k + \sum_{n \ge 1} Q_{\mu_{n,\omega_k}-4}(1) |u_n^k|^2 \right). $$

Then following the estimates above we obtain

$$
\begin{aligned}
\mathcal{G}(U, 0) &\le \mathcal{G}(u_{i(\Theta)}^0 b_{i(\Theta),\omega_0}(0) - \bar{\epsilon} \left( \mathcal{G}(u_{i(\Theta)}^0 b_{i(\Theta),\omega_0}(0) - \frac{\Theta}{2} \right)_+ \\
&\quad + \left(1 - \bar{\epsilon} - \frac{\eta}{2}\right) \sum_{n>i(\Theta)} Q_{\mu_{n,\omega_0}-4}(1) |u_n^0|^2 + \left(1 - \bar{\epsilon}\right) \sum_{n<i(\Theta)} Q_{\mu_{n,\omega_0}-4}(1) |u_n^0|^2 \\
&\quad + \sum_{k \ge 1} \left( \frac{1 - \bar{\epsilon}}{2} \omega_k + \left(1 - \bar{\epsilon} - \frac{\eta}{2}\right) \sum_{n \ge 1} Q_{\mu_{n,\omega_k}-4}(1) |u_n^k|^2 \right) \\
&\le \mathcal{G}(u, 0) - \bar{\epsilon} \left( \mathcal{G}(u, 0) - \frac{\Theta}{2} \right)_+ - \frac{\eta}{2} \sum_{(n,k): n>i(\Theta) \text{ or } k \ge 1} Q_{\mu_{n,\omega_k}-4}(1) |u_n^k|^2.
\end{aligned}
$$

We remind for the last line that if $(n, k)$ is such that $n > i(\Theta)$ or $k \ge 1$, then $\mu_{n,\omega_k} \ge 5$ so $Q_{\mu_{n,\omega_k}-4}(1) = \frac{\mu_{n,\omega_k}-4}{2} \ge \frac{1}{10}\mu_{n,\omega_k}$. We now conclude the proof with $a = \frac{\eta}{20}$, since

$$
\begin{aligned}
\sum_{(n,k): n>i(\Theta) \text{ or } k \ge 1} \mu_{n,\omega_k} |u_n^k|^2 &= \sum_{n>i(\Theta)} \mu_{n,\omega_0} |u_n^0|^2 + \sum_{n \ge 1} \sum_{k \ge 1} \mu_{n,\omega_k} |u_n^k|^2 \\
&= \left\| \partial_{\theta,\theta} \left( u^0 - \sum_{n=1}^{i(\Theta)} u_n^0 b_{n,\omega_0} \right) \right\|^2 + \sum_{k \ge 1} \| \partial_{\theta,\theta} u^k \|^2.
\end{aligned}
$$

<figure><img src="image_15.png" /></figure>
Next, we treat separately the case $\Theta = 2\pi$ where the support of $u$ is the union of two disjoint segments of length close to $\pi$. The proof follows the same computations, we have separated the two for clarity.

**Lemma 47.** There exist $c, \epsilon, a > 0$ such that for any $u \in H^2(\mathbb{S}^1)$ such that

$[c, \pi - c] \cup [\pi + c, 2\pi - c] \subset \text{Spt}(u)$, $[\pi - c, \pi + c] \not\subset \text{Spt}(u)$, $[2\pi - c, 2\pi + c] \not\subset \text{Spt}(u)$, there exists $U \in H_{\text{lin}}^2(C_0)$ such that $U(0, \theta) = u(\theta)$, $\partial_t U(0, \theta) = 0$ and

$$ \mathcal{G}(U, 0) \le \mathcal{G}(u, 0) - \epsilon (\mathcal{G}(u, 0) - \pi)_+ - a \left( \|\partial_{\theta,\theta} (u^0 - u_1^0 b_{1,\omega_0})\|^2 + \|\partial_{\theta,\theta} (u^1 - u_1^1 b_{1,\omega_1})\|^2 + \sum_{k \ge 2} \|\partial_{\theta,\theta} u^k\|^2 \right). \quad (4.17) $$

where $u^k$ is defined from the decomposition of $u$ in (4.15).

*Proof.* Let $\omega = |\text{Spt}(u)|$. Let $(\theta_k, \omega_k, u^k, u_n^k)$ be defined from the decomposition of $u$ (see (4.15)). By our hypothesis we have $|\omega_0 - \pi| \le 2c$, $|\omega_1 - \pi| \le 2c$, and $|\omega_2| \le 2c$: we suppose that $c$ is sufficiently small so that $|\omega_1 - \pi|, |\omega_2 - \pi| \le 0.1\pi$, and $\omega_3 \le \frac{1}{2}\pi$.

As previously, let $g$ be the function constructed from Lemma 44. Denote $T > 0$ the smallest value such that $g = 0$ on $[T, +\infty[$. Let

$$ f_n^k(t) = \begin{cases} g(t) & \text{if } \mu_{n,\omega_k} \ge 5 \text{ i.e. } k \ge 2 \text{ or } n \ge 2, \\ 1 & \text{if } |\mu_{n,\omega_k} - 4| < 1 \text{ i.e. } k \in \{0,1\} \text{ and } n = 1. \end{cases} $$

The last case $\mu_{n,\omega_k} \le 3$ never happens in this context. We define the competitor

$$ \tilde{U}(t, \theta) = \sum_{k \ge 1} \sum_{n \ge 1} f_n^k(t) u_n^k b_{n, \omega_k} (\theta - \theta_k). $$

We remind that from the estimates (4.7), we have exactly

$$ \tilde{U}(t, \theta) = u_1^0 b_{1, \omega_0} (\theta - \theta_0) + u_1^1 b_{1, \omega_1} (\theta - \theta_1) $$

for all $t \ge T$. Denote $\tilde{U}(t, \theta)$ the competitor built in Lemma 42 from the initial data

$$ \tilde{U}(T, \cdot + \theta_0) = u^{\text{double}} := u_1^0 b_{1, \omega_0} + u_1^1 b_{1, \omega_1} (\cdot + \theta_0 - \theta_1), $$

so that

$$ \mathcal{G}(\tilde{U}, 0) \le \mathcal{G}(u^{\text{double}}, 0) - \epsilon (\mathcal{G}(u^{\text{double}}, 0) - \pi)_+ . $$

Let then

$$ U(t, \theta) = \begin{cases} \tilde{U}(t, \theta) & \text{if } t < T, \\ \tilde{U}(t - T, \theta + \theta_0) & \text{if } t \ge T. \end{cases} $$

Following the computations from the previous lemma (denoting $\eta > 0$ the constant from Lemma 44, $\epsilon > 0$ the constant from Lemma 42), we get

$$ \begin{aligned} \mathcal{G}(U, 0) &= \mathcal{G}(\tilde{U}, 0) + (\mathcal{G}(U, 0) - \mathcal{G}(\tilde{U}, 0)) \\ &\le \mathcal{G}(u^{\text{double}}, 0) + (1 - \eta) \sum_{k=0}^{1} \sum_{n>1} Q_{\mu_{n,\omega_k}-4}(1) |u_n^k|^2 \\ &\quad + \sum_{k \ge 2} \left( \frac{1 - e^{-2T}}{2} \omega_k + (1 - \eta) \sum_{n \ge 1} Q_{\mu_{n,\omega_k}-4}(1) |u_n^k|^2 \right) \\ &\quad + e^{-2T} (\mathcal{G}(\tilde{U}, 0) - \mathcal{G}(u^{\text{double}}, 0)) \quad \text{by Lemmas 43, 44.} \end{aligned} $$
Then, by lemma 41, we have

$$
\begin{aligned}
\mathcal{G}(U, 0) \le \mathcal{G}(u^{\text{double}}, 0) - e^{-2T} \epsilon \left( \mathcal{G}(u^{\text{double}}, 0) - \pi \right)_+ & \\
& + (1-\eta) \sum_{k=0}^{1} \sum_{n>1} Q_{\mu_{n,\omega_k}-4}(1) |u_n^k|^2 \\
& + \sum_{k \ge 2} \left( \frac{1-e^{-2T}}{2} \omega_k + (1-\eta) \sum_{n \ge 1} Q_{\mu_{n,\omega_k}-4}(1) |u_n^k|^2 \right).
\end{aligned}
$$

Denote $\bar{\epsilon} = \min(e^{-2T}\epsilon, e^{-2T}, \frac{\eta}{2})$, and identify

$$
\mathcal{G}(u, 0) = \mathcal{G}(u^{\text{double}}, 0) + \sum_{k=0}^{1} \sum_{n>1} Q_{\mu_{n,\omega_k}-4}(1) |u_n^k|^2 + \sum_{k \ge 2} \left( \frac{1}{2}\omega_k + \sum_{n \ge 1} Q_{\mu_{n,\omega_k}-4}(1) |u_n^k|^2 \right).
$$

Then

$$
\begin{aligned}
\mathcal{G}(U, 0) &\le \mathcal{G}(u, 0) - \bar{\epsilon} (\mathcal{G}(u, 0) - \pi)_+ - \frac{\eta}{2} \sum_{(n,k):n>1 \text{ or } k \ge 2} Q_{\mu_{n,\omega_k}-4}(1) |u_n^k|^2 \\
&\le \mathcal{G}(u, 0) - \bar{\epsilon} (\mathcal{G}(u, 0) - \pi)_+ - \frac{\eta}{20} \sum_{(n,k):n>1 \text{ or } k \ge 2} \mu_{n,\omega_k} |u_n^k|^2.
\end{aligned}
$$

This ends the proof with $a = \frac{\eta}{20}$, since

$$
\begin{aligned}
\sum_{(n,k):n>1 \text{ or } k \ge 2} \mu_{n,\omega_k} |u_n^k|^2 &= \sum_{k=0}^{1} \sum_{n \ge 2} \mu_{n,\omega_k} |u_n^k|^2 + \sum_{k \ge 2} \sum_{n \ge 1} \mu_{n,\omega_k} |u_n^k|^2 \\
&= \|\partial_{\theta,\theta} (u^0 - u_1^0 b_{1,\omega_0})\|^2 + \|\partial_{\theta,\theta} (u^1 - u_1^1 b_{1,\omega_1})\|^2 + \sum_{k \ge 2} \|\partial_{\theta,\theta} u^k\|^2.
\end{aligned}
$$

<figure><img src="image_16.png" /></figure>

4.4 General case $v \neq 0$

In this subsection, we give the proof of Lemma 38, by handling general boundary data $(u, v)$
where $v$ is not necessarily zero. We prove it by reducing the analysis to the case where $v = 0$.
We first prove a purely technical lemma used below in the proof of Lemma 49.

**Lemma 48.** For any $\omega \in [0.5\pi, 2\pi]$, there exists $\chi_\omega \in C^\infty(\mathbb{R}_+, \mathbb{R})$ such that $\chi_\omega(t) = 1$ in a neighbourhood of 0, $\chi_\omega(t) = 0$ in $[1, +\infty[$, $\sup_{\omega \in [0.5\pi, 2\pi]} \|\chi_\omega\|_{C^2} \lesssim 1$ and

$$
\forall \omega \in [0.5\pi, 2\pi], \quad \forall n \in \{1, 2, 3\}, \quad \int_0^{+\infty} t e^{-(2+\mu_{n,\omega}^{1/3})t} \chi_\omega(t) dt = 0.
$$

Proof. Let $g_{n,\omega}(t) = te^{-(2+\mu_{n,\omega}^{1/3})t}$. Let $t_i = \frac{i}{4}$ for $i=1,2,3$. For any $\omega \in [0.5\pi, 2\pi]$, we have

$$
\det((g_{n,\omega}(t_m))_{1\le n,m\le 3}) \ne 0,
$$

since this is (up to a nonzero scalar factor) the determinant of the non-trivial Vandermonde
matrix

$$
\left( e^{-\frac{2+\mu_{n,\omega}^{1/3}}{4}k} \right)_{1\le k,n\le 3} .
$$
Functions $g_{n,\omega}$ depend continuously on $\omega$, so we may find some approximation of unity $\sigma_1, \sigma_2, \sigma_3 \in C_c^\infty((0,1))$ around $t_1, t_2, t_3$ respectively such that

$$ \inf_{\omega \in [0.5\pi, 2\pi]} \left| \det \left( \left( \int_0^1 g_{n,\omega}(t) \sigma_m(t) dt \right)_{1 \le n, m \le 3} \right) \right| > 0 $$

Let now $h \in C^\infty(\mathbb{R}_+, \mathbb{R})$ that is equal to 1 in a neighbourhood of 0, and to 0 on $[1, +\infty[$, then there exists continuously defined $a_1(\omega), a_2(\omega), a_3(\omega) \in \mathbb{R}$ such that for any $\omega \in [0.5\pi, 2\pi]$, $n \in \{1, 2, 3\}$, we have

$$ \int_0^1 (a_1(\omega)\sigma_1 + a_2(\omega)\sigma_2 + a_3(\omega)\sigma_3) g_{n,\omega} = \int_0^1 hg_{n,\omega}. $$

The function $\chi_\omega(t) = h(t) - a_1(\omega)\sigma_1(t) - a_2(\omega)\sigma_2(t) - a_3(\omega)\sigma_3(t)$ has all the required properties. $\square$

**Lemma 49.** Let $\omega \in (0, 2\pi]$, $u \in H^2(\mathbb{S}^1)$, $v \in H^1(\mathbb{S}^1)$, with

$$ \{v \neq 0\} \subset \text{Spt}(u) = (0, \omega), $$

then there exists $U \in H_{\text{lin}}^2(C_0)$ such that $U(0, \cdot) = u, \partial_t U(0, \cdot) = v, U(t, \cdot) = u$ for all $t \ge 1$ and

$$ \mathcal{G}(U, 0) \le \mathcal{G}(u, 0) + C (\|\partial_{\theta,\theta} u\| \|\partial_{\theta} v\| + \|\partial_{\theta} v\|^2). $$

Moreover, if $|\omega - \Theta| \le 0.1\Theta$ for some $\Theta \in \{\pi, t_1, 2\pi\}$, then we have the improved bound

$$ \mathcal{G}(U, 0) \le \mathcal{G}(u, 0) + C \left( \left\| \partial_{\theta,\theta} \left( u - \sum_{n=1}^{i(\Theta)} u_n b_{n,\omega} \right) \right\| \| \partial_{\theta} v \| + \| \partial_{\theta} v \|^2 \right) $$

where $(u_n)_{n \ge 1}$ are the coefficients in the decomposition $u = \sum_{n \ge 1} u_n b_{n,\omega}$.

**Proof.** For any $n \in \mathbb{N}^*$, we define

$$ f_{n,\omega}(t) = t e^{-\mu_{n,\omega}^{1/3} t}. $$

Note that $f_{n,\omega}(0) = 0$ and $f'_{n,\omega}(0) = 1$. Let now $\chi_\omega$ be defined from Lemma 48, such that

$$ \int_0^{+\infty} e^{-2t} f_{n,\omega}(t) \chi_\omega(t) dt = 0 $$

for $n = 1, 2, 3$ when $\omega \ge 0.5\pi$. When $\omega < 0.5\pi$ we just set $\chi_\omega := \chi_{0.5\pi}$. We define

$$ U(t, \theta) = \sum_{n \ge 1} (u_n + \chi_\omega(t) f_{n,\omega}(t) v_n) b_{n,\omega}(\theta). $$

Then $U$ verifies $U \in H_{\text{lin}}^2(C_0)$, $U(0, \cdot) = u, \partial_t U(0, \cdot) = v, U(t, \cdot) = u$ for $t \ge 1$, and

$$ \begin{aligned} \mathcal{G}(U, 0) &\le \frac{\omega}{2} + \int_0^{+\infty} e^{-2t} \left\{ \left\| \sum_{n \ge 1} (\chi_\omega f_{n,\omega})''(t) v_n b_{n,\omega} \right\|^2 + 2 \left\| \sum_{n \ge 1} (\chi_\omega f_{n,\omega})'(t) v_n \partial_{\theta} b_{n,\omega} \right\|^2 \right\} dt \\ &\quad + \int_0^{+\infty} e^{-2t} \left\{ \left\| \sum_{n \ge 1} (u_n + (\chi_\omega f_{n,\omega})(t) v_n) \partial_{\theta,\theta} b_{n,\omega} \right\|^2 - 4 \left\| \sum_{n \ge 1} (u_n + (\chi_\omega f_{n,\omega})(t) v_n) \partial_{\theta} b_{n,\omega} \right\|^2 \right\} dt \\ &= \frac{1}{2} (\omega + \|\partial_{\theta,\theta} u\|^2 - 4 \|\partial_{\theta} u\|^2) \\ &\quad + \sum_{n \ge 1} (\mu_{n,\omega} - 4) \left( \int_0^{+\infty} 2e^{-2t} \chi_\omega(t) f_{n,\omega}(t) dt \right) u_n v_n \\ &\quad + \sum_{n \ge 1} \left( \int_0^{+\infty} e^{-2t} \left\{ 2|(\chi_\omega f_{n,\omega})'(t)|^2 + (\mu_{n,\omega} - 4)|\chi_\omega f_{n,\omega}(t)|^2 \right\} dt \right) v_n^2 \\ &\quad + \sum_{n,m \ge 1} \left( \int_0^{+\infty} e^{-2t} (\chi_\omega f_{n,m,\omega})''(t) (\chi_\omega f_{m,\omega})''(t) dt \right) \langle b_{n,\omega}, b_{m,\omega} \rangle v_n v_m. \end{aligned} $$

We handle each of these four lines separately.
• The first line is exactly $\mathcal{G}(u, 0)$.

• When $\omega \ge 0.5\pi$ and $n = 1, 2, 3$, this term vanishes by the orthogonality condition we imposed on $\chi_\omega$ in Lemma 48. In all other cases we have

$$ \left| (\mu_{n,\omega} - 4) \int_{0}^{+\infty} 2e^{-2t} \chi_{\omega}(t) f_{n,\omega}(t) dt \right| \lesssim \mu_{n,\omega} \int_{0}^{+\infty} e^{-(2+\mu_{n,\omega}^{1/3})t} dt \lesssim \mu_{n,\omega}^{1/3}. $$

• Using the uniform bound on $\|\chi_\omega\|_{C^2}$, we have

$$ \int_{0}^{+\infty} e^{-2t} \left\{ 2\left| (\chi_{\omega} f_{n,\omega})'(t) \right|^2 + (\mu_{n,\omega} - 4) \left| \chi_{\omega} f_{n,\omega}(t) \right|^2 \right\} dt \lesssim \int_{0}^{+\infty} e^{-2t} \left\{ f'_{n,\omega}(t)^2 + \mu_{n,\omega} f_{n,\omega}(t)^2 \right\} dt \\ \lesssim \frac{1}{\mu_{n,\omega}^{1/3}} + 1 \lesssim 1 $$

by the definition of $f_{n,\omega}$.

• Similarly, we have

$$ \int_{0}^{+\infty} e^{-2t} |(\chi_{\omega} f_{n,\omega})''(t)|^2 dt \lesssim \int_{0}^{+\infty} e^{-2t} \left\{ f''_{n,\omega}(t)^2 + f'_{n,\omega}(t)^2 + f_{n,\omega}(t)^2 \right\} dt \\ \lesssim \mu_{n,\omega}^{1/3} + \mu_{n,\omega}^{-1/3} + \mu_{n,\omega}^{-1} \lesssim \mu_{n,\omega}^{1/3} $$

so by Cauchy-Schwarz inequality and using the bound $\|b_{n,\omega}\| \lesssim \mu_{n,\omega}^{-1/2}$ from Lemma 40, we have

$$ \left( \int_{0}^{+\infty} e^{-2t} (\chi_{\omega} f_{n,\omega})''(t) (\chi_{\omega} f_{m,\omega})''(t) dt \right) \langle b_{n,\omega}, b_{m,\omega} \rangle \lesssim \frac{1}{\mu_{n,\omega}^{1/3} \mu_{m,\omega}^{1/3}}. $$

We gather the three estimates. First for a general $\omega \in (0, 2\pi]$:

$$ \begin{aligned} \mathcal{G}(U, 0) - \mathcal{G}(u, 0) & \lesssim \sum_{n \ge 1} \mu_{n,\omega}^{1/3} |u_n v_n| + \sum_{n \ge 1} v_n^2 + \sum_{n,m \ge 1} \frac{|v_n v_m|}{\mu_{n,\omega}^{1/3} \mu_{m,\omega}^{1/3}} \\ & \lesssim \sum_{n \ge 1} \mu_{n,\omega}^{1/2} |u_n v_n| + \sum_{n \ge 1} v_n^2 + \left( \sum_{n \ge 1} \frac{|v_n|}{\mu_{n,\omega}^{1/3}} \right)^2 \\ & \lesssim \|\partial_{\theta,\theta} u\| \|\partial_{\theta} v\| + \left( 1 + \sum_{n \ge 1} \frac{1}{\mu_{n,\omega}^{2/3}} \right) \sum_{n \ge 1} |v_n|^2 \\ & \lesssim \|\partial_{\theta,\theta} u\| \|\partial_{\theta} v\| + \|\partial_{\theta} v\|^2. \end{aligned} $$

We used the fact that $\mu_{n,\omega} \gtrsim \frac{n^2}{\omega^2}$, so $\mu_{n,\omega}^{-2/3}$ is summable with uniformly bounded sum in $\omega$.

Then, when $|\omega - \Theta| \le 0.1\Theta$ for some $\Theta \in \{\pi, t_1, 2\pi\}$, in particular $\omega \ge 0.5\pi$ so using Lemma 48 we have the stronger estimate:

$$ \begin{aligned} \mathcal{G}(U, 0) - \mathcal{G}(u, 0) & \lesssim \sum_{n>i(\Theta)} \mu_{n,\omega}^{1/3} |u_n v_n| + \sum_{n \ge 1} v_n^2 + \sum_{n,m \ge 1} \frac{|v_n v_m|}{\mu_{n,\omega}^{1/3} \mu_{m,\omega}^{1/3}} \\ & \lesssim \sum_{n>i(\Theta)} \mu_{n,\omega}^{1/2} |u_n v_n| + \sum_{n \ge 1} v_n^2 + \left( \sum_{n \ge 1} \frac{|v_n|}{\mu_{n,\omega}^{1/3}} \right)^2 \\ & \lesssim \left\| \partial_{\theta,\theta} \left( u - \sum_{n \le i(\Theta)} u_n b_{n,\omega} \right) \right\| \|\partial_{\theta} v\| + \left( 1 + \sum_{n \ge 1} \frac{1}{\mu_{n,\omega}^{2/3}} \right) \sum_{n \ge 1} |v_n|^2 \\ & \lesssim \left\| \partial_{\theta,\theta} \left( u - \sum_{n \le i(\Theta)} u_n b_{n,\omega} \right) \right\| \|\partial_{\theta} v\| + \|\partial_{\theta} v\|^2. \end{aligned} $$

<figure><img src="image_17.png" /></figure>
We can now prove Lemma 38 as a combination of Lemma 49 on each interval of the support of $u$, and Lemma 46.

*Proof of Lemma 38.* As in the previous section, we first introduce the decomposition of $u$ on the connected component of its support: we write Spt($u$) as a countable union of disjoint intervals $[\theta_k, \theta_k + \omega_k]$ of lengths $\omega_0 \ge \omega_1 \ge \omega_2 \ge \dots \to 0$. We remind that either

$$|\omega_0 - \Theta| \le 2c, \omega_1 \le 2c,$$

or

$$\Theta = 2\pi, |\omega_0 - \pi| \le 2c, |\omega_1 - \pi| \le 2c, \omega_2 \le 2c.$$

We call (a) the first case, (b) the second case. We let $u^k(\theta) = u(\theta_k + \theta)\chi_{0<\theta<\omega_k}$ such that $u^k \in H_0^2([0, \omega_k])$. Likewise, we let $v^k(\theta) = v(\theta_k + \theta)\chi_{0<\theta<\omega_k} \in H_0^1([0, \omega_k])$.

We now construct the competitor $U$ as follows: we let $\bar{U}^k$ be the competitor defined in Lemma 49 from the initial condition $u^k, v^k, \bar{U}(t, \theta) = \sum_{k \ge 0} \bar{U}^k(t, \theta - \theta_k)$, and $\hat{U}$ the competitor defined in Lemma 46 (in the case (a)) or Lemma 47 (in the case (b)) from the initial condition $u$. Then we let

$$U(t, \theta) = \begin{cases} \bar{U}(t, \theta) & \text{if } t < 1, \\ \hat{U}(t-1, \theta) & \text{if } t \ge 1. \end{cases}$$

which verifies $U(0, \theta) = u(\theta)$, $\partial_t U(t, \theta) = v(\theta)$. We denote $c, \epsilon, a > 0$ the constants from Lemma 46, 47 (note that we may always take the minimum between each) and $C$ the constant from Lemma 49.

(a) We first suppose we are in the case (a) i.e. $|\omega_0 - \Theta| \le 2c, \omega_1 \le 2c$. Then by Lemma 46 we have

$$\begin{aligned} \mathcal{G}(U, 0) &= \mathcal{G}(\bar{U}, 0) + (\mathcal{G}(U, 0) - \mathcal{G}(\bar{U}, 0)) = \mathcal{G}(\bar{U}, 0) + e^{-2} (\mathcal{G}(\hat{U}, 0) - \mathcal{G}(u, 0)) \\ &\le \mathcal{G}(u, 0) + C \left( \|\partial_{\theta} v\|^2 + \|\partial_{\theta} v^0\| \left\| \partial_{\theta, \theta} \left( u^0 - \sum_{n \le i(\Theta)} u_n^0 b_{n, \omega_0} \right) \right\| + \sum_{k \ge 1} \|\partial_{\theta} v^k\| \|\partial_{\theta, \theta} u^k\| \right) \\ &\quad - e^{-2}\epsilon \left( \mathcal{G}(u, 0) - \frac{\Theta}{2} \right)_+ - e^{-2}a \left( \left\| \partial_{\theta, \theta} \left( u^0 - \sum_{n \le i(\Theta)} u_n^0 b_{n, \omega_0} \right) \right\|^2 + \sum_{k \ge 1} \|\partial_{\theta, \theta} u^k\|^2 \right). \end{aligned}$$

Now, we bound

$$C \|\partial_{\theta} v^0\| \left\| \partial_{\theta, \theta} \left( u^0 - \sum_{n \le i(\Theta)} u_n^0 b_{n, \omega_0} \right) \right\| \le \frac{C^2 e}{4a} \|\partial_{\theta} v^0\|^2 + e^{-2a} \left\| \partial_{\theta, \theta} \left( u^0 - \sum_{n \le i(\Theta)} u_n^0 b_{n, \omega_0} \right) \right\|^2, \\ \text{and} \\ C \|\partial_{\theta} v^k\| \|\partial_{\theta, \theta} u^k\| \le \frac{C^2 e}{4a} \|\partial_{\theta} v^k\|^2 + e^{-2a} \left\| \partial_{\theta, \theta} u^k \right\|^2 \text{ for } k \ge 1,$$

so

$$\mathcal{G}(U, 0) \le \mathcal{G}(u, 0) - e^{-2\epsilon} \left( \mathcal{G}(u, 0) - \frac{\Theta}{2} \right)_+ + \left( C + \frac{C^2 e^2}{4a} \right) \|\partial_{\theta} v\|^2,$$

which prove the result in case (a).
(b) This time we have

$$
\begin{align*}
\mathcal{G}(U, 0) &= \mathcal{G}(\bar{U}, 0) + (\mathcal{G}(U, 0) - \mathcal{G}(\bar{U}, 0)) = \mathcal{G}(\bar{U}, 0) + e^{-2} (\mathcal{G}(\hat{U}, 0) - \mathcal{G}(u, 0)) \\
&\le \mathcal{G}(u, 0) + C \left( \|\partial_{\theta} v\|^2 + \sum_{k=0}^{1} \|\partial_{\theta} v^k\| \|\partial_{\theta,\theta} (u^k - u_1^k b_{1,\omega_k})\| + \sum_{k \ge 2} \|\partial_{\theta} v^k\| \|\partial_{\theta,\theta} u^k\| \right) \\
&\quad - e^{-2}\epsilon \left( \mathcal{G}(u, 0) - \frac{\Theta}{2} \right)_+ - e^{-2}a \left( \sum_{k=0}^{1} \|\partial_{\theta,\theta} (u^k - u_1^k b_{1,\omega_k})\|^2 + \sum_{k \ge 2} \|\partial_{\theta,\theta} u^k\|^2 \right).
\end{align*}
$$

Now, we bound

$$
C \|\partial_{\theta} v^k\| \left\| \partial_{\theta,\theta} \left( u^k - \sum_{n \le i(\Theta)} u_n^k b_{n,\omega_0} \right) \right\| \le \frac{C^2 e}{4a} \|\partial_{\theta} v^k\|^2 + e^{-2a} \left\| \partial_{\theta,\theta} \left( u^k - \sum_{n \le i(\Theta)} u_n^k b_{n,\omega_0} \right) \right\|^2 \\
\text{for } k=0,1
$$

$$
\text{and} \quad C \|\partial_{\theta} v^k\| \|\partial_{\theta,\theta} u^k\| \le \frac{C^2 e}{4a} \|\partial_{\theta} v^k\|^2 + e^{-2a} \|\partial_{\theta,\theta} u^k\|^2 \quad \text{for } k \ge 2,
$$

so

$$
\mathcal{G}(U, 0) \le e^{-2\epsilon\pi} + (1 - e^{-2\epsilon})\mathcal{G}(\bar{U}, 0) + \left(C + \frac{C^2 e^2}{4a}\right) \|\partial_{\theta} v\|^2,
$$

which prove the result in case (b).

<figure><img src="image_18.png" /></figure>

4.5 Estimates on biharmonic extensions

We now prove Lemma 39, which is an analog of Lemma 38 in the case Spt(u) = S¹. Our method in this case is a more elementary construction of competitors: we consider the competitor corresponding to the biharmonic extension in the disk coordinates. We follow closely the energy difference between a biharmonic function in D₁ and its 2-homogeneous extension.

In what follows, for any $f \in L^1(\mathbb{S}^1)$, we denote

$$
c_n[f] = \frac{1}{2\pi} \int_{\mathbb{S}^1} e^{-in\theta} f(e^{i\theta}) d\theta,
$$

its Fourier coefficients: by the convention we chose we have

$$
\|f\|_{L^2(\partial \mathbb{D}_1)}^2 = 2\pi \sum_{n \in \mathbb{Z}} |c_n[f]|^2.
$$

*Proof of Lemma 39.* We remind that $v$ being a minimizer of $\mathcal{G}(\cdot, 0)$ means that

$$
u(re^{i\theta}) := r^2 v(-\log(r), \theta)
$$

belongs to $\mathcal{M}(\mathbb{D}_1)$. Consider $\tilde{u}$ the biharmonic extension of the boundary data $(u|_{\partial\mathbb{D}_1}, \partial_r u|_{\partial\mathbb{D}_1})$ in $\mathbb{D}_1$, which can be decomposed as

$$
\tilde{u} (re^{i\theta}) = \sum_{n \in \mathbb{Z}} (a_n r^{|n|+2} + b_n r^{|n|}) e^{in\theta},
$$
for some coefficients $a_n, b_n$ (we recall this is called Goursat's decomposition, also discussed at (3.13)). As we already noticed in the proof of Lemma 21, the energy $\int_{\mathbb{D}_1} |\Delta \tilde{u}|^2$ only depends on the $(a_n)_n$ coefficients, and more precisely $\Delta \tilde{u} = \sum_{n \in \mathbb{Z}} 4(|n| + 1)a_n r^{|n|} e^{in\theta}$, so

$$ \int_{\mathbb{D}_1} |\Delta \tilde{u}|^2 = \int_0^1 2\pi r \left| \sum_{n \in \mathbb{Z}} 4(|n| + 1) r^{|n|} e^{in\theta} \right|^2 dr = 16\pi \sum_{n \in \mathbb{Z}} (|n| + 1) |a_n|^2. $$

We may identify

$$ \begin{aligned} c_n[u] &= c_n[\tilde{u}] = a_n + b_n, \\ c_n[\partial_r u] &= c_n[\partial_r \tilde{u}] = (|n| + 2)a_n + |n|b_n, \end{aligned} $$

so $a_n = \frac{c_n[\partial_r u] - |n|c_n[u]}{2}$ and

$$ E(\tilde{u}; \mathbb{D}_1) = \int_{\mathbb{D}_1} (|\Delta \tilde{u}|^2 + \chi_{\tilde{u} \neq 0}) = \pi + 2\pi \sum_{n \in \mathbb{Z}} 2(|n| + 1) |c_n[\partial_r u] - |n|c_n[u]|^2. $$

We identify $u(e^{i\theta}) = v(0, \theta)$, $\partial_r u(e^{i\theta}) = 2v(0, \theta) - \partial_t v(0, \theta)$, so $c_n[u] = c_n[v(0, \cdot)]$, $c_n[\partial_r u] = 2c_n[v] - c_n[\partial_t v(0, \cdot)]$, meaning

$$ E(\tilde{u}; \mathbb{D}_1) = \pi + \pi \sum_{n \in \mathbb{Z}} 4(|n| + 1) |c_n[\partial_t v(0, \cdot)] + (|n| - 2)c_n[v(0, \cdot)]|^2. $$

Since $u \in \mathcal{M}(\mathbb{D}_1)$, we have $E(u; \mathbb{D}_1) \le E(\tilde{u}; \mathbb{D}_1)$, so by Lemma 16, we get

$$ \begin{aligned} \mathcal{G}(v, 0) \le \pi + \pi \sum_{n \in \mathbb{Z}} 4(|n| + 1) |c_n[\partial_t v(0, \cdot)] + (|n| - 2)c_n[v(0, \cdot)]|^2 \\ - 2 \int_{\partial \mathcal{C}_\tau} (2(\partial_t v)^2 - 4v\partial_t v + 4v^2 - (\partial_\theta v)^2 + \partial_{t,\theta} v \partial_\theta v) (0, \cdot) d\theta. \end{aligned} $$

We express everything in terms of Fourier coefficients of $v(0, \cdot)$, $\partial_t v(0, \cdot)$. To simplify notations we write $x_n = c_n[v(0, \cdot)]$, $y_n = c_n[\partial_t v(0, \cdot)]$:

$$ \begin{aligned} \mathcal{G}(v, 0) - \pi \le \pi \sum_{n \in \mathbb{Z}} 4(|n| + 1) |y_n + (|n| - 2)x_n|^2 \\ + \pi \sum_{n \in \mathbb{Z}} -8|y_n|^2 - 4(n^2 - 4)\Re(\overline{x_n}y_n) + 4(n^2 - 4)|x_n|^2, \end{aligned} $$

so

$$ \mathcal{G}(v, 0) - \pi \le \pi \sum_{n \in \mathbb{Z}} 4n^2(|n| - 2)|x_n|^2 + 4|n|(|n| - 2)\Re(\overline{x_n}y_n) + 4(|n| - 1)|y_n|^2. \quad (4.18) $$

We can also express in terms of Fourier coefficients:

$$ \mathcal{G}(v(0, \cdot), 0) - \pi = \frac{1}{2} \|\partial_{\theta,\theta} v(0, \cdot)\|^2 - 2 \|\partial_{\theta} v(0, \cdot)\|^2 = \pi \sum_{n \in \mathbb{Z}} n^2(n^2 - 4) |x_n|^2. \quad (4.19) $$

Here we used the fact that the support of $v(0, \cdot)$ has measure $2\pi$. Now we compare the $n$-th term of (4.18) and (4.19):

* if $|n| \ge 2$, then we bound $4|n|(|n| - 2)\Re(\overline{x_n}y_n) \le \frac{1}{2}n^2(|n| - 2)|x_n|^2 + 8(|n| - 2)|y_n|^2$, so
$$ \begin{aligned} 4n^2(|n| - 2)|x_n|^2 + 4|n|(|n| - 2)\Re(\overline{x_n}y_n) + 4(|n| - 1)|y_n|^2 \\ \le \frac{9}{2}n^2(|n| - 2)|x_n|^2 + (12|n| - 20)|y_n|^2 \\ \le 0.9n^2(n^2 - 4)|x_n|^2 + (12|n| - 4)|y_n|^2. \end{aligned} $$

Here the factor “0.9” is sharp for $n = 3$ in the last inequality.
• For $n = 0$, then the 0-th terms of (4.18) and (4.19) are respectively $-4|y_0|^2$, 0.

• For $n = \pm 1$, we write

$$-4|x_{\pm 1}|^2 - 4\Re(\overline{x_{\pm 1}}y_{\pm 1}) \leq -3|x_{\pm 1}|^2 + 4|y_{\pm 1}|^2 \leq (1^2(1^2 - 4))|x_{\pm 1}|^2 + (12 - 4)|y_{\pm 1}|^2.$$

Summing every contributions, we obtain

$$\begin{aligned} \mathcal{G}(v, 0) - \mathcal{G}(v(0, \cdot), 0) &\leq -0.1\pi \sum_{n \in \mathbb{Z} \setminus \{-1, 1\}} n^2(n^2 - 4)|x_n|^2 + \pi \sum_{n \in \mathbb{Z}} (12|n| - 4)|y_n|^2 \\ &\leq -0.1 \left( \pi \sum_{n \in \mathbb{Z}} n^2(n^2 - 4)|x_n|^2 \right)_+ + 2\pi \sum_{n \in \mathbb{Z}} (6|n| - 2)|y_n|^2 \\ &\leq -0.1 (\mathcal{G}(v(0, \cdot), 0) - \pi)_+ + 6\|\partial_{t,\theta}v(0, \cdot)\|^2 - 2\|\partial_t v(0, \cdot)\|^2, \end{aligned}$$

which implies the result.

<figure><img src="image_19.png" /></figure>

## 4.6 Proof of the epiperimetric inequality

As a consequence of Lemma 38 and 39, we obtain the following corollary.

**Corollary 50.** Let $\Theta \in \{\pi, t_1, 2\pi\}$. There exist constants $c, \epsilon, C > 0$ such that, if $u \in H_{\text{lin}}^2(C_0)$ is a minimizer of $\mathcal{G}(\cdot, 0)$ that verifies the support condition

$$\begin{cases} (c, \Theta - c) \subset \text{Spt}(u(0, \cdot)) \subset (-c, \Theta + c) & \text{when } \Theta \in \{\pi, t_1\} \\ (c, \pi - c) \cup (\pi + c, 2\pi - c) \subset \text{Spt}(u(0, \cdot)) & \text{when } \Theta = 2\pi, \end{cases}$$

then

$$\mathcal{G}(u, 0) \leq \mathcal{G}(u(0, \cdot), 0) - \epsilon \left( \mathcal{G}(u(0, \cdot), 0) - \frac{\Theta}{2} \right)_+ + C \|\partial_{t,\theta} u(0, \cdot)\|_{L^2(\mathbb{S}^1)}^2 - 2 \|\partial_t u(0, \cdot)\|_{L^2(\mathbb{S}^1)}^2. \quad (4.20)$$

*Proof.* Denote $c, \epsilon, C$ the constants from Lemma 38. If $u$ verifies the support condition and $\text{Spt}(u) \neq \mathbb{S}^1$, then by Lemma 38 applied to $u$ we get

$$\begin{aligned} \mathcal{G}(u, 0) &\leq \mathcal{G}(u(0, \cdot), 0) - \epsilon \left( \mathcal{G}(u(0, \cdot), 0) - \frac{\Theta}{2} \right)_+ + C \|\partial_{t,\theta} u(0, \cdot)\|_{L^2(\mathbb{S}^1)}^2 \\ &\leq \mathcal{G}(u(0, \cdot), 0) - \epsilon \left( \mathcal{G}(u(0, \cdot), 0) - \frac{\Theta}{2} \right)_+ + (C + 8) \|\partial_{t,\theta} u(0, \cdot)\|_{L^2(\mathbb{S}^1)}^2 - 2 \|\partial_t u(0, \cdot)\|_{L^2(\mathbb{S}^1)}^2 \end{aligned}$$

where we used the Poincaré inequality $\|f\|_{L^2(\mathbb{S}^1)}^2 \leq 4\|\partial_\theta f\|_{L^2(\mathbb{S}^1)}^2$, which is valid since $\text{Spt}(u(0, \cdot)) \neq \mathbb{S}^1$.

Suppose now that $\text{Spt}(u) = \mathbb{S}^1$ (so in particular we are in the case $\Theta = 2\pi$), then the result is directly implied by Lemma 39 (for any $\epsilon \leq 0.1$, $C \geq 6$). $\square$

We now have all the tools to prove the epiperimetric inequality. We will make use of the following interpolation inequality: for any smooth bounded set $D \subset \mathbb{R}^2$, there exists $c(D) > 0$ such that for any $u \in W^{2,4}(D)(\subset C^1(D))$:

$$\|u\|_{C^1(D)} \leq c(D) \|u\|_{H^1(D)}^{\frac{1}{2}} \|u\|_{W^{2,4}(D)}^{\frac{1}{2}}. \quad (4.21)$$

This is a consequence of the following classical Gagliardo-Nirenberg inequality applied to $f = \nabla u$:

$$\|f\|_{L^\infty(D)} \leq C(D) \|f\|_{L^2(D)}^{\frac{1}{2}} \|f\|_{W^{1,4}(D)}^{\frac{1}{2}}.$$
Proof of Theorem 37. Let $c_0 > 0$ be some quantity that we fix (arbitrarily small) later such that

$$ \|u - b\|_{H^1(c_0 \setminus c_1)} \le c_0 \|b\|_{H^1(\mathbb{S}^1)}. $$

By the Cacciopoli inequality of Lemma 10 and the BMO estimate of Lemma 11, we have

$$ \|u\|_{W^{2,4}(c_1 \setminus c_3)} \lesssim \|u\|_{H^1(c_0 \setminus c_1)} \lesssim \|b\|_{H^1(\mathbb{S}^1)}. $$

By the interpolation inequality (4.21) we then have

$$ \|u - b\|_{C^1(c_1 \setminus c_3)} \lesssim \|u - b\|_{H^1(c_1 \setminus c_3)}^{\frac{1}{2}} \|u - b\|_{W^{2,4}(c_1 \setminus c_3)}^{\frac{1}{2}} \lesssim c_0^{\frac{1}{2}} \|b\|_{H^1(\mathbb{S}^1)}. \quad (4.22) $$

We claim that for a sufficiently small $c_0$, $u(t, \cdot)$ verifies the support hypothesis of Corollary 50 for almost every $t$, up to a rotation in $\theta$. We denote by $c$ the constant from Corollary 50.

First note that $\{\partial_t u \neq 0\} \subset \text{Spt}(u)$, so $\{\partial_t u(t, \cdot) \neq 0\} \subset \text{Spt}(u(t, \cdot))$ for almost every $t$. Then we differentiate whether $\Theta \in \{\pi, t_1\}$ or $\Theta = 2\pi$.

• $\Theta \in \{\pi, t_1\}$. Suppose without loss of generality that $b = b^1$ or $b = b^{II}$. Then

$$ \inf_{\theta \in [c, \Theta-c]} (|b(\theta)| + |\partial_{\theta} b(\theta)|) \gtrsim 1. $$

So by the $C^1$ estimate of $u - b$ in (4.22), we have for a small enough $c_0$:

$$ \forall t \in \left[\frac{1}{4}, \frac{3}{4}\right], \quad \inf_{\theta \in [c, \Theta-c]} (|u(t, \theta)| + |\partial_{\theta} u(t, \theta)|) \gtrsim 1. $$

As a consequence, $(c, \Theta - c) \subset \text{Spt}(u(t, \cdot))$ for any $t \in [\frac{1}{4}, \frac{3}{4}]$.

On the other hand, for any disk $D = \mathbb{D}_{p,c}$ with $p \in [\frac{1}{4}, \frac{3}{4}] \times (\mathbb{S}^1 \setminus [-c, \Theta + c])$, we have

$$ \|u\|_{H^1(D)} = \|u - b\|_{H^1(c_0 \setminus c_1)} \lesssim c_0. $$

For a small enough $c_0$ we may apply Lemma 14, so $u(p) = 0$. This implies that for every $t \in [\frac{1}{4}, \frac{3}{4}]$, we have $\text{Spt}(u(t, \cdot)) \subset (-c, \Theta + c)$.

• $\Theta = 2\pi$. Up to a rotation (and change of sign) there exists $(\alpha, \beta) \in \mathbb{R}_+^2$, such that

$$ b(\theta) = \alpha - \beta \cos(2\theta). $$

In particular

$$ \begin{aligned} |b(\theta)|^2 + \frac{1}{4}|b'(\theta)|^2 &= \alpha^2 + \beta^2 - 2\alpha\beta \cos(2\theta) \\ &\ge \frac{1 - \cos(2\theta)}{2}(\alpha + \beta)^2 \\ &\gtrsim (1 - \cos(2\theta))\|b\|_{H^1(\mathbb{S}^1)}^2, \end{aligned} $$

so for any $\theta \in (c, \pi - c) \cup (\pi + c, 2\pi - c)$, we have $|b(\theta)| + |b'(\theta)| \gtrsim \|b\|_{H^1(\mathbb{S}^1)}$. By the $C^1$ estimate (4.22) we obtain for a small enough $c_0$:

$$ \forall t \in \left[\frac{1}{4}, \frac{3}{4}\right], \quad \inf_{\theta \in [c, \pi-c] \cup [\pi+c, 2\pi-c]} (|u(t, \theta)| + |u'(t, \theta)|) \gtrsim \|b\|_{H^1(\mathbb{S}^1)}, $$

so the support hypothesis of Corollary 50 is verified.
Let $\phi \in C_c^\infty([\frac{1}{4}, \frac{3}{4}], [0, 1])$, such that

$$|\phi'| \lesssim 1, \quad \phi|_{[\frac{1}{3}, \frac{2}{3}]} \equiv 1.$$

Then

$$
\begin{align*}
W(u, 0) - W(u, 1) &= \int_0^1 -W'(u, t)dt \geq \int_0^1 -\phi(t)^2 W'(u, t)dt \\
&= \int_0^1 (-\phi(t)^2 \mathcal{G}'(u, t) - 2\phi(t)^2 \partial_t \langle \partial_t u, \partial_t u - \partial_{t,t} u \rangle) dt \\
&= \int_0^1 \phi(t)^2 (\|\partial_{t,t} u\|^2 + 2\|\partial_{t,\theta} u\|^2 + \|\partial_{\theta,\theta} u\|^2 - 4\|\partial_{\theta} u\|^2 + |\{u(t, \cdot) \neq 0\}| - 2\mathcal{G}(u, t)) dt \\
&\quad - 4 \int_0^1 \phi(t) (\phi(t) + \phi'(t)) \langle \partial_t u, \partial_{t,t} u \rangle dt.
\end{align*}
$$

Since $\mathcal{G}(u(t, \cdot), t) = \frac{1}{2}(\|\partial_{\theta,\theta} u\|^2 - 4\|\partial_{\theta} u\|^2 + |\{u(t, \cdot) \neq 0\}|)$, we may rewrite this as

$$
\begin{align}
W(u, 0) - W(u, 1) &\ge \int_0^1 \phi(t)^2 (\|\partial_{t,t} u\|^2 + 2\|\partial_{t,\theta} u\|^2 + 2\mathcal{G}(u(t, \cdot), t) - 2\mathcal{G}(u, t)) dt \tag{4.23} \\
&\quad - 4 \int_0^1 \phi(t) (\phi(t) + \phi'(t)) \langle \partial_t u, \partial_{t,t} u \rangle dt. \nonumber
\end{align}
$$

Now we apply Corollary 50 to $u(t+\cdot, \cdot)$ (which verifies the hypothesis for almost every $t \in [\frac{1}{4}, \frac{3}{4}]$, from the previous discussion). The estimate of Corollary 50 may be rewritten as

$$\mathcal{G}(u(t, \cdot), t) - \epsilon \left( \mathcal{G}(u(t, \cdot) - \frac{\Theta}{2})_+ - \mathcal{G}(u, t) \right) \geq -C \|\partial_{t,\theta} u(0, \cdot)\|_{L^2(\mathbb{S}^1)}^2 + 2 \|\partial_t u(0, \cdot)\|_{L^2(\mathbb{S}^1)}^2,$$

which implies (supposing without loss of generality that $\epsilon$ is less that $\frac{1}{2}$, so that $-\frac{1}{1-\epsilon} \geq -2$):

$$
\begin{align*}
\mathcal{G}(u(t, \cdot), t) - \mathcal{G}(u, t) &\geq \frac{\epsilon}{1-\epsilon} \left( \mathcal{G}(u, t) - \frac{\Theta}{2} \right) - 2C \|\partial_{t,\theta} u(t, \cdot)\|_{L^2(\mathbb{S}^1)}^2 + \frac{2}{1-\epsilon} \|\partial_t u(t, \cdot)\|_{L^2(\mathbb{S}^1)}^2 \\
&\geq \epsilon \left( W(u, t) - \frac{\Theta}{2} \right) + 2 \|\partial_t u(t, \cdot)\|^2 - 4\epsilon |\langle \partial_t u, \partial_{t,t} u \rangle| - 2C \|\partial_{t,\theta} u(t, \cdot)\|_{L^2(\mathbb{S}^1)}^2.
\end{align*}
$$

We inject this estimate in (4.23):

$$
\begin{align}
W(u, 0) - W(u, 1) &\ge 2\epsilon \left( W(u, 1) - \frac{\Theta}{2} \right) \int_0^1 \phi(t)^2 dt + \int_0^1 4\phi(t)^2 \|\partial_t u\|^2 dt \nonumber \\
&\quad + \int_0^1 \phi(t)^2 (\|\partial_{t,t} u\|^2 + 2\|\partial_{t,\theta} u\|^2 - 8\epsilon |\langle \partial_{t,t} u, \partial_t u \rangle| - 4C \|\partial_{t,\theta} u\|^2) dt \tag{4.24} \\
&\quad - 4 \int_0^1 \phi(t) (\phi(t) + \phi'(t)) \langle \partial_t u, \partial_{t,t} u \rangle dt \nonumber
\end{align}
$$

We have $-8\epsilon |\langle \partial_{t,t} u, \partial_t u \rangle| \geq -\|\partial_{t,t} u\|^2 - \|\partial_t u\|^2$, for a sufficiently small $\epsilon > 0$ (which we can assume without loss of generality). Similarly, we bound

$$-4\phi(t)(\phi(t) + \phi'(t))\langle \partial_t u, \partial_{t,t} u \rangle \geq -\phi(t)^2\|\partial_t u\|^2 - 8(\phi(t)^2 + \phi'(t)^2)\|\partial_{t,t} u\|^2.$$
We obtain (for some universal constant $C' > 0$):

$$ \mathcal{W}(u, 0) - \mathcal{W}(u, 1) \geq 2\epsilon \left( \mathcal{W}(u, 1) - \frac{\Theta}{2} \right) \int_{0}^{1} \phi(t)^2 dt + \int_{0}^{1} 2\phi(t)^2 \|\partial_t u\|^2 dt \\ - C' \int_{0}^{1} \left( \phi(t)^2 + \phi'(t)^2 \right) \left( \|\partial_{t,\theta} u\|^2 + \|\partial_{t,t} u\|^2 \right) dt. \quad (4.25) $$

We remind that $\phi|_{[\frac{1}{3}, \frac{2}{3}]} \equiv 1$ and $|\phi'| \lesssim 1$, so

$$ \mathcal{W}(u, 0) - \mathcal{W}(u, 1) \geq \frac{2\epsilon}{3} \left( \mathcal{W}(u, 1) - \frac{\Theta}{2} \right) + 2 \|\partial_t u\|_{L^2(c_1 \setminus c_2)}^2 \\ - C'' \int_0^1 \left( \|\partial_{t,\theta} u\|^2 + \|\partial_{t,t} u\|^2 \right) dt, \quad (4.26) $$

for some constant $C'' > 0$. Finally, $f \mapsto \sqrt{\|f\|_{L^2(c_1 \setminus c_2)}^2 + \|\nabla f\|_{L^2(c_0 \setminus c_1)}^2}$ is a norm for $H^1(C_0 \setminus C_1)$, so

$$ \|f\|_{L^2(c_1 \setminus c_2)}^2 + \|\nabla f\|_{L^2(c_0 \setminus c_1)}^2 \gtrsim \|f\|_{L^2(c_0 \setminus c_1)}^2 + \|\nabla f\|_{L^2(c_0 \setminus c_1)}^2, $$

and there is some universal constant $d \in (0, 1)$, $D > 0$ such that

$$ \mathcal{W}(u, 0) - \mathcal{W}(u, 1) \geq d \left( \left( \mathcal{W}(u, 1) - \frac{\Theta}{2} \right) + \|\partial_t u\|_{H^1(c_0 \setminus c_1)}^2 \right) - 4D \int_0^1 \left( \|\partial_{t,\theta} u\|^2 + \|\partial_{t,t} u\|^2 \right) dt. \quad (4.27) $$

Using the expression of $\mathcal{W}'(u, t)$ (see Theorem 18), we have

$$ 4 \int_{0}^{1} \left( \|\partial_{t,t} u\|^2 + \|\partial_{t,\theta} u\|^2 \right) dt = \mathcal{W}(u, 0) - \mathcal{W}(u, 1), \quad (4.28) $$

so we may replace the last term of (4.27) with $D(\mathcal{W}(u, 0) - \mathcal{W}(u, 1))$. We obtain

$$ \mathcal{W}(u, 0) - \mathcal{W}(u, 1) \geq \frac{d}{D+1} \left( \left( \mathcal{W}(u, 1) - \frac{\Theta}{2} \right) + \|\partial_t u\|_{H^1(c_0 \setminus c_1)}^2 \right), $$

which can be rearranged into

$$ \mathcal{W}(u, 1) \leq \mathcal{W}(u, 0) - \frac{d}{D+1-d} \left( \left( \mathcal{W}(u, 0) - \frac{\Theta}{2} \right) + \|\partial_t u\|_{H^1(c_0 \setminus c_1)}^2 \right), $$

which is the result. $\square$

We also need the following lemma to control the variation of the blow-up sequence $(u_r)$ by the variation of $\mathcal{W}$.

**Lemma 51.** There exist $c, C > 0$ such that, for any $u \in H_{\text{lin}}^2(C_0)$ a local minimizer of $\mathcal{G}(\cdot; 0)$, assume that there exists $b \in \mathcal{B}_{\text{hom}}^{\Theta}$ if $\Theta \in \{\pi, t_1\}$ (resp. $b \in \text{Span}(1, \cos(2\theta), \sin(2\theta))$ if $\Theta = 2\pi$) such that

$$ \|u - b\|_{H^1(c_0 \setminus c_1)} \leq c \|b\|_{H^1(\mathbb{S}^1)}, \quad \mathcal{W}(u, +\infty) \geq \frac{\Theta}{2}, $$

then

$$ \|u - u(1 + \cdot, \cdot)\|_{H^1(c_0 \setminus c_1)} \leq C \sqrt{\mathcal{W}(u, 0) - \mathcal{W}(u, +\infty)}. $$
Proof. For any $\tau \in (0, 1)$, we write

$$
\begin{aligned}
\int_{\mathbb{S}^1} |\nabla u(\tau + 1, \theta) - \nabla u(\tau, \theta)|^2 d\theta &= \int_{\mathbb{S}^1} \left| \int_{\tau}^{\tau+1} \partial_t \nabla u(s, \theta) ds \right|^2 d\theta \\
&\leq \int_{\mathbb{S}^1} \int_{\tau}^{\tau+1} |\partial_t \nabla u(s, \theta)|^2 ds d\theta \\
&\leq \frac{W(u, 0) - W(u, +\infty)}{4} \quad \text{by Theorem 18.}
\end{aligned}
$$

Likewise,

$$
\begin{aligned}
\int_{\mathbb{S}^1} |u(\tau + 1, \theta) - u(\tau, \theta)|^2 d\theta &= \int_{\mathbb{S}^1} \left| \int_{\tau}^{\tau+1} \partial_t u(s, \theta) ds \right|^2 d\theta \\
&\leq \int_{\mathbb{S}^1} \int_{\tau}^{\tau+1} |\partial_t u(s, \theta)|^2 ds d\theta \\
&= \int_{\mathbb{S}^1} \int_{\tau}^{\tau+1} \left| \partial_t u(\tau, \theta) + \int_{\tau}^{s} \partial_{t,t} u(s', \theta) ds' \right|^2 ds d\theta \\
&\leq 2 \int_{\mathbb{S}^1} |\partial_t u(\tau, \theta)|^2 d\theta + 2 \int_{\mathbb{S}^1} \int_{\tau}^{\tau+1} |\partial_{t,t} u(s, \theta)|^2 ds d\theta \\
&\leq 2 \int_{\mathbb{S}^1} |\partial_t u(\tau, \theta)|^2 d\theta + \frac{W(u, 0) - W(u, +\infty)}{2}
\end{aligned}
$$

so

$$
\int_{C_0 \setminus C_1} \int_{\mathbb{S}^1} |u(\tau + 1, \theta) - u(\tau, \theta)|^2 d\theta d\tau \le 2 \int_{C_0 \setminus C_1} |\partial_t u(\tau, \theta)|^2 d\theta d\tau + \frac{W(u, 0) - W(u, +\infty)}{2}.
$$

When $\frac{\|u-b\|_{H^1(c_0\setminus c_1)}}{\|b\|_{H^1(\mathbb{S}^1)}}$ is small enough, then by the epiperimetry inequality of Theorem 37 we have
$\int_{c_0\setminus c_1} |\partial_t u|^2 \lesssim W(u,0)-W(u,1)(\le W(u,0)-W(u, +\infty))$, which concludes the proof. $\square$

We now prove the main result of this section.

Proof of Theorem 36. We let $v(t, \theta) = e^{2t}u(e^{-t+i\theta})$, so that $v$ is a minimizer of $\mathcal{G}(\cdot, 0)$. We let also $b \in \mathcal{B}_{\text{hom}}^{\Theta}$ be such that

$$
\bar{u}(re^{i\theta}) = r^2 b(\theta).
$$

We remind that $W(v, t) = W(u, e^{-t})$ (by Theorem 19). Let $c_1 > 0$ to be fixed small enough later.
The bound on $\|u - \bar{u}\|_{H^1(\mathbb{D}_1 \setminus \mathbb{D}_{e-1})}$ implies

$$
\|v - b\|_{H^1(c_0 \setminus c_1)} \lesssim c_1 \|b\|_{H^1(\mathbb{S}^1)}.
$$

Then for a small enough $c_1$, Theorem 37 applies and we have

$$
W(u, e^{-1}) \leq \eta \frac{\Theta}{2} + (1 - \eta) W(u, 1).
$$

Similarly, for a small enough $c_1$, we may apply Lemma 51, which implies

$$
\|u - u_{e^{-1}}\|_{H^1(\mathbb{D}_1 \setminus \mathbb{D}_{e-1})} \lesssim \|v - v(1 + \cdot, \cdot)\|_{H^1(c_0 \setminus c_1)} \le C_1 \sqrt{W(u, 1) - \frac{\Theta}{2}}.
$$

Since $W(u, 0) \ge \frac{\Theta}{2}$, this concludes the proof. $\square$

<figure><img src="image_20.png" /></figure>
# 5 Description of flat, angular and isolated boundary points.

In this section, we prove the main Theorems 4, 7, 6, 5 (in this order), and conclude with the proof of Theorem 3.

The proof of Theorem 4 and 7 is based on the epiperimetric inequality and variation control given by Theorem 36: we will see in subsection 5.1 this implies an explicit polynomial rate of convergence of $u_r$ to a (unique) limit, given in Proposition 52.

We then prove in subsection 5.2 $\epsilon$-regularity results near flat boundary points (Proposition 53) and angular boundary points (Proposition 54); assuming $u$ is sufficiently close to a flat or angular blow-up, we prove that the support of $u$ is a $C^{1,\alpha}$ perturbation of the support of the associated homogeneous solution. This will imply Theorem 6.

In subsection 5.3, we make a conformal change of variable to transform an overdetermined Stokes equation on a $C^{1,\alpha}$ free boundary into an overdetermined elliptic equation on a disk: from this we obtain the higher regularity of the boundary and Theorem 5.

Finally, we prove the main Theorem 3 in subsection 5.4 by gathering all the previous results.

## 5.1 Uniqueness and speed of convergence of blow-ups

**Proposition 52.** Let $\Theta \in \{\pi, t_1, 2\pi\}$. There exist constants $c_2, C_2 > 0$, $\gamma \in (0,1)$ such that the following holds: let $\bar{u} \in \mathcal{M}_{\text{hom}}^{\Theta}$ if $\Theta \in \{\pi, t_1\}$ (resp. $\bar{u} \in \text{Span}(x^2 + y^2, x^2 - y^2, xy)$ if $\Theta = 2\pi$), $u \in \mathcal{M}(\mathbb{D}_1)$ such that

$$W(u, 0) \geq \frac{\Theta}{2}, \quad \|u - \bar{u}\|_{H^1(\mathbb{D}_1)} \leq c_2 \|\bar{u}\|_{H^1(\mathbb{D}_1)}.$$

Then there exists $\hat{u} \in \mathcal{M}_{\text{hom}}^{\Theta}$ such that for any $r \in (0,1]$:

$$\|u_r - \hat{u}\|_{H^1(\mathbb{D}_1)} \leq C_2 \min \left( r, \frac{\|u - \bar{u}\|_{H^1(\mathbb{D}_1)}}{\|\bar{u}\|_{H^1(\mathbb{D}_1)}} \right)^{\gamma} \|\bar{u}\|_{H^1(\mathbb{D}_1)},$$

and

$$\|\bar{u} - \hat{u}\|_{H^1(\mathbb{D}_1)} \leq C_2 \|u - \bar{u}\|_{H^1(\mathbb{D}_1)}^{\gamma} \|\bar{u}\|_{H^1(\mathbb{D}_1)}^{1-\gamma}.$$

Note that the hypothesis $W(u, 0) \geq \frac{\Theta}{2}$ is implied by the existence of a blow-up of opening $\Theta$ (i.e. type I if $\Theta = \pi$, II if $\Theta = t_1$, III, IV if $\Theta = 2\pi$).

In the proof, we will use the following scaling property: for any $r \in (0,1)$, for any $u \in H^1(rD)$ for some domain $D \subset \mathbb{R}^2$,

$$r^{-2}\|u\|_{H^1(rD)} \leq \|u_r\|_{H^1(D)} \leq r^{-3}\|u\|_{H^1(rD)} \quad (5.1)$$

Indeed, this is a consequence of the change of variable $\|u_r\|_{H^1(D)} = (\int_{rD} r^{-4}u^2 + r^{-6}|\nabla u|^2)^{\frac{1}{2}}$.

*Proof.* Let $c_2 > 0$ be some constant that will be fixed arbitrarily small later. Let $c_1, C_1, \eta$ be the constants from Theorem 36. By Lemma 20, we have

$$W\left(u, \frac{1}{2}\right) \leq C \|\bar{u}\|_{H^1(\mathbb{D}_1)}^2,$$

for some universal constant $C > 0$. We define the auxiliary constants

$$\alpha = \left( \frac{c_1 (1 - \sqrt{1 - \eta})}{2C_1} \right)^2, \quad K = \left[ \frac{\log(\alpha/C)}{\log(1 - \eta)} \right]_+^2, \quad \bar{r} = \frac{1}{2} e^{-K},$$
i.e. $K \in \mathbb{N}$ is the smallest integer such that $(1 - \eta)^K C \le \alpha$. We suppose $c_2$ verifies

$$c_2 \le \frac{\bar{r}^3 c_1}{2}. \qquad (5.2)$$

For any $k \in \{0, 1, 2, \dots, K-1\}$, we have (by the scaling formula (5.1)):

$$
\begin{aligned}
\|u_{\frac{1}{2}e^{-k}} - \bar{u}\|_{H^1(\mathbb{D}_1 \setminus \mathbb{D}_{e-1})} &= \|(u - \bar{u})_{\frac{1}{2}e^{-k}}\|_{H^1(\mathbb{D}_1 \setminus \mathbb{D}_{e-1})} && \text{since } \bar{u} \text{ is homogeneous} \\
&\le 8e^{3k}\|u - \bar{u}\|_{H^1(\mathbb{D}_1)} && \text{by the scaling (5.1)} \\
&\le \bar{r}^{-3}c_2\|\bar{u}\|_{H^1(\mathbb{D}_1)} && \text{by definition of } \bar{r} \\
&\le \frac{c_1}{2}\|\bar{u}\|_{H^1(\mathbb{D}_1)} && \text{by assumption (5.2).}
\end{aligned}
$$

As a consequence, Theorem 36 applies to $u_{\frac{1}{2}e^{-k}}$, and we get

$$\forall k = 0, 1, 2, \dots, K-1, \quad W\left(u, \frac{1}{2}e^{-k-1}\right) - \frac{\Theta}{2} \le (1-\eta)\left(W\left(u, \frac{1}{2}e^{-k}\right) - \frac{\Theta}{2}\right).$$

By induction on $k = 0, 1, \dots, K-1$, we obtain

$$
\begin{aligned}
W(u, \bar{r}) - \frac{\Theta}{2} &= W\left(u, \frac{1}{2}e^{-K}\right) - \frac{\Theta}{2} \le (1-\eta)^K \left(W\left(u, \frac{1}{2}\right) - \frac{\Theta}{2}\right) \\
&\le C(1-\eta)^K \|\bar{u}\|_{H^1(\mathbb{D}_1)}^2 && \text{by Lemma 20} \\
&\le \alpha \|\bar{u}\|_{H^1(\mathbb{D}_1)}^2 && \text{by definition of } \alpha.
\end{aligned}
$$

We claim now that for any $k \in \mathbb{N}$, we have

$$W(u, e^{-k}\bar{r}) \le \frac{\Theta}{2} + (1-\eta)^k \alpha \|\bar{u}\|_{H^1(\mathbb{D}_1)}^2, \quad \frac{\|u_{e^{-k}\bar{r}} - \bar{u}\|_{H^1(\mathbb{D}_1 \setminus \mathbb{D}_{e-1})}}{\|\bar{u}\|_{H^1(\mathbb{D}_1)}} \le \frac{c_1}{2} + C_1 \sqrt{\alpha} \sum_{j=0}^{k-1} (1-\eta)^{\frac{j}{2}} \quad (5.3)$$

We prove this by induction on $k$: this is verified at $k = 0$, since we verified $W(u, \bar{r}) \le \frac{\Theta}{2} + \alpha \|\bar{u}\|_{H^1(\mathbb{D}_1)}^2$ and

$$\frac{\|u_{\bar{r}} - \bar{u}\|_{H^1(\mathbb{D}_1 \setminus \mathbb{D}_{e-1})}}{\|\bar{u}\|_{H^1(\mathbb{D}_1)}} \le \bar{r}^{-3}c_2 \le \frac{1}{2}c_1.$$

Suppose now that (5.3) is verified for some $k \in \mathbb{N}$. By our definition of $\alpha$, we have

$$\frac{c_1}{2} + C_1 \sqrt{\alpha} \sum_{j=0}^{k-1} (1-\eta)^{\frac{j}{2}} \le \frac{c_1}{2} + \frac{C_1 \sqrt{\alpha}}{1 - \sqrt{1-\eta}} = c_1,$$

so Theorem 36 applies to $u_{e^{-k}\bar{r}}$. This gives the two conclusions of (5.3) for $k + 1$:

$$
\begin{aligned}
W(u, e^{-k-1}\bar{r}) - \frac{\Theta}{2} &\le (1-\eta)\left(W(u, e^{-k}\bar{r}) - \frac{\Theta}{2}\right) \\
&\le (1-\eta)^{k+1}\alpha \|\bar{u}\|_{H^1(\mathbb{D}_1)}^2
\end{aligned}
$$

and

$$
\begin{aligned}
\frac{\|u_{e^{-k-1}\bar{r}} - \bar{u}\|_{H^1(\mathbb{D}_1 \setminus \mathbb{D}_{e-1})}}{\|\bar{u}\|_{H^1(\mathbb{D}_1)}} &\le \frac{\|u_{e^{-k}\bar{r}} - \bar{u}\|_{H^1(\mathbb{D}_1 \setminus \mathbb{D}_{e-1})} + \|u_{e^{-k-1}\bar{r}} - u_{e^{-k}\bar{r}}\|_{H^1(\mathbb{D}_1 \setminus \mathbb{D}_{e-1})}}{\|\bar{u}\|_{H^1(\mathbb{D}_1)}} \\
&\le \frac{c_1}{2} + C_1 \sqrt{\alpha} \sum_{j=0}^{k-1} (1-\eta)^{\frac{j}{2}} + C_1 \sqrt{W(u, e^{-k}\bar{r}) - \frac{\Theta}{2}} && \text{by (5.3)} \\
&\le \frac{c_1}{2} + C_1 \sqrt{\alpha} \sum_{j=0}^{k} (1-\eta)^{\frac{j}{2}} && \text{by (5.3).}
\end{aligned}
$$
As a consequence, we have for every $k \in \mathbb{N}$:

$$W(u, e^{-k\bar{r}}) \leq \frac{\Theta}{2} + (1-\eta)^k \alpha \|\bar{u}\|_{H^1(\mathbb{D}_1)}^2, \quad \|u_{e^{-k\bar{r}}}-\bar{u}\|_{H^1(\mathbb{D}_1\setminus\mathbb{D}_{e-1})} \leq c_1 \|\bar{u}\|_{H^1(\mathbb{D}_1)}$$

so using again Theorem 36:

$$\|u_{e^{-k\bar{r}}} - u_{e^{-(k-1)\bar{r}}}\|_{H^1(\mathbb{D}_1\setminus\mathbb{D}_{e-1})} \leq C_1 \sqrt{\alpha(1-\eta)^{\frac{k}{2}}} \|\bar{u}\|_{H^1(\mathbb{D}_1)}$$

and

$$\begin{aligned}
\|u_{e^{-k\bar{r}}} - u_{e^{-(k-1)\bar{r}}}\|_{H^1(\mathbb{D}_1)} &= \sqrt{\sum_{p \in \mathbb{N}} \|u_{e^{-k\bar{r}}} - u_{e^{-(k-1)\bar{r}}}\|_{H^1(\mathbb{D}_{e-p} \setminus \mathbb{D}_{e-p-1})}^2} \\
&\leq \sqrt{\sum_{p \in \mathbb{N}} e^{-4p} \|u_{e^{-(k+p)\bar{r}}} - u_{e^{-(k+p-1)\bar{r}}}\|_{H^1(\mathbb{D}_1 \setminus \mathbb{D}_{e-1})}^2} \quad \text{by (5.1)} \\
&\leq \sqrt{\sum_{p \in \mathbb{N}} e^{-4p} C_1^2 \alpha(1-\eta)^{k+p} \|\bar{u}\|_{H^1(\mathbb{D}_1)}} \\
&\lesssim (1-\eta)^{\frac{k}{2}} \|\bar{u}\|_{H^1(\mathbb{D}_1)},
\end{aligned}$$

The quantity $\|u_{e^{-k\bar{r}}} - u_{e^{-(k-1)\bar{r}}}\|_{H^1(\mathbb{D}_1)}$ is summable in $k$: as a consequence there exists $\hat{u} \in \mathcal{M}_{\text{hom}}$ a limit of $u_{e^{-k\bar{r}}}$ such that for any $k \geq 0$:

$$\|u_{e^{-k\bar{r}}} - \hat{u}\|_{H^1(\mathbb{D}_1)} \leq \sum_{p \geq k} \|u_{ep\bar{r}} - u_{e^{-(p-1)\bar{r}}}\|_{H^1(\mathbb{D}_1)} \lesssim \sum_{p \geq k} (1-\eta)^{\frac{p}{2}} \|\bar{u}\|_{H^1(\mathbb{D}_1)} \lesssim (1-\eta)^{\frac{k}{2}} \|\bar{u}\|_{H^1(\mathbb{D}_1)}.$$

Denoting $\nu = -\log \sqrt{1-\eta}$, we obtain, for any $r \in [0, 1]$:

$$\|u_r - \hat{u}\|_{H^1(\mathbb{D}_1)} \lesssim r^{\nu} \|\bar{u}\|_{H^1(\mathbb{D}_1)}. \quad (5.4)$$

Consider now any $r \in (0, 1]$, and let $s \in (0, 1)$ to be fixed. Then

$$\begin{aligned}
\|u_r - \hat{u}\|_{H^1(\mathbb{D}_1)} &\leq \|u_r - \bar{u}\|_{H^1(\mathbb{D}_1)} + \|\bar{u} - u_s\|_{H^1(\mathbb{D}_1)} + \|u_s - \hat{u}\|_{H^1(\mathbb{D}_1)} \\
&\lesssim \left(\frac{1}{r^3} + \frac{1}{s^3}\right) \|u - \bar{u}\|_{H^1(\mathbb{D}_1)} + s^{\nu} \|\bar{u}\|_{H^1(\mathbb{D}_1)}.
\end{aligned}$$

We optimize this quantity in $s$ by fixing $s = \left( \frac{\|u-\bar{u}\|_{H^1(\mathbb{D}_1)}}{\|\bar{u}\|_{H^1(\mathbb{D}_1)}} \right)^{\frac{1}{3+\nu}}$ (which is smaller than 1 since $c_1 \leq 1$), so

$$\|u_r - \hat{u}\|_{H^1(\mathbb{D}_1)} \lesssim \frac{\|u - \bar{u}\|_{H^1(\mathbb{D}_1)}}{r^3} + \|u - \bar{u}\|_{H^1(\mathbb{D}_1)}^{\frac{\nu}{3+\nu}} \|\bar{u}\|_{H^1(\mathbb{D}_1)}^{\frac{3}{3+\nu}}.$$

We gather this estimate and (5.4):

$$\begin{aligned}
\|u_r - \hat{u}\|_{H^1(\mathbb{D}_1)} &\lesssim \min \left( r^{\nu} \|\bar{u}\|_{H^1(\mathbb{D}_1)}, \frac{\|u - \bar{u}\|_{H^1(\mathbb{D}_1)}}{r^3} + \|u - \bar{u}\|_{H^1(\mathbb{D}_1)}^{\frac{\nu}{3+\nu}} \|\bar{u}\|_{H^1(\mathbb{D}_1)}^{\frac{3}{3+\nu}} \right) \\
&\lesssim \min \left( r^{\nu} \|\bar{u}\|_{H^1(\mathbb{D}_1)}, \|\bar{u}\|_{H^1(\mathbb{D}_1)}^{\frac{3}{3+\nu}} \|u - \bar{u}\|_{H^1(\mathbb{D}_1)}^{\frac{\nu}{3+\nu}} \right) \\
&\lesssim \min \left( r, \frac{\|u - \bar{u}\|_{H^1(\mathbb{D}_1)}}{\|\bar{u}\|_{H^1(\mathbb{D}_1)}} \right)^{\gamma} \|\bar{u}\|_{H^1(\mathbb{D}_1)},
\end{aligned}$$

with $\gamma = \frac{\nu}{3+\nu}$. Evaluating this at $r=1$ we get

$$\begin{aligned}
\|\bar{u} - \hat{u}\|_{H^1(\mathbb{D}_1)} &\leq \|\bar{u} - u_r\|_{H^1(\mathbb{D}_1)} + \|u_r - \hat{u}\|_{H^1(\mathbb{D}_1)} \\
&\lesssim \|u - \bar{u}\|_{H^1(\mathbb{D}_1)} + \|u - \bar{u}\|_{H^1(\mathbb{D}_1)}^{\gamma} \|\bar{u}\|_{H^1(\mathbb{D}_1)}^{1-\gamma} \\
&\lesssim (c_2 + c_2^{\gamma}) \|\bar{u}\|_{H^1(\mathbb{D}_1)},
\end{aligned}$$

so for a sufficiently small constant $c_2$ we obtain that $\hat{u} \in \mathcal{M}_{\text{hom}}^{\Theta}$ (by Lemma 27: the $H^1(\mathbb{D}_1)$ distance between $\mathcal{M}_{\text{hom}}^{\pi}$, $\mathcal{M}_{\text{hom}}^{t_1}$, $\mathcal{M}_{\text{hom}}^{2\pi}$ is positive). This concludes the proof. $\square$
As a corollary, we obtain Theorem 7 and 4.

Proof of Theorem 7. This is a direct consequence of the previous Proposition 52 for $\Theta = 2\pi$. $\square$

Proof of Theorem 4. Let $u \in \mathcal{M}(\mathbb{D}_1)$ such that $0 \in \text{Spt}(u)$. By Proposition 52, it is sufficient to prove the following claims:

(i) In the case $\liminf_{r \to 0} \frac{|\text{Spt}(u) \cap \mathbb{D}_{p,r}|}{\pi r^2} < 1$, then there exists a sequence $r_n \to 0$ and $\bar{u} \in \mathcal{B}_{\text{hom}}^{\pi} \cup \mathcal{B}_{\text{hom}}^{t_1}$ such that $u_{r_n}$ converges to $\bar{u}$. Indeed this implies $W(u, 0) \ge \frac{\Theta}{2}$ (with $\Theta = 2|\text{Spt}(\bar{u}) \cap \mathbb{D}_1|$) and for some sufficiently large $n$ we have

$$ \frac{\|u_{r_n} - \bar{u}\|_{H^1(\mathbb{D}_1)}}{\|\bar{u}\|_{H^1(\mathbb{D}_1)}} \le c_2, $$

where $c_2$ is the constant from Proposition 52, and this proposition implies the result.

(ii) In the case $W(u, 0) \ge \pi$, then there exists $\bar{u} \in \text{Span}(x^2 + y^2, x^2 - y^2, xy)$ and $r \in (0, 1)$ such that

$$ \|u_r - \bar{u}\|_{H^1(\mathbb{D}_1)} \le c_2 \|\bar{u}\|_{H^1(\mathbb{D}_1)}. $$

Indeed, by Proposition 52, this implies the result.

The first claim is an immediate consequence of Proposition 30. Assume now we are in the second case. If some sequence $u_{r_n}$ is bounded in $H^1(\mathbb{D}_1)$ as $r_n \to 0$, then by Lemma 29, $u_{r_n}$ converges to some 2-homogeneous limit $v$. Since $W(v, 1) \ge \pi$, necessarily $v$ is of type III or IV, and we obtain the result.

Assume that $\|u_r\|_{H^1(\mathbb{D}_1)} \xrightarrow{r \to 0} +\infty$ instead. By Proposition 31, there exists some extracted sequence $r_n \to 0$ and some nonzero 2-homogeneous biharmonic function $v$ such that

$$ \frac{u_{r_n}}{\|u_{r_n}\|_{H^1(\mathbb{D}_1)}} \xrightarrow{n \to +\infty} v \text{ in } H_{\text{lin}}^2(\mathbb{R}^2). $$

In particular, for some large enough $n$ we have

$$ \frac{\|u_{r_n} - \|u_{r_n}\|_{H^1(\mathbb{D}_1)}v\|_{H^1(\mathbb{D}_1)}}{\|u_{r_n}\|_{H^1(\mathbb{D}_1)}} = \left\| \frac{u_{r_n}}{\|u_{r_n}\|_{H^1(\mathbb{D}_1)}} - v \right\|_{H^1(\mathbb{D}_1)} \le c_2, $$

which concludes the proof. $\square$

## 5.2 $C^{1,\alpha}$ regularity of the boundary

We are now ready to prove a weaker form (with only $C^{1,\gamma}$ bound) of Theorem 5. We will use the following interpolation inequalities: for any smooth bounded open set $D \subset \mathbb{R}$ or $\mathbb{R}^2$, $0 < \alpha < \beta < 1$, there exists $C = C(D, \alpha, \beta) \ge 0$ such that for any $f \in C^{1,\beta}(D, \mathbb{R})$:

$$ \|f\|_{C^1(D)} \le C \|f\|_{L^\infty(D)}^{\frac{\beta}{1+\beta}} \|f\|_{C^{1,\beta}(D)}^{\frac{1}{1+\beta}}, \quad (5.5) $$

$$ \|f\|_{C^{1,\alpha}(D)} \le C \|f\|_{C^1(D)}^{\frac{\beta-\alpha}{\beta}} \|f\|_{C^{1,\beta}(D)}^{\frac{\alpha}{\beta}}, \quad (5.6) $$

$$ \|f\|_{C^{1,\alpha}(D)} \le C \|f\|_{L^\infty(D)}^{\frac{\beta-\alpha}{\beta+1}} \|f\|_{C^{1,\beta}(D)}^{\frac{\alpha+1}{\beta+1}}. \quad (5.7) $$
**Proposition 53.** There exist $c_3 > 0$, $\alpha, \kappa \in (0,1)$, with the following property: for any $u \in \mathcal{M}(\mathbb{D}_1)$ such that

$$ \|u - u^I\|_{H^1(\mathbb{D}_1)} \leq c_3, $$

there exists a function $h \in C^{1,\alpha}([-\frac{1}{2}, \frac{1}{2}], \mathbb{R})$ such that $\|h\|_{L^\infty([-\frac{1}{2}, \frac{1}{2}])} \leq \sqrt[6]{\|u - u^I\|_{H^1(\mathbb{D}_1)}}$ and

$$ \|h\|_{C^{1,\alpha}([-\frac{1}{2}, \frac{1}{2}])} \lesssim \|u - u^I\|_{H^1(\mathbb{D}_1)}^{\kappa} $$

and

$$ \{(x, y) \in \mathbb{D}_4^3 : |x| \leq \frac{1}{2}, u(x, y) \neq 0\} = \{(x, y) \in \mathbb{D}_4^3 : |x| \leq \frac{1}{2}, y > h(x)\}. $$

Moreover, $\Delta u \in C_{\text{loc}}^{1,\alpha}(\mathbb{D}_{\frac{1}{2}} \cap \overline{\text{Spt}(u)})$ and $\Delta u = 1$ in $\mathbb{D}_{\frac{1}{2}} \cap \partial \text{Spt}(u)$.

*Proof.* In this proof, when $u_{p,r}$ has a unique limit as $r \to 0$, we write its limit $u_{p,0}$. We let $\epsilon > 0$ be such that $\|u - u^I\|_{H^1(\mathbb{D}_1)} \leq \epsilon$, and $\epsilon$ will be supposed arbitrarily small. We define the cone

$$ C = \{(x, y) \in \mathbb{R}^2 : y < -|x|/2\}. $$

The angular opening of the cone $C$ is approximately $0.7\pi$, which is strictly larger than $2\pi - t_1 \approx 0.57\pi$. We let

$$ G = \left\{ p \in \partial \text{Spt}(u) \cap \left[-\frac{1}{2}, \frac{1}{2}\right]^2 : (p + C) \cap \mathbb{D}_4^3 \subset \{u=0\} \right\}. $$

Note that the square $[-\frac{1}{2}, \frac{1}{2}]^2$ is fully contained in the open disk $\mathbb{D}_4^3$.

We now prove a sequence of claims on $G$ and $\partial \text{Spt}(u)$, to prove that $G$ is exactly $\partial \text{Spt}(u) \cap [-\frac{1}{2}, \frac{1}{2}]^2$ and that it is a $C^{1,\alpha}$ graph (over the $x$ coordinate) for some $\alpha \in (0,1)$ to be defined.

(Claim a) For a sufficiently small $\epsilon$, which we will suppose in the rest of the proof, we have

$$ \mathbb{D}_4^3 \cap \partial \text{Spt}(u) \subset \{(x, y) \in \mathbb{R}^2 : |y| \leq \sqrt[6]{\epsilon}\}. $$

Indeed, let $r = \sqrt[6]{\epsilon}$, $p = (x, y) \in \mathbb{D}_4^3$ such that $y \leq -r$, then $u_{p,r}^I = 0$ in $\mathbb{D}_1$ so

$$ \|u_{p,r}\|_{H^1(\mathbb{D}_1)} = \|(u - u^I)_{p,r}\|_{H^1(\mathbb{D}_1)} \leq \frac{1}{r^3} \|u - u^I\|_{H^1(\mathbb{D}_1)} \leq \frac{\epsilon}{r^3} = \sqrt{\epsilon} $$

so for a small enough $\epsilon$ by the non-degeneracy of Lemma 14 we have $u = 0$ in $\mathbb{D}_{p,r/2}$: $p$ cannot be in the boundary of the support.

Assume now instead that $y \geq r$. Lemma 10 gives a bound $\|u\|_{H^2(\mathbb{D}_4^3)} \lesssim 1$, so

$$ \begin{aligned} \|u - u^I\|_{C^0(\mathbb{D}_4^3)} & \lesssim \|u - u^I\|_{W^{1,4}(\mathbb{D}_4^3)} \lesssim \|u - u^I\|_{H^2(\mathbb{D}_4^3)}^{\frac{1}{2}} \|u - u^I\|_{H^1(\mathbb{D}_4^3)}^{\frac{1}{2}} \\ & \lesssim \|u - u^I\|_{H^1(\mathbb{D}_4^3)}^{\frac{1}{2}} \lesssim \sqrt{\epsilon} \end{aligned} $$

by Gagliardo-Nirenberg interpolation inequality. Thus, $u(x, y) \geq \frac{1}{2}y^2 - C\sqrt{\epsilon}$ for some universal constant $C$. Since $y \geq \sqrt[6]{\epsilon}$, then $u(x, y) > 0$ for a small enough $\epsilon$. This concludes claim (a).

(Claim b) For any $p \in \mathbb{D}_4^3 \cap \partial \text{Spt}(u)$, we claim $\|u_{p,\frac{1}{8}} - u^I\|_{H^1(\mathbb{D}_1)} \lesssim \sqrt[12]{\epsilon}$. Indeed, using Lemma 10, 12, we have a uniform bound

$$ \|u\|_{C^{1, \frac{1}{2}}(\mathbb{D}_{7/8})} \lesssim 1. $$