Let $p = (x, y) \in G$, by the claim (a) we have $|y| \le \sqrt[6]{\epsilon}$, so

$$
\begin{align*}
\|u_{p, \frac{1}{8}} - u^I\|_{H^1(\mathbb{D}_1)} &\le \|u_{(x,y), \frac{1}{8}} - u_{(x,0), \frac{1}{8}}\|_{H^1(\mathbb{D}_1)} + \|u_{(x,0), \frac{1}{8}} - u^I\|_{H^1(\mathbb{D}_1)} \\
&\lesssim \sqrt{|y|} + \|(u - u^I)_{(x,0), \frac{1}{8}}\|_{H^1(\mathbb{D}_1)} \text{ by the } \mathcal{C}^{1, \frac{1}{2}}(\mathbb{D}_8) \text{ bound} \\
&\le \sqrt[12]{\epsilon} + \epsilon \lesssim \sqrt[12]{\epsilon}
\end{align*}
$$

We obtain claim (b).

(Claim c) $G$ is a graph in the variable $x \in [-\frac{1}{2}, \frac{1}{2}]$: by definition two distinct points of $G$ cannot share the same $x$ coordinate, so it is sufficient to prove that for any $x_0 \in [-\frac{1}{2}, \frac{1}{2}]$ there exists $y_0 \in \mathbb{R}$ such that $(x_0, y_0) \in G$.

To prove this, consider the largest possible value of $y_0$ such that $((x_0, y_0)+\mathcal{C})\cap\mathbb{D}_{\frac{3}{4}} \subset \{u=0\}$.
Since $y_0$ is maximal, there must exist some contact point

$$
q = (x, y) \in \partial \left[ \left( (x_0, y_0) + \mathcal{C} \right) \cap \mathbb{D}_{\frac{3}{4}} \right] \cap \partial \mathrm{Spt}(u).
$$

By the claim (a), necessarily $|y| \le \sqrt[6]{\epsilon}$.

Suppose $q$ is **not** $(x_0, y_0)$. Then $\mathrm{Spt}(u)$ has Lebesgue density at most $\frac{1}{2}$ at $q$, so there exists a unique blow-up $u_{q,0}$ that is necessarily of the form

$$
u_{q,0} = s_1 u^I \circ \mathrm{rot}_{s_2 \theta_0}
$$

where $\theta_0 = \arctan(1/2)$, $s_1, s_2 \in \{-1, +1\}$. However by the claim (b) we have

$$
\|u_{q, \frac{1}{8}} - u^I\|_{H^1(\mathbb{D}_1)} \lesssim \sqrt[12]{\epsilon},
$$

so Proposition 52 applied to $u_{q, \frac{1}{8}}$ gives

$$
\|s_1 u^I \circ \mathrm{rot}_{s_2 \theta_0} - u^I\|_{H^1(\mathbb{D}_1)} \lesssim \epsilon^{\frac{\gamma}{12}}.
$$

Here $\gamma$ is the constant from Proposition 52. This is a contradiction when $\epsilon$ is small enough:
as a consequence $G$ is a graph, and it is automatically the graph of a 1-Lipschitz function
(since the slope of $\mathcal{C}$ is smaller than 1), denoted

$$
g : \left[-\frac{1}{2}, \frac{1}{2}\right] \to \left[-\sqrt[6]{\epsilon}, \sqrt[6]{\epsilon}\right].
$$

(Claim d) For every $p \in G$, $u$ admits at $p$ a unique blow-up of type I, denoted $u_{p,0}$, such that for every $r \in (0, \frac{1}{8})$ we have

$$
\|u_{p,r} - u_{p,0}\|_{H^1(\mathbb{D}_1)} \lesssim \min \left( \epsilon^{\frac{1}{12}}, r \right)^{\gamma}.
$$

Indeed, the existence of a blow-up of type I is a consequence of Proposition 30, since the
density of the support of $u$ at points of $G$ is strictly less than $\frac{t_1}{2\pi}$. The uniqueness and rate of
convergence follows from applying Proposition 52 to $u_{p,\frac{1}{8}}$, which verifies $\|u_{p,\frac{1}{8}} - u^I\|_{H^1(\mathbb{D}_1)} \lesssim
\sqrt[12]{\epsilon}$ by the claim (b).

(Claim e) $\|g\|_{C^{1,\beta}([-\frac{1}{2}, \frac{1}{2}])} \lesssim 1$, where $\beta = \frac{\gamma}{2\gamma+1} \in (0, 1)$.

To prove this, consider $p, q$ two points in $G$, $d := |p - q|$, and $r = d^{\frac{1}{2\gamma+1}}$. We suppose $d$ is small enough so that $r \le \frac{1}{16}$ and $\frac{d}{r} \le \frac{1}{4}$. We apply claim (d) above to $u$ at $p$ and $q$:

$$
\begin{align*}
\|u_{p,2r} - u_{p,0}\|_{H^1(\mathbb{D}_1)} &\lesssim r^{\gamma} = d^{\frac{\gamma}{2\gamma+1}} \\
\|u_{q,r} - u_{q,0}\|_{H^1(\mathbb{D}_1)} &\lesssim r^{\gamma} = d^{\frac{\gamma}{2\gamma+1}}.
\end{align*}
$$
By the Cacciopoli-type and higher regularity estimates from Lemma 10, 12, we have

$$ \|u_{p,2r}\|_{C^{1, \frac{1}{2}}(\mathbb{D}_{\frac{3}{4}})} \lesssim 1 $$

Then,

$$ \|u_{p,r} - u_{q,r}\|_{H^1(\mathbb{D}_1)} = \left\| u_{p,r} - u_{p,r} \left( \cdot + \frac{q-p}{r} \right) \right\|_{H^1(\mathbb{D}_1)} \lesssim \|u_{p,r}\|_{C^{1, \frac{1}{2}}(\mathbb{D}_{\frac{3}{2}})} \sqrt{\frac{|q-p|}{r}} \lesssim d^{\frac{\gamma}{2\gamma+1}}. $$

This implies $\|u_{q,0} - u_{p,0}\|_{H^1(\mathbb{D}_1)} \lesssim d^{\frac{\gamma}{2\gamma+1}}$, so $g$ (the function defining the graph $G$) is differentiable at every point $x \in [-\frac{1}{2}, \frac{1}{2}]$, with $|g'(x) - g'(y)| \lesssim (|x-y| + |g(x) - g(y)|)^\beta \lesssim |x-y|^\beta$.

(Claim f) $\{(x, y) \in \mathbb{D}_{\frac{3}{4}} : |x| \le \frac{1}{2}\} \cap \partial \text{Spt}(u) \subset G$. Indeed, suppose that there exists some point

$$ q \in \left\{ (x, y) \in \mathbb{D}_{\frac{3}{4}} : |x| \le \frac{1}{2} \right\} \cap \partial \text{Spt}(u) \setminus G. $$

Let $p$ be a projection of $q$ on $G$ ($p$ may not be unique). Since the $y$-coordinate of $p$ and $q$ are bounded by $\sqrt[6]{\epsilon}$, we have necessarily $|p-q| \lesssim \sqrt[6]{\epsilon}$. Let $r = 4|p-q|$, then by application of claim (d) to $u$ at the point $p$ with radius $r$, we have

$$ \|u_{p,r} - u^I\|_{H^1(\mathbb{D}_1)} \lesssim \epsilon^{\frac{\gamma}{12}}. $$

Let $(\tilde{u}, \tilde{q}) = (u_{p,r}, \frac{q-p}{r})$. Let $\tilde{G}$ be the graph associated to $\tilde{u}$ (defined in the same way that $G$ is defined for $u$), note that $0 \in \tilde{G}$ (since $p \in G$).

All the above claims apply to $\tilde{u}$: by claims (a,e) applied to $\tilde{u}$, we obtain that $\tilde{G}$ is the graph of a function $\tilde{g} \in C^{1,\alpha}([-\frac{1}{2}, \frac{1}{2}])$ such that $\|\tilde{g}\|_{L^\infty([-\frac{1}{2}, \frac{1}{2}])} \lesssim \epsilon^{\frac{\gamma}{72}}$. By definition of $r$, we have $|\tilde{q}| = \frac{1}{4}$. By claim (a), we have $|\tilde{q}_y| \lesssim \epsilon^{\frac{\gamma}{36}}$. At the same time, the origin is a projection of $\tilde{q}$ on $\frac{G-p}{r}$: this is a contradiction for a small enough $\epsilon$, which proves the claim (f).

Thus, we have proved that

$$ \{(x, y) \in \mathbb{D}_{\frac{3}{4}} : |x| \le \frac{1}{2}\} \cap \text{Spt}(u) = \{(x, y) \in \mathbb{D}_{\frac{3}{4}} : |x| \le \frac{1}{2}, y > g(x)\} $$

for some function $g : [-\frac{1}{2}, \frac{1}{2}] \to \mathbb{R}$ that verifies

$$ \|g\|_{C^{1,\beta}}([-\frac{1}{2}, \frac{1}{2}]) \lesssim 1, \quad \|g\|_{L^\infty}([-\frac{1}{2}, \frac{1}{2}]) \le \sqrt[6]{\epsilon}. $$

We define $\alpha = \frac{\beta}{2}$, $\kappa = \frac{\alpha}{12\alpha+4}$. By the interpolation properties (5.7), this implies

$$ \|h\|_{C^{1,\alpha}}([-\frac{1}{2}, \frac{1}{2}]) \lesssim \epsilon^{\kappa}. $$

Now we prove the $C_{\text{loc}}^{1,\alpha}(\mathbb{D}_{\frac{1}{2}})$ regularity of $\Delta u$ on its support. Let $q \in \text{Spt}(u) \cap \mathbb{D}_{\frac{1}{2}}$, and let $p$ be a projection of $q$ on $G$. We suppose $r := 2|p-q|$ is sufficiently small such that $r \le \frac{1}{8}$. Then we may apply claim (d) to $u$ at the point $p$:

$$ \|u_{p,r} - u_{p,0}\|_{H^1(\mathbb{D}_1)} \lesssim \epsilon^{\frac{\gamma}{12}}. $$
Moreover, since $p$ is a projection of $q$ on $G$, then for a small enough $\epsilon$ we have $\mathbb{D}_{\tilde{q}, \frac{1}{2}} \subset \text{Spt}(u_{p,0})$, where $\tilde{q} = \frac{q-p}{r}$. Let now $\chi \in C_c^\infty(\mathbb{D}_{\tilde{q}, \frac{1}{2}}, \mathbb{R})$ a radial function with integral 1 with $|\nabla^k \chi| \lesssim 1$ for $k=0,1$. Since $\Delta u$ is harmonic on its support, then

$$ \begin{aligned} |\Delta u(q) - 1| &= |\Delta u_{p,r}(\tilde{q}) - \Delta u_{p,0}(\tilde{q})| = \left| \int_{\mathbb{D}_{\tilde{q}, \frac{1}{2}}} (\Delta u_{p,r} - \Delta u_{p,0}) \chi(\cdot - \tilde{q}) \right| \\ &= \left| \int_{\mathbb{D}_{\tilde{q}, \frac{1}{2}}} \nabla (u_{p,r} - u_{p,0}) \cdot \nabla \chi(\cdot - \tilde{q}) \right| \lesssim \epsilon^{\frac{\gamma}{12}} \end{aligned} \quad (5.8) $$

Thus $\Delta u - 1$ is harmonic on its support and continuous up to the boundary $G$, with a homogeneous Dirichlet condition. Since $G$ has a regularity $C^{1,\alpha}$, then by classical boundary elliptic regularity we have $\Delta u \in C_{\text{loc}}^{1,\alpha}(\mathbb{D}_{\tilde{q}, \frac{1}{2}} \cap \overline{\text{Spt}(u)})$. $\square$

**Proposition 54.** There exist $c_4, C_4 > 0$, $\alpha \in (0,1)$, with the following property: for any $u \in \mathcal{M}(\mathbb{D}_1)$ such that

$$ \|u - u^{\text{II}}\|_{H^1(\mathbb{D}_1)} \le c_4, $$

there exist two functions $\theta^-, \theta^+ \in C^{1,\alpha}([\frac{1}{4}, \frac{3}{4}])$ such that

$$ \|\theta^-\|_{C^{1,\alpha}([\frac{1}{4}, \frac{3}{4}])} + \|\theta^+\|_{C^{1,\alpha}([\frac{1}{4}, \frac{3}{4}])} \lesssim 1, $$

$$ \|\theta^-\|_{L^\infty([\frac{1}{4}, \frac{3}{4}])} + \|\theta^+\|_{L^\infty([\frac{1}{4}, \frac{3}{4}])} \lesssim \epsilon^{\frac{\gamma}{2}} \|u - u^{\text{II}}\|_{H^1(\mathbb{D}_1)}, $$

and

$$ \{re^{i\theta} \in \text{Spt}(u) \cap \mathbb{D}_{\frac{3}{4}} \setminus \mathbb{D}_{\frac{1}{4}}\} = \{re^{i\theta} \in \mathbb{D}_{\frac{3}{4}} \setminus \mathbb{D}_{\frac{1}{4}} : \theta^{-}(r) \le \theta \le t_1 + \theta^{+}(r)\}. $$

*Proof.* We denote

$$ \epsilon = \|u - u^{\text{II}}\|_{H^1(\mathbb{D}_1)}, \quad S_0 = \left[\frac{1}{4}e_1, \frac{3}{4}e_1\right], \quad S_{t_1} = \text{rot}_{t_1}(S_0), $$

where $\text{rot}_{t_1}$ is the rotation of angle $t_1$. We let $\epsilon > 0$ that will be fixed sufficiently small later. We let also for any $\delta > 0$:

$$ S_0^\delta = \{p \in \mathbb{D}_{\frac{3}{4}} \setminus \mathbb{D}_{\frac{1}{4}} : \text{dist}(p, S_0) \le \delta, S_{t_1}^\delta = \text{rot}_{t_1}(S_0^\delta)\}. $$

Let $c_2, C_2, \gamma$ be the constants from Proposition 52, and $c_3, C_3, \alpha, \kappa$ be the constants from Proposition 53. Let $r \in [\frac{1}{4}, \frac{3}{4}]$, $\theta \in [-\frac{\pi}{2}, \frac{\pi}{2}]$, then

$$ u^{\text{II}}(re^{i\theta}) - u^{\text{I}}(re^{i\theta}) = -\frac{r^2}{2t_1} \left( \theta - \frac{\sin(2\theta)}{2} \right). $$

Observe the last factor is of the form $\mathcal{O}_{\theta \to 0}(|\theta|^3)$, so for any $\rho \in [0, \frac{1}{4}]$ we get

$$ \|u_{re_1, \rho}^{\text{II}} - u^{\text{I}}\|_{H^1(\mathbb{D}_1)} \lesssim \rho^3. $$

Thus,

$$ \|u_{re_1, \rho} - u^{\text{I}}\|_{H^1(\mathbb{D}_1)} \le \|u_{re_1, \rho} - u_{re_1, \rho}^{\text{II}}\|_{H^1(\mathbb{D}_1)} + \|u_{re_1, \rho}^{\text{II}} - u^{\text{I}}\|_{H^1(\mathbb{D}_1)} \lesssim \frac{\epsilon}{\rho^3} + \rho^3. \quad (5.9) $$

We may fix a sufficiently small $\bar{\rho} \gtrsim 1$ such that, when $\epsilon$ is sufficiently small, then

$$ \forall r \in \left[\frac{1}{4}, \frac{3}{4}\right], \quad \|u_{re_1, \bar{\rho}} - u^{\text{I}}\|_{H^1(\mathbb{D}_1)} \le c_3. $$
We apply Proposition 53 to every $r \in [\frac{1}{4}, \frac{3}{4}]$. As a consequence, we get that

$$S_0^{\bar{\rho}/2} \cap \partial \mathrm{Spt}(u) = \{te^{i\theta^{-}(t)}, t \in [\frac{1}{4}, \frac{3}{4}]\}$$

with $\|\theta^{-}\|_{C^{1,\alpha}([\frac{1}{4}, \frac{3}{4}])} \lesssim 1$. Moreover, using again (5.9) with $\rho := \sqrt[6]{\epsilon}$ and Proposition 53, we have

$$\|\theta^{-}\|_{L^{\infty}([\frac{1}{4}, \frac{3}{4}])} \lesssim \sqrt[6]{\epsilon}.$$

We define similarly $\theta^{+}$ from the rotation $u \circ \mathrm{rot}_{t_1}$, and we obtain

$$S_{t_1}^{\bar{\rho}/2} \cap \partial \mathrm{Spt}(u) = \{te^{i(t_1+\theta^+(t))}, t \in [\frac{1}{4}, \frac{3}{4}]\}$$

with

$$\|\theta^{+}\|_{C^{1,\alpha}([\frac{1}{4}, \frac{3}{4}])} \lesssim 1, \quad \|\theta^{+}\|_{L^{\infty}([\frac{1}{4}, \frac{3}{4}])} \lesssim \sqrt[6]{\epsilon}.$$

By the interpolation inequality (4.21), and Lemmas 10, 11:

$$\begin{aligned}
\|u - u^{\mathrm{II}}\|_{C^{1}(\mathbb{D}_{4}^{\prime})} &\lesssim \|u - u^{\mathrm{II}}\|_{H^{1}(\mathbb{D}_{4}^{\prime})}^{\frac{1}{2}} \|u - u^{\mathrm{II}}\|_{W^{2,4}(\mathbb{D}_{4}^{\prime})}^{\frac{1}{2}} \\
&\lesssim \|u - u^{\mathrm{II}}\|_{H^{1}(\mathbb{D}_{1})}^{\frac{1}{2}} (\|u\|_{H^{1}(\mathbb{D}_{1})} + \|u^{\mathrm{II}}\|_{H^{1}(\mathbb{D}_{1})})^{\frac{1}{2}} \\
&\lesssim \epsilon^{\frac{1}{2}}.
\end{aligned}$$

In particular, for a sufficiently small $\epsilon$, we obtain

$$\partial \mathrm{Spt}(u) \cap (\mathbb{D}_{4}^{\prime} \setminus \mathbb{D}_{4}) \subset S_{0}^{\bar{\rho}/2} \cup S_{t_{1}}^{\bar{\rho}/2},$$

and this proves the result.

<figure><img src="image_21.png" /></figure>

We are now ready to prove Theorem 6.

*Proof of Theorem 6.* Let $\epsilon = \mathrm{dist}_{H^1(\mathbb{D}_1)}(u, \mathcal{M}_{\mathrm{hom}}^{t_1})$, which we will suppose to be arbitrarily small.

We let $c_2, C_2 > 0$, $\gamma \in (0, 1)$ be the constants from Proposition 52, and $c_4, C_4 > 0$, $\alpha \in (0, 1)$ be the constants from Proposition 54. By Proposition 52 there exists $\hat{u} \in \mathcal{M}_{\mathrm{hom}}^{t_1}$ such that

$$\|u_r - \hat{u}\|_{H^1(\mathbb{D}_1)} \lesssim \min(r, \epsilon)^{\gamma}$$

for any $r \in (0, 1)$. Up to a rotation and a change of sign, we may assume without loss of generality that $\hat{u} = u^{\mathrm{II}}$. By Proposition 54 applied to $u_r$ for every $r \in (0, 1)$, there exists two functions $\theta^{-}, \theta^{+} \in C^{1,\alpha}((0, \frac{3}{4}], \mathbb{R})$ such that

$$\begin{aligned}
\|\theta^{-}(r \cdot)\|_{C^{1,\alpha}([\frac{1}{4}, \frac{3}{4}])} + \|\theta^{+}(r \cdot)\|_{C^{1,\alpha}([\frac{1}{4}, \frac{3}{4}])} &\lesssim 1, \\
\|\theta^{-}(r \cdot)\|_{L^{\infty}([\frac{1}{4}, \frac{3}{4}])} + \|\theta^{+}(r \cdot)\|_{L^{\infty}([\frac{1}{4}, \frac{3}{4}])} &\lesssim \min(r, \epsilon)^{\frac{\gamma}{6}},
\end{aligned}
\quad (5.10)$$

for every $r \in (0, 1)$ and

$$\{re^{i\theta} \in \mathrm{Spt}(u) \cap \mathbb{D}_{4}^{\prime} \setminus \{0\}\} = \{re^{i\theta} \in \mathbb{D}_{4}^{\prime} : r > 0, \theta^{-}(r) \le \theta \le t_{1} + \theta^{+}(r)\}.$$

Functions $\theta_{\pm}$ extends to $r = 0$ with the value 0. We now define the diffeomorphism (seeing it as a complex-valued function)

$$\Phi(re^{i\theta}) = re^{i\left(\theta+\theta^{-}(r)+\frac{\theta^{+}(r)-\theta^{-}(r)}{t_1}\theta\right)},$$

so that

$$\Phi(\mathbb{D}_{4}^{\prime} \cap \mathrm{Spt}(u^{\mathrm{II}})) = \mathbb{D}_{4}^{\prime} \cap \mathrm{Spt}(u).$$

We now verify that $\Phi - \mathrm{id}$ is small in a suitable Hölder space:
• By (5.10), we have $||\Phi - \text{id}||_{L^{\infty}(\mathbb{D}_1)} \lesssim \epsilon^{\frac{\gamma}{6}}$.

• For any $x \in \mathbb{D}_{\frac{1}{2}} \setminus \{0\}$, then by (5.10) applied to $r := 2|x|$ we have

$$|D\Phi(x) - I_2| \lesssim |x|^{\frac{\gamma}{6} - \frac{\alpha}{\alpha+1}}$$

We used here the interpolation inequality (5.5).

• For any distinct $x, y \in \mathbb{D}_{\frac{1}{2}}$, suppose that $|x| \le |y| \le 3|x|$ i.e. there exists $r \in (0, 1)$ such that $\frac{1}{4}r \le |x| \le |y| \le \frac{3}{4}r$. We let

$$\beta := \frac{\frac{\gamma}{6}\alpha}{\alpha + 1 + \frac{\gamma}{6}}.$$

Then by (5.10) applied with the radius $r$ we have

$$\begin{align*}
|D\Phi(x) - D\Phi(y)| &\lesssim \left( \|\theta_+(r\cdot)\|_{C^{1,\beta}([\frac{1}{4}, \frac{3}{4}])} + \|\theta_-(r\cdot)\|_{C^{1,\beta}([\frac{1}{4}, \frac{3}{4}])} \right) \left|\frac{x-y}{r}\right|^\beta \\
&\lesssim r^{\frac{\gamma}{6} - \frac{\alpha-\beta}{\alpha+1}} \left|\frac{x-y}{r}\right|^\beta \quad \text{by (5.7), (5.10)} \\
&= |x-y|^\beta \quad \text{by definition of } \beta.
\end{align*}$$

The second and third point imply $||\Phi||_{C^{1,\beta}(\mathbb{D}_{\frac{1}{2}})} \lesssim 1$. Since $||\Phi - \text{id}||_{L^{\infty}(\mathbb{D}_{\frac{1}{2}})} \lesssim \epsilon^{\frac{\gamma}{6}}$, then we may apply interpolation inequality (5.7); we let

$$\nu = \frac{\beta}{2}, \quad \mu = \frac{\beta}{\beta + 1} \frac{\gamma}{12},$$

then

$$||\Phi - \text{id}||_{C^{1,\nu}(\mathbb{D}_{\frac{1}{2}})} \lesssim \epsilon^{\mu}.$$

□

## 5.3 Higher regularity of the boundary

In the classical Alt-Caffarelli problem, higher order regularity of the boundary is usually obtained by some form of partial hodograph transform (i.e. seeing the state function $u$ as a coordinate, and finding a nonlinear, overdetermined elliptic equation verified in this coordinate system), based on [18, Th. 2]. This was generalized for elliptic systems in [19], and for higher order systems in [20]: in particular our boundary condition appears verbatim in [20, Th. 4.2] with $n = m = 2$, $F(p, u, \nabla u, \dots, \nabla^4 u) = \Delta^2 u$, $g(x, M) = \text{Tr}(M) - 1$. However, in [20, Th. 4.2] $u$ is assumed to be of class $C^4$ up to the free boundary, so we cannot apply it directly. A similar issue for the Alt-Caffarelli problem (regarding the results from [19]) was pointed out in the appendix of [21].

For this reason, we give an independent proof of higher regularity based on a conformal transform of the boundary. We find that the Stokes system associated to the biharmonic equation is conformally transported to an overdetermined elliptic equation, where the additional boundary condition (from the optimality condition) allows a bootstrap procedure. A similar method was used for a second order problem in [22].

The key argument is that after a conformal change of variable to flatten the boundary, the optimality conditions imply that the pressure (which is harmonic) verifies a homogeneous Neumann condition, so it is automatically smooth up to the boundary. From there, we treat the system as a complex-valued linear elliptic equation.
**Lemma 55.** Let $\Omega \subset D_1$ an open set such that $\Gamma := D_1 \cap \partial\Omega$ is a $C^{1,\alpha}$ curve (for some $\alpha \in (0,1)$) that connects two points of $\partial D_1$. Let $u \in H^2(\Omega)$ such that $\Delta u \in C^0(\Omega \cup \Gamma)$ and

$$ egin{cases} \Delta^2 u = 0 & (\Omega) \\ u = |\nabla u| = 0 & (\Gamma) \\ \Delta u = c & (\Gamma) \end{cases} $$

for some nonzero constant $c$. Then $D_1 \cap \partial\Omega$ is an analytic curve.

*Proof.* We identify $\mathbb{R}^2$ with $\mathbb{C}$ (so vector fields are complex-valued functions) and write the Wirtinger derivative

$$ \partial_z = \frac{1}{2} (\partial_x - i \partial_y), \quad \partial_{\bar{z}} = \frac{1}{2} (\partial_x + i \partial_y). $$

Since $D_1 \cap \partial\Omega$ is a $C^{1,\alpha}$ curve and $\Delta u - c$ is a harmonic function that verify a (homogeneous) Dirichlet condition on $D_1 \cap \partial\Omega$, then $\Delta u \in C_{\text{loc}}^{1,\alpha}(\Omega \cup \Gamma)$. Let $v : \Omega \to \mathbb{C}$ and $p : \Omega \to \mathbb{R}$ two functions defined by

$$ v = i \partial_{\bar{z}} u, \quad \partial_{\bar{z}} p = i \partial_z \partial_{\bar{z}} u $$

i.e. $p$ is a harmonic conjugate (uniquely defined up to an additive constant) of the harmonic function $\frac{1}{4}\Delta u$, that belongs to $C_{\text{loc}}^{1,\alpha}(\Omega \cup \Gamma)$. ($v, p$) verifies the Stokes equation with Dirichlet (or no-slip) boundary condition

$$ egin{cases} \partial_z \partial_{\bar{z}} v = \partial_{\bar{z}} p & (\Omega) \\ v = 0 & (\Gamma). \end{cases} $$

Moreover, the condition $\Delta u|_{\Gamma} = c$ gives an additional boundary condition on $p$ and $v$: since $p$ is the harmonic conjugate of $\Delta u$, which is constant in $\Gamma$, then $p$ verifies a homogeneous Neumann boundary condition in $\Gamma$, and since $\partial_z v = \frac{i}{4}\Delta u$ we have:

$$ \partial_z v = \frac{1}{4} ic \text{ in } \Gamma. $$

Let $\Phi : D_1 \to \Omega$ be a bijective conformal map (that is uniquely defined up to some conformal automorphism of the disk): by the Carathéodory Theorem [8] (see [28, Th. 2.6] for a modern reference), $\Phi$ extends as a homeomorphism $\bar{D}_1 \to \bar{\Omega}$, and we let $\Gamma'$ be the reciprocal image of $\Gamma$ through $\Phi$. By the Kellogg-Warschawski theorem [37] (see also [28, Th. 3.6], or [28, Prop. 3.4] for a local version), we have $\Phi \in C_{\text{loc}}^{1,\alpha}(D_1 \cup \Gamma')$.

Let

$$ \hat{p} = p \circ \Phi, \quad \hat{v} = v \circ \Phi $$

Then $\hat{p}$ is harmonic and verifies a homogeneous Neumann boundary condition on $\Gamma'$ (since this property is transported through the conformal mapping). Thus $\hat{p}$ is analytic in $D_1 \cup \Gamma'$.

Then we compute the equation verified by $(\hat{v}, \hat{p})$ in $D_1$:

$$ \partial_z \hat{v} = \Phi'(\partial_z v) \circ \Phi, \quad \partial_{\bar{z}} \hat{p} = \bar{\Phi}'(\partial_{\bar{z}} p) \circ \Phi, $$

so

$$ \partial_z \partial_{\bar{z}} \hat{v} = |\Phi'|^2 (\partial_z \partial_{\bar{z}} v) \circ \Phi = |\Phi'|^2 (\partial_{\bar{z}} p) \circ \Phi = \Phi' \partial_{\bar{z}} \hat{p}. $$

We obtain that $\hat{v}$ verifies the overdetermined elliptic equation

$$ egin{cases} \partial_{\bar{z}} \partial_z \hat{v} = \Phi' \partial_{\bar{z}} \hat{p} & (D_1) \\ \hat{v} = 0 & (\Gamma') \\ \partial_z \hat{v} = \frac{1}{4} i c \Phi' & (\Gamma'). \end{cases} $$
Since $\partial_{\bar{z}}\hat{p}$ is analytic up to the boundary $\Gamma'$, and $\Phi' \in C_{\text{loc}}^{0,\alpha}(\mathbb{D}_1 \cup \Gamma')$, by standard Schauder theory (using the first two equations only) we have $\hat{v} \in C_{\text{loc}}^{2,\alpha}(\mathbb{D}_1 \cup \Gamma')$. Using the third line, we see $\Phi' \in C_{\text{loc}}^{1,\alpha}(\Gamma')$, so $\Phi \in C_{\text{loc}}^{2,\alpha}(\mathbb{D}_1 \cup \Gamma')$.

Iterating this, for any $k \in \mathbb{N}_{>0}$, if $\Phi \in C_{\text{loc}}^{k,\alpha}(\mathbb{D}_1 \cup \Gamma')$, then $\hat{v} \in C_{\text{loc}}^{k+1,\alpha}(\mathbb{D}_1 \cup \Gamma')$, so $\Phi'|_{\Gamma'} \in C_{\text{loc}}^{k,\alpha}(\Gamma')$, which implies $\Phi \in C_{\text{loc}}^{k+1,\alpha}(\mathbb{D}_1 \cup \Gamma')$.

Thus, we obtain $\Phi \in C^{\infty}(\mathbb{D}_1 \cup \Gamma')$. From here on, the analyticity of $\Phi$ up to the boundary $\Gamma'$ may be obtained through tracking constants in this bootstrapping procedure, or simply by application of [20, Th. 4.2] with $n = m = 2$, $F(p, u, \nabla u, \dots, \nabla^4 u) = \Delta^2 u$, $g(x, M) = \text{Tr}(M) - c$ to the function $u$. $\square$

From this we obtain the main Theorem 5.

*Proof of Theorem 5.* This is a direct combination of Proposition 53, and Lemma 55. $\square$

## 5.4 Proof of the main result

We may now conclude with the proof of Theorem 3.

*Proof of Theorem 3.* Let $u \in \mathcal{M}(D)$ for some open set $D \subset \mathbb{R}^2$. We define

* $\mathcal{R}_u$ the points of $D \cap \partial\text{Spt}(u)$, with density $\frac{1}{2}$ in $\text{Spt}(u)$.

* $\mathcal{A}_u$ the points of $D \cap \partial\text{Spt}(u)$, with density $\frac{t_1}{2\pi}$ in $\text{Spt}(u)$.

* $\mathcal{N}_u$ the points $p \in D \cap \partial\text{Spt}(u)$ such that for some sufficiently small $r > 0$, we have $|\mathbb{D}_{p,r} \setminus \text{Spt}(u)| = 0$.

* $\mathcal{J}_u$ is the set of points $p \in D \cap \partial\text{Spt}(u) \setminus (\mathcal{R}_u \cup \mathcal{A}_u \cup \mathcal{N}_u)$ such that $\liminf_{r \to 0} \|u_{p,r}\|_{H^1(\mathbb{D}_1)} < \infty$.

* $\mathcal{E}_u := D \cap \partial\text{Spt}(u) \setminus (\mathcal{R}_u \cup \mathcal{A}_u \cup \mathcal{N}_u \cup \mathcal{J}_u)$

These sets are a partition of $D \cap \partial\text{Spt}(u)$ by construction. Now, we prove the claim of Theorem 3 case-by-case.

* Let $p \in \mathcal{R}_u$: by Theorem 4, $u_{p,r}$ converges in $H^1(\mathbb{D}_1)$ as $r \to 0$ to some homogeneous solution of type I. By Theorem 5 applied to $u_{p,2r}$ for a sufficiently small $r$, we obtain that $\text{Spt}(u) \cap \mathbb{D}_{p,r}$ is, after rotation, the epigraph of some analytic function. In particular, every point of $\partial\text{Spt}(u) \cap \mathbb{D}_{p,r}$ is also regular.

* Let $p \in \mathcal{A}_u$. By the same reasoning, by Theorem 4 and 6, there exists a sufficiently small $r > 0$ such that $\text{Spt}(u) \cap \mathbb{D}_{p,r}$ is the image of the angular cone $\text{Spt}(u^{\text{II}}) \cap \mathbb{D}_1$ through a $C^{1,\gamma}$ diffeomorphism (sending 0 to $p$, $\partial\mathbb{D}_1$ to $\partial\mathbb{D}_{p,r}$). As a consequence, every point of $\partial\text{Spt}(u) \cap \mathbb{D}_{p,r} \setminus \{p\}$ belongs to $\mathcal{R}_u$, $\mathcal{A}_u$ is a discrete set where two connected components of $\mathcal{R}_u$ join with an angle $t_1$ and $\Delta u|_{\mathcal{R}_u}$ takes opposite signs on each.

* Let $p \in \mathcal{N}_u$. Then $u$ is biharmonic in a neighbourhood of $p$. Since biharmonic functions are analytic, this implies that for a sufficiently small $r > 0$, $\partial\text{Spt}(u) \cap \mathbb{D}_{p,r}$ is an intersection of zeros of analytic function. Since $u(p) = |\nabla u(p)| = 0$, the blow-up $u_{p,0}$ is well-defined and non-zero by the nondegeneracy Lemma 14, and it is a blow-up either of type III (in which case $|\Delta u(p)| \ge 1$) or of type IV (in which case $\partial\text{Spt}(u) \cap \mathbb{D}_{p,r} = \{p\}$ for a sufficiently small $r$).

* Let $p \in \mathcal{J}_u$. By Lemma 29 and the fact that $p \notin \mathcal{R}_u \cup \mathcal{A}_u$, $u$ admits a blow-up of type III or IV, denoted $\hat{u}$. By Theorem 7, we have $\|u_{p,r} - \hat{u}\|_{H^1(\mathbb{D}_1)} \to 0$ as $r$ tends to 0. If $\hat{u}$ was of type IV, then $p$ would be an isolated point of $\partial\text{Spt}(u)$ and in particular it would be in the nodal set $\mathcal{N}_u$: thus $\hat{u}$ is necessarily of type III.
• Let $p \in \mathcal{E}_u$. Since $p$ belongs to none of the sets above we know that $p$ neither has density $\frac{1}{2}$ or $\frac{t_1}{2\pi}$ in Spt($u$) - meaning by Proposition 30 that it has density 1 in Spt($u$) - nor is it isolated in $\partial$Spt($u$) - meaning it does not admit a blow-up of type IV - nor does it admit a blow-up of type III since it would otherwise belong to $\mathcal{N}_u \cup \mathcal{J}_u$. Since $u$ admits no blow-up at $p$, necessarily we have

$$ \|u_{p,r}\|_{H^1(\mathbb{D}_{p,r})} \xrightarrow{r \to 0} +\infty. $$

By Theorem 4, we have necessarily $W(u(p + \cdot), 0) < \pi$, since otherwise there would be a blow-up of type III or IV. Finally, since $\|u_{p,r}\|_{H^1(\mathbb{D}_1)} \to +\infty$, then for any subsequence $r_n \to 0$ we may apply Proposition 31 to the sequence $u_{p,r_n}$: there is some subsequence $(n_i)$ such that $\frac{u_{p,r_n}}{\|u_{p,r_n}\|_{H^1(u_{p,r_n})}}$ converges in $H_{\text{loc}}^2(\mathbb{R}^2)$ norm to some nonzero 2-homogeneous biharmonic function.

<figure><img src="image_22.png" /></figure>

# 6 Quasi-minimizers

We explain here how the method we developed for the energy $E(\cdot, D)$ extends to a more general setting, with applications to the buckling eigenvalue minimization (1.4). We do not attempt to find the weakest optimality condition that would still imply the main regularity results: for the sake of clarity we instead give some mild optimality condition that contains the buckling problem.

## 6.1 Regularity of quasi-minimizers

**Definition 56.** Let $D \subset \mathbb{R}^2$ be an open set, we say $u$ is a quasiminimiser of $E(\cdot, D)$, and we write

$$ u \in \mathcal{Q}\mathcal{M}(D), $$

if for any $v \in H^2(D)$ such that $\{u \neq v\} \subseteq D$, we have

$$ E(u; D) \le E(v; D) + \|v - u\|_{L^2(D)}. \quad (6.1) $$

More generally, for $\mu \ge 0$ we say $u$ is a $\mu$-quasiminimizer of $E(\cdot; D)$, we write $u \in \mathcal{Q}\mathcal{M}^\mu(D)$ if for any $v \in H^2(D)$ such that $\{u \neq v\} \subseteq D$ we have

$$ E(u; D) \le E(v; D) + \mu \|v - u\|_{L^2(D)}. \quad (6.2) $$

This notion of quasi-minimizer scales as follows.

**Lemma 57.** Let $D \subset \mathbb{R}^2$ be an open set, $\mu \ge 0$, $r > 0$, and $u \in \mathcal{Q}\mathcal{M}^\mu(D)$. Then $u_r \in \mathcal{Q}\mathcal{M}^{r\mu}(D/r)$.

In other words, it is sufficient to prove the minimality condition (6.2) for some (possibly large) constant $\mu > 0$ to know that some rescaling of $u$ is a quasiminimizer in the sense of (6.1).

*Proof.* Consider $v \in H^2(D/r)$ such that $\{v \neq u_r\} \subseteq D/r$. This means that $v = w_r$, for some $w \in H^2(D)$ such that $\{w \neq u\} \subseteq D$. By the minimality condition (6.2) we have

$$ E(u; D) \le E(w; D) + \mu \|v - u\|_{L^2(D)}. $$

After a change of variable, we obtain

$$ E(u_r; D/r) \le E(v; D/r) + r\mu \|v - u_r\|_{L^2(D/r)}. $$

So $u_r \in \mathcal{Q}\mathcal{M}^{r\mu}(D)$.

<figure><img src="image_23.png" /></figure>

We now explain the different changes to adapt the proofs to quasi-minimizers, following the sections of the rest of the paper.
### 6.1.1 Cacciopoli inequality

Extending Lemma 10 to elements of $u \in \mathcal{M}(\mathbb{D}_1)$ is a direct consequence of the quasiminimality condition (6.1) with $v = e^{-t\eta^4}u_r$ for any $r \in (0, 1)$, $\eta \in C_c^\infty(\mathbb{D}_1, [0, 1])$, $t \in \mathbb{R}$. Indeed, this implies

$$E(u_r; \mathbb{D}_1) - E(e^{t\eta^4}u_r; \mathbb{D}_1) \le r\|(e^{t\eta^4} - 1)u_r\|_{L^2(\mathbb{D}_1)},$$

so for $t \to 0^+$ we get

$$\int_{\mathbb{D}_1} 2\nabla^2 u_r : \nabla^2(\eta^4 u_r) \le r\|\eta^4 u_r\|_{L^2(\mathbb{D}_1)}.$$

We bound

$$\begin{aligned} r\|\eta^4 u_r\|_{L^2(\mathbb{D}_1)} &\le r\|\eta^2 u_r\|_{L^2(\mathbb{D}_1)} \quad \text{since } \eta \le 1 \\ &\lesssim r\|\nabla^2(\eta^2 u_r)\|_{L^2(\mathbb{D}_1)} \quad \text{by Poincaré inequality} \\ &\lesssim r^2 + \|\nabla(\eta^2) \otimes \nabla u_r\|_{L^2(\mathbb{D}_1)}^2 + \|\nabla^2(\eta^2) u_r\|_{L^2(\mathbb{D}_1)}^2 + \|\eta^2 \nabla^2 u_r\|_{L^2(\mathbb{D}_1)}^2. \end{aligned}$$

The rest of the proof follows with the same computation as in Lemma 10. We obtain

**Lemma 58.** Let $u \in \mathcal{M}(\mathbb{D}_1)$, $r \in (0, 1)$, $\rho \in [\frac{1}{2}, 1)$, then

$$\|u_r\|_{H^2(\mathbb{D}_\rho)} \lesssim \frac{r^2 + \|u_r\|_{H^1(\mathbb{D}_1 \setminus \mathbb{D}_\rho)}}{(1-\rho)^2}. \quad (6.3)$$

### 6.1.2 BMO estimate

The BMO estimate is proved similarly to Lemma 11: we compare $u$ with its biharmonic replacement on any disk, and deduce a BMO bound on $\Delta u$.

**Lemma 59.** Let $u \in \mathcal{M}(\mathbb{D}_1)$. Then we have

$$[\nabla^2 u]_{\text{BMO}(\mathbb{D}_1^2)} \lesssim 1 + \|u\|_{H^2(\mathbb{D}_1)} \quad (6.4)$$

*Proof.* For any $\mathbb{D}_{p,r} \in D$, we let $u_{\mathbb{D}_{p,r}}$ be the biharmonic extension of $u$ in $\mathbb{D}_{p,r}$. Then the quasiminimality of $u$ gives

$$\int_{\mathbb{D}_{p,r}} |\Delta(u - u_{\mathbb{D}_{p,r}})|^2 \le |\mathbb{D}_{p,r}| + \|u - u_{\mathbb{D}_{p,r}}\|_{L^2(\mathbb{D}_{p,r})}.$$

We bound $\|u - u_{\mathbb{D}_{p,r}}\|_{L^2(\mathbb{D}_{p,r})} \lesssim r^2 \|\Delta(u - u_{\mathbb{D}_{p,r}})\|_{L^2(\mathbb{D}_{p,r})}$ by Poincaré inequality, so by Young inequality:

$$\int_{\mathbb{D}_{p,r}} |\Delta(u - u_{\mathbb{D}_{p,r}})|^2 \lesssim |\mathbb{D}_{p,r}| + r^4.$$

We divide by the measure of $\mathbb{D}_{p,r}$:

$$\int_{\mathbb{D}_{p,r}} |\Delta(u - u_{\mathbb{D}_{p,r}})|^2 \lesssim 1 + r^2.$$

The rest of the proof is the same as the one of Lemma 11. □
### 6.1.3 Nondegeneracy lemma

The extension of Lemma 14 to the quasi-minimal context has no major changes.

**Lemma 60.** There exists $\epsilon > 0$ such that the following holds: let $u \in \mathcal{M}(\mathbb{D}_1)$, such that $\text{Spt}(u) \cap \mathbb{D}_{\frac{1}{2}} \neq \emptyset$, then

$$ \|u\|_{H^1(\mathbb{D}_1)} \geq \epsilon. \qquad (6.5) $$

*Proof.* It is sufficient to prove that when $\|u\|_{H^1(\mathbb{D}_1)}$ is sufficiently small, then $u(0) = |\nabla u(0)| = 0$ (since we may apply this to $u_{p, \frac{1}{2}}$ for $p \in \mathbb{D}_{\frac{1}{2}}$). Let $r \in (0, 1)$, we have the first (non-linear) estimate:

$$ \|u_r\|_{H^1(\mathbb{D}_{\frac{1}{2}})}^2 \lesssim |\text{Spt}(u_r) \cap \mathbb{D}_{\frac{3}{4}}|^{\frac{1}{2}} \|u_r\|_{W^{1,4}(\mathbb{D}_{\frac{3}{4}})}^2 \quad \text{by Hölder inequality} $$

$$ \lesssim |\text{Spt}(u_r) \cap \mathbb{D}_{\frac{3}{4}}|^{\frac{1}{2}} \|u_r\|_{H^2(\mathbb{D}_{\frac{3}{4}})}^2 \quad \text{by Sobolev embedding } H^2 \to W^{1,4} $$

$$ \lesssim |\text{Spt}(u_r) \cap \mathbb{D}_1|^{\frac{1}{2}} (r^2 + \|u_r\|_{H^1(\mathbb{D}_1)}^2) \quad \text{by Lemma 58.} $$

Then we have an estimate of the area of the support, obtained by the minimality condition applied to $v = (1 - \eta)u_r$, where $\eta \in C_c^\infty(\mathbb{D}_{\frac{3}{4}})$ verifies $\eta = 1$ in $\mathbb{D}_{\frac{1}{2}}$:

$$ \begin{aligned} |\text{Spt}(u_r) \cap \mathbb{D}_{\frac{1}{2}}| &\le \int_{\mathbb{D}_{\frac{1}{2}}} (\chi_{u_r \ne 0} - \chi_{v \ne 0}) \\ &\le \int_{\mathbb{D}_{\frac{3}{4}}} (|\Delta(u_r - \eta u_r)|^2 - |\Delta u_r|^2) + r \|\eta u_r\|_{L^2(\mathbb{D}_1)} \quad \text{by (6.1)} \\ &\lesssim \int_{\mathbb{D}_{\frac{3}{4}}} (|\Delta(\eta u_r)|^2 - 2\Delta(\eta u_r)\Delta u_r) + r \|\eta u_r\|_{L^2(\mathbb{D}_1)} \\ &\lesssim \|u_r\|_{H^1(\mathbb{D}_1)}^2 + r^2 \quad \text{by Lemma 58.} \end{aligned} $$

Denoting $g(r) = \|u_r\|_{H^1(\mathbb{D}_1)}^2$, we obtain that for some constant $C_0 > 0$:

$$ \forall r > 0 : \mathbb{D}_{p,r} \subset D, \text{ we have } g\left(\frac{r}{4}\right) \le C_0 \left(r^3 + g(r)^{\frac{3}{2}}\right). $$

Let $a_0 = (4C_0)^{-2}$, $r_0 := \min(1, C_0^{-\frac{1}{2}}, \frac{3}{4}a_0)$, then for any $r \in (0, r_0)$, assume $g(r) \le a_0$, then we have

$$ g\left(\frac{r}{4}\right) \le r + \frac{1}{4}g(r) \le a_0 $$

and by induction $g(4^{-k}r) \le k4^{1-k}r + 4^{-k}g(r)$ for any $k \in \mathbb{N}^*$. This tends to 0 as $k \to +\infty$.

Assume $\|u\|_{H^1(\mathbb{D}_1)} \le r_0^3 a_0^{\frac{1}{2}}$, then $\|u_{r_0}\|_{H^1(\mathbb{D}_1)} \le a_0^{\frac{1}{2}}$, so $u(0) = |\nabla u(0)| = 0$. This concludes the proof. $\square$

From the previous results we can deduce a weaker (but easier to work with) optimality condition.

**Lemma 61.** There exists $Q > 0$ such that for any $u \in \mathcal{M}(\mathbb{D}_1)$ such that $0 \in \partial \text{Spt}(u)$, for any $r \in (0, \frac{1}{2})$, $v \in u + H_0^2(\mathbb{D}_r)$, we have

$$ E(u; \mathbb{D}_r) \le E(v; \mathbb{D}_r) + Q \|u\|_{H^1(\mathbb{D}_1)} r^{\frac{5}{2}}. $$
*Proof.* We assume without loss of generality that $E(v; \mathbb{D}_r) \le E(u; \mathbb{D}_r)$ (otherwise there is nothing to prove). By the non-degeneracy lemma we have

$$ \|u\|_{H^1(\mathbb{D}_1)} \gtrsim 1. \qquad (6.6) $$

By combining Lemma 59, 58, we have for any $p \in \mathbb{D}_{\frac{3}{2}r}$:

$$ \frac{|u(p)|}{|p|} + |\nabla u(p)| \lesssim r^{\frac{1}{2}} [\nabla u]_{C^{0, \frac{1}{2}}(\mathbb{D}_{\frac{3}{4}})} \lesssim r^{\frac{1}{2}} \|u\|_{H^1(\mathbb{D}_1)}. $$

So

$$ \|u_r\|_{H^2(\mathbb{D}_1)} \lesssim \|u_r\|_{H^1(\mathbb{D}_{\frac{3}{2}})} \lesssim r^{-\frac{1}{2}}. \qquad (6.7) $$

Then, by the quasi-minimality condition on $u$ we have

$$ E(u; \mathbb{D}_r) \le E(v; \mathbb{D}_r) + \|u - v\|_{L^2(\mathbb{D}_r)} $$

and we bound the remainder term:

$$
\begin{align*}
\|u - v\|_{L^2(\mathbb{D}_r)} &\lesssim r^2 \|\Delta(u - v)\|_{L^2(\mathbb{D}_r)} \quad \text{by Poincaré inequality} \\
&\lesssim r^2 \left( \|\Delta u\|_{L^2(\mathbb{D}_r)} + \|\Delta v\|_{L^2(\mathbb{D}_r)} \right) \\
&\lesssim r^2 E(u; \mathbb{D}_r)^{\frac{1}{2}} \quad \text{since we assume } E(v; \mathbb{D}_r) \le E(u; \mathbb{D}_r) \\
&\lesssim r^2 \left( r^2 + \|\Delta u\|_{L^2(\mathbb{D}_r)}^2 \right)^{\frac{1}{2}} \\
&= r^3 \left( 1 + \|\Delta u_r\|_{L^2(\mathbb{D}_1)}^2 \right)^{\frac{1}{2}} \\
&\lesssim r^{\frac{5}{2}} \left( 1 + \|u\|_{H^1(\mathbb{D}_1)}^2 \right)^{\frac{1}{2}} \quad \text{by (6.7)} \\
&\lesssim r^{\frac{5}{2}} \|u\|_{H^1(\mathbb{D}_1)} \quad \text{by (6.6).}
\end{align*}
$$

And so

$$ E(u; \mathbb{D}_r) - E(v; \mathbb{D}_r) \le \|u - v\|_{L^2(\mathbb{D}_r)} \lesssim r^{\frac{5}{2}} \|u\|_{H^1(\mathbb{D}_1)}. $$

This concludes the proof.

<figure><img src="image_24.png" /></figure>

### 6.1.4 Sequence of minimizers

For the next lemma, we write (for $\lambda, \mu \ge 0$) $u \in \mathcal{Q}\mathcal{M}_{\lambda}^{\mu}(D)$ if for every $\mathbb{D}_{p,r} \in D$, $v \in u + H_0^2(\mathbb{D}_{p,r})$, we have

$$ E_{\lambda}(u; \mathbb{D}_{p,r}) \le E_{\lambda}(v; \mathbb{D}_{p,r}) + \mu \|u - v\|_{L^2(\mathbb{D}_{p,r})}. $$

**Lemma 62.** Let $\lambda^{(n)} \in \mathbb{R}_+$ some sequence that converges in $C^0(\mathbb{D}_1)$ to $\lambda \ge 0$, $\mu^{(n)}$ a sequence of $[0, 1]$ that converge to $\mu \ge 0$. For each $n$, let $u^{(n)} \in \mathcal{Q}\mathcal{M}_{\lambda^{(n)}}^{\mu}(\mathbb{D}_1)$ such that

$$ \limsup_{n \to +\infty} \|u^{(n)}\|_{H^1(\mathbb{D}_1)} < +\infty. $$

Then there is a subsequence $u^{(n_k)}$ that converges in $H_{\text{loc}}^2(\mathbb{D}_1)$ to $u \in \mathcal{Q}\mathcal{M}_{\lambda}^{\mu}(\mathbb{D}_1)$, with a strong $L^1(\mathbb{D}_1)$ convergence of $\lambda^{(n_k)}\chi_{u^{(n_k)} \neq 0}$ to $\lambda \chi_{u \neq 0}$.
Proof. By Lemma 58, $(u^{(n)})_{n \in \mathbb{N}}$ is bounded in $H_{\text{loc}}^2(\mathbb{D}_1)$. Consider $u$ a $H_{\text{loc}}^2(\mathbb{D}_1)$-weak limit of $(u^{(n)})_{n \in \mathbb{N}}$ (after extraction, which we do not write here for clarity), $v \in H^2(\mathbb{D}_1)$ such that $\{u \neq v\} \subset \mathbb{D}_1$, $\eta \in C_c^\infty(\mathbb{D}_1)$ such that $\eta = 1$ on $\text{Spt}(u-v)$, then we apply the quasi-optimality condition to $(1-\eta)u^{(n)} + \eta v$:

$$E_{\lambda^{(n)}}(u^{(n)}; \mathbb{D}_1) \le E_{\lambda^{(n)}}((1-\eta)u^{(n)} + \eta v; \mathbb{D}_1) + \mu^{(n)} \|\eta(v - u^{(n)})\|_{L^2(\mathbb{D}_1)} \quad (6.8)$$

Following the proof of Lemma 15, with $v = u$ we obtain

$$\|u\|_{H^2(\mathbb{D}_r)} = \lim_{n \to +\infty} \|u^{(n)}\|_{H^2(\mathbb{D}_r)}$$

for any $r \in (0, 1)$, thus the convergence $u^{(n)} \to u$ is strong in $H_{\text{loc}}^2(\mathbb{D}_1)$. Then Lemma 62 for a general $v$ gives as $n \to +\infty$:

$$E_{\lambda}(u; \mathbb{D}_1) \le E_{\lambda}(v; \mathbb{D}_1) + \mu \|v - u\|_{L^2(\mathbb{D}_1)},$$

so $u \in \mathcal{M}_\lambda^\mu(\mathbb{D}_1)$.

### 6.1.5 Monotonicity

In this section, we consider a minimizer $u \in \mathcal{M}(\mathbb{D}_1)$ with $0 \in \partial \text{Spt}(u)$. We prove that up to adding a sufficiently large polynomial term to $W(u, r)$, we still obtain an increasing function.

**Theorem 63.** There exists $\lambda_0 > 0$ such that the following holds. For any $u \in \mathcal{M}(\mathbb{D}_1)$ such that $0 \in \partial \text{Spt}(u)$, we let

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

Then the function

$$r \in (0, \frac{1}{2}) \mapsto W(u, r) + \lambda_0 \|u\|_{H^1(\mathbb{D}_1)} \sqrt{r}$$

is non-decreasing. More precisely, for any $\tau \ge 0$, we have $W(u, e^{-\tau}) = W(v, \tau)$ and

$$\frac{d}{d\tau} \left( W(v, \tau) + \lambda_0 \|u\|_{H^1(\mathbb{D}_1)} e^{-\frac{\tau}{2}} \right) \le -4 \|\partial_t \nabla_{t,\theta} v\|_{L^2(\partial C_\tau)}^2.$$

Since $W(u, r) + \lambda_0 \|u\|_{H^1(\mathbb{D}_1)} \sqrt{r}$ is nondecreasing, this means that $W(u, r)$ admits a limit - that we denote $W(u, 0)$ - as $r \to 0$.

**Proof.** By Lemma 59, we have $v \in H_{\text{lin}}^2(C_0)$. By a change of variable (see Lemma 16), the quasi-minimality of $u$ implies the following: for any $\tau \ge 0$, any $w \in H_{\text{lin}}^2(C_0)$ such that $\{v \neq w\} \subset C_\tau$, we have

$$\mathcal{G}(v, \tau) \le \mathcal{G}(w, \tau) + e^{2\tau} \sqrt{\int_{C_\tau} e^{-6t}(v-w)(t, \theta)^2 dt d\theta}.$$

The minimality condition simplifies to

$$\mathcal{G}(v, \tau) \le \mathcal{G}(w, \tau) + e^{-\tau} \|v-w\|_{L^2(C_\tau, e^{-2t} dtd\theta)} \quad (6.9)$$

As in the proof of Theorem 18, let $\tau > 0$, consider the vector field $\phi(t, \theta) = (f(t), 0)$ for some function $f \in C^\infty(\mathbb{R}_+)$ with $f = 0$ on $[0, \tau]$, $\|f\|_{W^{1,\infty}(\mathbb{R}_+)} < \infty$, $\psi_\epsilon$ the inverse of $t \mapsto t + \epsilon f(t)$ and $w(t, \theta) = v(\psi_\epsilon(t), \theta)$. Then the optimality condition (6.9) applied for $\epsilon \to 0$ gives the condition

$$\begin{aligned} & \int_0^{+\infty} e^{-2t} (2f - f') \left\{ \|\partial_{t,t} v\|^2 + 2\|\partial_{t,\theta} v\|^2 + \|\partial_{\theta,\theta} v\|^2 - 4\|\partial_{\theta} v\|^2 + \int_{\partial C_t} \chi_{v \neq 0} \right\} dt \\ & \qquad + 2 \int_0^{+\infty} e^{-2t} f'' \langle \partial_{t,t} v, \partial_{t,t} v \rangle dt \\ & \le -4 \int_0^{+\infty} e^{-2t} f' \left\{ \|\partial_{t,t} v\|^2 + \|\partial_{t,\theta} v\|^2 \right\} dt + e^{-\tau} \|f \partial_{t,t} v\|_{L^2(C_\tau, e^{-2t} dtd\theta)} \end{aligned}$$
By Lemma 59, we bound $|\partial_t v| \lesssim e^{\frac{t}{2}} \|u\|_{H^1(\mathbb{D}_1)}$. So taking formally $f \to \chi_{t \ge \tau}$ (or using the weak derivative characterization as in the proof of Theorem 18) we obtain

$$W'(v, \tau) + 4 (\|\partial_{t,t}v(\tau)\|^2 + \|\partial_{t,\theta}v(\tau)\|^2) \lesssim \|u\|_{H^1(\mathbb{D}_1)}e^{-\frac{\tau}{2}}.$$

So for some sufficiently large $\lambda_0 \gtrsim 1$, we have

$$\frac{d}{d\tau} \left( W(v, \tau) + \lambda_0 \|u\|_{H^1(\mathbb{D}_1)} e^{-\frac{\tau}{2}} \right) \le -4 \left( \|\partial_{t,t}v(\tau)\|^2 + \|\partial_{t,\theta}v(\tau)\|^2 \right).$$

The rest follows exactly the proof of Theorem 19 with the additional corrective term. □

We obtain as a consequence the following blow-up results, that is a combination of Lemma 29 and Propositions 30, 31.

**Lemma 64.** Let $u \in \mathcal{M}(\mathbb{D}_1)$ such that $0 \in \partial \text{Spt}(u)$, then

* Any converging subsequence of $u_r$ as $r \to 0$ converges to a 2-homogeneous minimizer of $E(\cdot; \mathbb{D}_1)$.

* If $\liminf_{r \to 0} \frac{|\text{Spt}(u) \cap \mathbb{D}_r|}{|\mathbb{D}_r|} < 1$, then there exists some sequence $r_n \to 0$ such that $u_{r_n}$ converges to an element of $\mathcal{M}_{\text{hom}}^{\pi} \sqcup \mathcal{M}_{\text{hom}}^{t_1}$.

* If $\limsup_{r \to 0} \|u_r\|_{H^1(\mathbb{D}_1)} = +\infty$ and $0 \in \partial \text{Spt}(u)$, then there exists a sequence $r_n \to 0$ such that $\frac{u_{r_n}}{\|u_{r_n}\|_{H^1(\mathbb{D}_1)}}$ converges in $H_{\text{loc}}^2(\mathbb{R}^2)$ to a non-zero 2-homogeneous biharmonic function.

*Proof.* This is a direct consequence of Lemma 62 and Theorem 63.

* With the same computation as for Lemma 33, we have for any $u \in \mathcal{M}(\mathbb{D}_1)$, $0 < s < r < 1$:

$$N(u, s) \ge \left(\frac{s}{r}\right)^{\kappa} N(u, r) - \frac{1 - \left(\frac{s}{r}\right)^{\kappa}}{\kappa} \left(W(u, 1) + \lambda_0 \|u\|_{H^1(\mathbb{D}_1)}\right) \quad (6.10)$$

which implies by contradiction that for any sequence $r_n \to 0$ such that $\limsup_n |\text{Spt}(u_{r_n}) \cap \mathbb{D}_1| < \pi$, we have

$$\limsup_{n \to +\infty} \|u_{r_n}\|_{H^1(\mathbb{D}_1)} < \infty.$$

Indeed, otherwise $u_{r_n}/\|u_{r_n}\|_{H^1(\mathbb{D}_1)}$ would converge to a non-zero biharmonic function $v$, such that $\{v = 0\}$ has a positive measure, which is a contradiction.

Since $u_{r_n}$ is bounded in $H^1(\mathbb{D}_1)$ and belongs to $\mathcal{M}^{r_n}(\mathbb{D}_1)$ (by Lemma 57), then up to extracting some subsequence we may suppose (by Lemma 62) that $u_{r_n}$ converges to some $v \in \mathcal{M}^0(\mathbb{D}_1) = \mathcal{M}(\mathbb{D}_1)$, which is non-zero by the nondegeneracy Lemma 60.

Since $W(u, r)$ converges to a constant as $r \to 0$, then $W(v, r)$ is constant with respect to $r$, so $v$ is a 2-homogeneous element of $\mathcal{M}(\mathbb{D}_1)$. Since the density of its support is less than 1, necessarily

$$v \in \mathcal{M}_{\text{hom}}^{\pi} \sqcup \mathcal{M}_{\text{hom}}^{t_1}.$$

* This follows the same proof as Lemma 31. Let $\lambda_n = \|u_{r_n}\|_{H^1(\mathbb{D}_1)}^{-1}$, the renormalized function $\lambda_n u_{r_n}$ belongs to $\mathcal{M}_{\lambda_n^2}^{\lambda_n}(\mathbb{D}_1)$: by Lemma 62 we may suppose (after extraction) that $\|u_{r_n}\|_{H^1(\mathbb{D}_1)}^{-1} u_{r_n}$ converges to $v \in \mathcal{M}^0(\mathbb{D}_1)$, meaning some biharmonic function $v$. Function $v$ is non-zero by the growth estimate (6.10) and we conclude as in Proposition 31. □
### 6.1.6 Epiperimetry

The central cornerstone of the epiperimetry formula is the estimate of Corollary 50: we do not need to re-prove this result, indeed we can just use the competitor of 50 in the minimality characterization of Lemma 61.

**Theorem 65.** Let $\Theta \in \{\pi, t_1, 2\pi\}$. There exist constants $c_1, C_1, \lambda_1 > 0$, $\eta \in (0, 1)$ such that the following holds: let $u \in \mathcal{Q}\mathcal{M}(\mathbb{D}_1)$ such that $0 \in \partial\text{Spt}(u)$, $\bar{u} \in \mathcal{M}_{\text{hom}}^{\Theta}$ if $\Theta \in \{\pi, t_1\}$ (resp. $\bar{u} \in \text{Span}(x^2 + y^2, x^2 - y^2, xy)$ if $\Theta = 2\pi$), $r \in (0, \frac{1}{2})$, assume

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

Then, denoting

$$ W^{\text{quasi}}(u, r) := W(u, r) + \lambda_1 \|u\|_{H^1(\mathbb{D}_1)} \sqrt{r}, \qquad (6.11) $$

we have

$$ W^{\text{quasi}}(u, e^{-1}r) - \frac{\Theta}{2} \le (1 - \eta) \left( W^{\text{quasi}}(u, r) - \frac{\Theta}{2} \right) $$

and

$$ \|u_r - u_{e^{-1}r}\|_{H^1(\mathbb{D}_1 \setminus \mathbb{D}_{e-1})} \le C_1 \sqrt{W^{\text{quasi}}(u, r) - \frac{\Theta}{2}}. $$

**Proof.** We let $v(t, \theta) = e^{2t}u(e^{-t+i\theta})$ and proceed as in the proof of Theorem 18. We fix $\tau = -\log(r)$. When the ratio

$$ \frac{\|u_r - \bar{u}\|_{H^1(\mathbb{D}_1 \setminus \mathbb{D}_{e-1})}}{\|\bar{u}\|_{H^1(\mathbb{D}_1)}} $$

is sufficiently small, we obtain (by the same argument as in the proof of Theorem 37) that $(v(t, \cdot), \partial_t v(t, \cdot))$ verifies the support hypothesis of 50 for almost every $t \in [\tau + \frac{1}{4}, \tau + \frac{3}{4}]$.

We let $\phi \in C_c^\infty(\mathbb{R}, [0, 1])$ such that

$$ \{\phi \neq 0\} \subset [\tau + \frac{1}{4}, \tau + \frac{3}{4}], \quad |\phi'| \lesssim 1, \quad \phi = 1 \text{ in } [\tau + \frac{1}{3}, \tau + \frac{2}{3}]. $$

Then following the computations of the proof of Theorem 37, we have

$$ \begin{aligned} W(v, \tau) - W(v, \tau + 1) &\ge \int_{\tau}^{\tau+1} \phi(t)^2 (\|\partial_{t,t}v\|^2 + 2\|\partial_{t,\theta}v\|^2 + 2\mathcal{G}(v(t, \cdot), t) - 2\mathcal{G}(v, t)) dt \\ &\quad - 4 \int_{\tau}^{\tau+1} \phi(t) (\phi(t) + \phi'(t)) \langle \partial_{t,t}v, \partial_{t,t}v \rangle dt. \end{aligned} $$

By the minimality condition of Lemma 61, applied to the competitor built in Corollary 50 with the initial conditions $(v(t, \cdot), \partial_t v(t, \cdot))$, we have for every $t \in [\tau + \frac{1}{4}, \tau + \frac{3}{4}]$:

$$ \begin{aligned} \mathcal{G}(v(t, \cdot), t) - \mathcal{G}(v, t) &\ge \epsilon \left( W(v, t) - \frac{\Theta}{2} \right) + 2 \|\partial_t v(t, \cdot)\|^2 - 4\epsilon |\langle \partial_t v, \partial_{t,t} v \rangle| - 2C \|\partial_{t,\theta} v(t, \cdot)\|_{L^2(\mathbb{S}^1)}^2 \\ &\quad - Q e^{-\frac{\tau}{2}} \|u\|_{H^1(\mathbb{D}_1)} \end{aligned} $$

for some constant $C > 0$. Here $\mathcal{G}$ is as defined in (3.2). Injecting this in the previous estimate, and rearranging the terms as in the proof of Theorem 37, we obtain for some constants $d, D, C' > 0$:

$$ \begin{aligned} W(v, \tau) - W(v, \tau + 1) &\ge d \left( \left( W(v, \tau + 1) - \frac{\Theta}{2} \right) + \|\partial_t v\|_{H^1(c_0 \setminus c_1)}^2 \right) - 4D \int_0^1 \left( \|\partial_{t,\theta} v\|^2 + \|\partial_{t,t} v\|^2 \right) dt \\ &\quad - C' e^{-\frac{\tau}{2}} \|u\|_{H^1(\mathbb{D}_1)}. \end{aligned} $$
By the monotonicity formula of Theorem 63 (we denote $\lambda_0$ the constant from this result), we have

$$4 \int_0^1 (\|\partial_{t,\theta} v\|^2 + \|\partial_{t,t} v\|^2) dt \geq W(v, \tau) - W(v, \tau + 1) - (1 - e^{-\frac{1}{2}}) \lambda_0 \|u\|_{H^1(\mathbb{D}_1)} e^{-\frac{\tau}{2}}.$$

And so we obtain that for some constant $C''$:

$$W(v, \tau + 1) \leq W(v, \tau) - \frac{d}{D+1-d} \left( W(u, \tau) - \frac{\Theta}{2} + \|\partial_t v\|_{H^1(C_\tau \setminus C_{\tau+1})}^2 \right) + C'' \|u\|_{H^1(\mathbb{D}_1)} e^{-\frac{\tau}{2}}.$$

Going back to the disk coordinate, and handling the estimate of $\|u_{e^{-1}r} - u_r\|_{H^1(\mathbb{D}_1 \setminus \mathbb{D}_{e^{-1}})}$ as in the proof of Lemma 51, we obtain the result. □

### 6.1.7 Uniqueness and speed of convergence of blow-ups

The adaptation of Proposition 52 to the quasiminimal setting is as follows.

**Proposition 66.** Let $\Theta \in \{\pi, t_1, 2\pi\}$, $M \ge 1$. There exist constants $c_2, C_2 > 0$, $\gamma \in (0,1)$ such that the following holds: let $u \in \mathcal{Q}\mathcal{M}(\mathbb{D}_1)$ such that $0 \in \partial\text{Spt}(u)$, $\bar{u} \in \mathcal{M}_{\text{hom}}^{\Theta}$ if $\Theta \in \{\pi, t_1\}$ (resp. $\bar{u} \in \text{Span}(x^2 + y^2, x^2 - y^2, xy)$ if $\Theta = 2\pi$), assume

$$W(u, 0) \geq \frac{\Theta}{2}, \quad \|u - \bar{u}\|_{H^1(\mathbb{D}_1)} \leq c_2 \|\bar{u}\|_{H^1(\mathbb{D}_1)}.$$

Then there exists $\hat{u} \in \mathcal{M}_{\text{hom}}^{\Theta}$ such that for any $r \in (0,1]$:

$$\|u_r - \hat{u}\|_{H^1(\mathbb{D}_1)} \le C_2 \min \left( r, \frac{\|u - \bar{u}\|_{H^1(\mathbb{D}_1)}}{\|\bar{u}\|_{H^1(\mathbb{D}_1)}} \right)^{\gamma} \|\bar{u}\|_{H^1(\mathbb{D}_1)},$$

and

$$\|\bar{u} - \hat{u}\|_{H^1(\mathbb{D}_1)} \le C_2 \|u - \bar{u}\|_{H^1(\mathbb{D}_1)}^{\gamma} \|\bar{u}\|_{H^1(\mathbb{D}_1)}^{1-\gamma}.$$

*Proof.* The proof is the same as the proof of Proposition 52, by replacing $W$ with $W^{\text{quasi}}$, and the constant $c_1, C_1$ from Theorem 36 by $c_1, C_1$ from Theorem 65. □

The $\epsilon$-regularity results near flat and angular boundary point also follow.

**Proposition 67.** There exist $c_3 > 0$, $\alpha, \kappa \in (0,1)$, with the following property: for any $u \in \mathcal{M}(\mathbb{D}_1)$, $\bar{u} \in \mathcal{M}_{\text{hom}}^{\pi} \cup \mathcal{M}_{\text{hom}}^{t_1}$ such that

$$\|u - \bar{u}\|_{H^1(\mathbb{D}_1)} \le c_3,$$

then

* If $\bar{u} = u^1$, then there exists a function $h \in C^{1,\alpha}([-\frac{1}{2}, \frac{1}{2}], \mathbb{R})$ such that $\|h\|_{L^\infty([-\frac{1}{2}, \frac{1}{2}])} \le \sqrt[6]{\|u - u^1\|_{H^1(\mathbb{D}_1)}}$ and
$$\|h\|_{C^{1,\alpha}([-\frac{1}{2}, \frac{1}{2}])} \lesssim \|u - u^1\|_{H^1(\mathbb{D}_1)}^{\kappa}$$

and

$$\{(x, y) \in \mathbb{D}_{\frac{3}{4}} : |x| \le \frac{1}{2}, u(x, y) \ne 0\} = \{(x, y) \in \mathbb{D}_{\frac{3}{4}} : |x| \le \frac{1}{2}, y > h(x)\}.$$

Moreover, $\Delta u \in C_{\text{loc}}^{0,\alpha}(\mathbb{D}_{\frac{1}{2}} \cap \overline{\text{Spt}(u)})$ and $\Delta u = 1$ in $\mathbb{D}_{\frac{1}{2}} \cap \partial\text{Spt}(u)$.
• If $u = u^{\text{II}}$ and $W(u, 0) \ge \frac{t_1}{2}$, then there exists a diffeomorphism $\Phi \in \mathcal{C}^{1, \gamma}(\mathbb{D}_{\frac{1}{2}}, \mathbb{D}_{\frac{1}{2}})$ such that

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

and

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

In the first case, the only difference with Proposition 53 is that $\Delta u$ is only proved to be $\mathcal{C}^{0,\alpha}$ (instead of $\mathcal{C}^{1,\alpha}$) up to the boundary. This is because we do not know that $\Delta u$ is harmonic in $\text{Spt}(u)$ anymore.

*Proof.* We only detail the part that differs (in the proof of Proposition 53) in the quasi-minimal setting: the regularity of $\Delta u$ up to the boundary. Indeed, starting over from equation (5.8), we have

$$ |\Delta u(q) - 1| = |\Delta(u_{p,r} - u_{p,0})(\tilde{q})|. $$

Since $u_{p,r} \in \mathcal{Q}\mathcal{M}(\mathbb{D}_1)$ and $\mathbb{D}_{\tilde{q}, \frac{1}{2}} \subset \text{Spt}(u_{p,r})$, in particular we have for every $\varphi \in \mathcal{C}_c^\infty(\mathbb{D}_{\tilde{q}, \frac{1}{2}}, \mathbb{R})$, $t \in \mathbb{R}$,

$$ \int_{\mathbb{D}_{\tilde{q}, \frac{1}{2}}} |\Delta u_{p,r}|^2 \le \int_{\mathbb{D}_{\tilde{q}, \frac{1}{2}}} |\Delta(u_{p,r} + t\varphi)|^2 + |t| \|\varphi\|_{L^2(\mathbb{D}_{\tilde{q}, \frac{1}{2}})}. $$

This implies that for every such $\varphi$, we have,

$$ \left| \int_{\mathbb{D}_{\tilde{q}, \frac{1}{2}}} \Delta u_{p,r} \Delta \varphi \right| \le \frac{1}{2} \|\varphi\|_{L^2(\mathbb{D}_{\tilde{q}, \frac{1}{2}})}. $$

Since $\Delta^2 u_{p,0} = 0$ in $\mathbb{D}_{\tilde{q}, \frac{1}{2}}$, we obtain

$$ \|\Delta^2(u_{p,r} - u_{p,0})\|_{L^2(\mathbb{D}_{\tilde{q}, \frac{1}{2}})} \le \frac{1}{2}. $$

We remind that $\|u_{p,r} - u_{p,0}\|_{H^1(\mathbb{D}_1)} \lesssim \epsilon^{\frac{\gamma}{12}}$, so $u_{p,r} - u_{p,0}$ is bounded in $H^4(\mathbb{D}_{\tilde{q}, \frac{1}{2}})$. By Gagliardo-Nirenberg inequality (interpolating $W^{2,\infty}(\mathbb{D}_{\tilde{q}, \frac{1}{2}})$ between $H^1(\mathbb{D}_{\tilde{q}, \frac{1}{2}})$ and $H^4(\mathbb{D}_{\tilde{q}, \frac{1}{2}})$) we obtain

$$ \begin{aligned} |\Delta u(q) - 1| &= |\Delta(u_{p,r} - u_{p,0})(\tilde{q})| \\ &\lesssim \|u_{p,r} - u_{p,0}\|_{H^1(\mathbb{D}_{\tilde{q}, \frac{1}{2}})}^{\frac{1}{3}} \|u_{p,r} - u_{p,0}\|_{H^4(\mathbb{D}_{\tilde{q}, \frac{1}{2}})}^{\frac{2}{3}} \\ &\lesssim r^{\frac{\gamma}{36}}. \end{aligned} $$

This concludes the proof of the Hölder regularity of $\Delta u$ up to the regular boundary. $\square$

### 6.1.8 Boundary decomposition for quasiminimizers

*Proof of Theorem 8.* The proof is identical to the proof of Theorem 3: we define $\mathcal{R}_u, \mathcal{A}_u, \mathcal{N}_u, \mathcal{J}_u, \mathcal{E}_u$ the same way. Proposition 67 implies the claim on $\mathcal{R}_u, \mathcal{A}_u$. The claim on $\mathcal{J}_u$ is a consequence of Proposition 66. Finally, the description of $\mathcal{E}_u$ is a consequence of Lemma 64 and Proposition 66. $\square$
## 6.2 Application to buckling eigenvalue

We remind that the definition of $\Lambda_1(\Omega)$, for an open set $\Omega$ with finite area, is given in (1.4). We write $u_\Omega$ an (arbitrary) choice of non-zero eigenfunction on $\Omega$ associated to $\Lambda_1(\Omega)$. Weinberger and Willms's argument may be decomposed in the following steps (we remind a detailed proof may be found in [17, Prop 4.4] and [2]).

a) If $\Omega \subset \mathbb{R}^2$ has finite measure, $C^2$ boundary, and is a critical point of the functional $|\Omega|\Lambda_1(\Omega)$ with respect to shape derivatives, then $|\Delta u_\Omega|$ must be constant in $\partial\Omega$.

b) If $\Omega \subset \mathbb{R}^2$ is a bounded simply connected set and $|\Delta u_\Omega|$ is constant on $\partial\Omega$, then $\Delta u_\Omega + \Lambda_1(\Omega)u_\Omega$ is constant in $\Omega$.

c) If $\Omega \subset \mathbb{R}^2$ is open with finite area such that $\Delta u_\Omega + \Lambda_1(\Omega)u_\Omega$ is constant in $\Omega$, then $\Lambda_1(\Omega) \ge \Lambda_1(\Omega^*)$, where $\Omega^*$ is a disk with the same area as $\Omega$.

These three steps imply that if $\Omega$ is a minimizer of $|\Omega|\Lambda_1(\Omega)$ that is bounded, simply connected, with $C^2$ boundary, then $\Omega$ must be the disk.

As a direct consequence of Theorem 8, we may lower the initial hypothesis to obtain this in two ways. We first prove that the first eigenfunction of a minimizers of the buckling eigenvalue problem (1.4) under area constraint belong to the class of quasi-minimizers given in Definition 56.

We remind that since $\Lambda_1(t\Omega) = t^{-2}\Lambda_1(\Omega)$, then up to a dilation, minimizing $\Lambda_1(\Omega)$ under an area constraint on $\Omega$ is equivalent to minimizing either $|\Omega|\Lambda_1(\Omega)$ or $\Lambda_1(\Omega) + |\Omega|$ among every set.

**Lemma 68.** Let $u \in H^2(\mathbb{R}^2)$ be a minimizer of $E(\cdot; \mathbb{R}^2)$ under the constraint $\int_{\mathbb{R}^2} |\nabla u|^2 = 1$. Then there exists $\bar{r} > 0$ such that for any $p \in \mathbb{R}^2$, we have $u_{\bar{r}} \in \mathcal{Q}\mathcal{M}(\mathbb{D}_{p,1})$.

Similar computation may be found in [33] (as well as the existence of such a minimizer, obtained via a concentration-compactness procedure).

*Proof.* The function $u$ is a minimizer of the functional

$$v \in H^2(\mathbb{R}^2) \setminus \{0\} \mapsto \frac{\int_{\mathbb{R}^2} |\Delta v|^2}{\int_{\mathbb{R}^2} |\nabla v|^2} + |\text{Spt}(v)|.$$

We write $\Lambda := \int_{\mathbb{R}^2} |\Delta u|^2$. Let $v \in H^2(\mathbb{R}^2)$ such that $\{u \neq v\} \subset \mathbb{D}_{p,r}$ for some disk $\mathbb{D}_{p,r}$, suppose without loss of generality that

$$E(v; \mathbb{D}_{p,r}) \leq E(u; \mathbb{D}_{p,r}),$$

otherwise the quasiminimality condition (6.1) is directly verified. Then

$$\begin{aligned} \left|1 - \int_{\mathbb{R}^2} |\nabla v|^2\right| &= \left|\int_{\mathbb{D}_{p,r}} |\nabla u|^2 - |\nabla v|^2\right| = \left|\int_{\mathbb{D}_{p,r}} (u - v)\Delta(u + v)\right| \\ &\leq \|u - v\|_{L^2(\mathbb{D}_{p,r})} \|\Delta(u + v)\|_{L^2(\mathbb{D}_{p,r})} \\ &\lesssim r^2 \|\Delta(u - v)\|_{L^2(\mathbb{D}_{p,r})} \|\Delta(u + v)\|_{L^2(\mathbb{D}_{p,r})} \text{ by Poincaré inequality} \\ &\lesssim r^2 \left(\|\Delta u\|_{L^2(\mathbb{D}_{p,r})}^2 + \|\Delta v\|_{L^2(\mathbb{D}_{p,r})}^2\right) \\ &\lesssim r^2 \left(\|\Delta u\|_{L^2(\mathbb{R}^2)}^2 + |\text{Spt}(u)|\right) \text{ since } E(v; \mathbb{D}_{p,r}) \leq E(u; \mathbb{D}_{p,r}). \end{aligned}$$

So for a small enough $r > 0$, we have $\int_{\mathbb{R}^2} |\nabla v|^2 \neq 0$. Then by the minimality of $u$ we obtain

$$\Lambda + |\text{Spt}(u)| \leq \frac{\Lambda + \int_{\mathbb{D}_{p,r}} (|\Delta v|^2 - |\Delta u|^2)}{1 + \int_{\mathbb{D}_{p,r}} (|\nabla v|^2 - |\nabla u|^2)} + |\text{Spt}(v)|.$$
This simplifies to

$$E(u; \mathbb{D}_{p,r}) - E(v; \mathbb{D}_{p,r}) \leq (\Lambda + |\mathrm{Spt}(u)| - |\mathrm{Spt}(v)|) \int_{\mathbb{D}_{p,r}} (|\nabla u|^2 - |\nabla v|^2).$$

We bound the last term:

$$\begin{aligned} \int_{\mathbb{D}_{p,r}} (|\nabla u|^2 - |\nabla v|^2) &= \int_{\mathbb{D}_{p,r}} \Delta(v+u)(v-u) \\ &\lesssim \|\Delta(v+u)\|_{L^2(\mathbb{D}_{p,r})} \|v-u\|_{L^2(\mathbb{D}_{p,r})} \\ &\lesssim (\|\Delta u\|_{L^2(\mathbb{R}^2)} + |\mathrm{Spt}(u)|^{\frac{1}{2}}) \|v-u\|_{L^2(\mathbb{D}_{p,r})} \quad \text{since } E(v; \mathbb{D}_{p,r}) \leq E(u; \mathbb{D}_{p,r}). \end{aligned}$$

And $\Lambda + |\mathrm{Spt}(u)| - |\mathrm{Spt}(v)| \leq \Lambda + \pi r^2$. This proves the result, for some sufficiently small $\bar{r}$. $\square$

**Corollary 69.** Let $\Omega \subset \mathbb{R}^2$ be a minimizer of $\Omega \mapsto \Lambda_1(\Omega)$ among open sets of area 1. Let $u_\Omega$ be a buckling eigenfunction associated to $\Lambda_1(\Omega)$. Then $\Omega$ is bounded and $u_\Omega$ satisfies the conclusion of Theorem 8; in particular for any $p \in \partial\Omega$ such that $\liminf_{r \to 0} \frac{|\Omega \cap \mathbb{D}_{p,r}|}{|\mathbb{D}_{p,r}|} < 1$, then there exists $r > 0$ such that $\mathbb{D}_{p,r} \cap \Omega$ is either a $C^{1,\alpha}$ curve, or two $C^{1,\alpha}$ curves meeting with an angle $t_1$.

*Proof.* Up to a dilation, we assume that $\Omega$ is a minimizer (among open sets of finite area) of $\Omega \mapsto \Lambda_1(\Omega) + |\Omega|$. We assume $u_\Omega$ is normalized by $\int_\Omega |\nabla u_\Omega|^2 = 1$. This means that $u_\Omega$ is a minimizer of $E(\cdot; \mathbb{R}^2)$ under the constraint $\int_{\mathbb{R}^2} |\nabla u_\Omega|^2 = 1$. By Lemma 68, there exists $\bar{r} > 0$ such that for every $p \in \mathbb{R}^2$ we have

$$(u_\Omega)_{p,\bar{r}} \in \mathcal{Q}\mathcal{M}(\mathbb{D}_1).$$

By the non-degeneracy Lemma 60, there exists $c > 0$ such that for any $p \in \mathbb{R}^2$, if $u(p) \neq 0$ then

$$\int_{\mathbb{D}_{p,\bar{r}}} (|\nabla u_{\Omega}|^2 + u_{\Omega}^2) \geq c.$$

Consider a sequence of disjoint disks $(\mathbb{D}_{p_i, \bar{r}})_{i=1,...,N}$ (for some $N \in \mathbb{N}^*$) such that $u_\Omega(p_i) \neq 0$ for every $i$, then applying the above inequality to each disk we obtain

$$\int_{\mathbb{R}^2} (|\nabla u_{\Omega}|^2 + |u_{\Omega}|^2) \geq Nc.$$

Thus the number $N$ of such disjoint disks is bounded from above, meaning that $\Omega$ must be a bounded set.

We now apply Theorem 8 to $(u_\Omega)_{\bar{r}}$, which gives the rest of the conclusions. Note that $\partial\Omega$ is, a priori, only a subset of $\partial\mathrm{Spt}(u_\Omega)$. $\square$

**Corollary 70.** Let $\Omega \subset \mathbb{R}^2$ be a minimizer of $\Omega \mapsto \Lambda_1(\Omega)$ among open sets of area 1. Let $u_\Omega$ be a buckling eigenfunction associated to $\Lambda_1(\Omega)$ and assume either that

$$\Omega \text{ is simply connected and for all } p \in \partial\Omega, \liminf_{r \to 0} \frac{|\mathbb{D}_{p,r} \cap \Omega|}{|\mathbb{D}_{p,r}|} < \frac{t_1}{2\pi} (\approx 0.715), \quad (6.12)$$

or

$$\Delta u_{\Omega} + \Lambda_{1}(\Omega) u_{\Omega} \geq 0 \text{ in } \Omega \text{ and for all } p \in \partial\Omega, \text{ we have } \liminf_{r \to 0} \frac{|\mathbb{D}_{p,r} \cap \Omega|}{|\mathbb{D}_{p,r}|} < 1. \quad (6.13)$$

Then $\Omega$ is a disk.
*Proof.* We first apply corollary 69: $\Omega$ is bounded, and $u_\Omega$ verifies the conclusion of Theorem 8.

If the first hypothesis (6.12) is verified, then we know that every point of $\partial\Omega$ belongs to $\mathcal{R}_{u_\Omega}$.
Since $\Omega$ is bounded and simply connected, then $\mathcal{R}_{u_\Omega}$ is connected. $\Delta u_\Omega$ is continuous up to the
boundary with values in $\{-1, +1\}$, so by the connectedness of the boundary it is constant. By
maximum principle, the (harmonic) function $\Delta u_\Omega + \Lambda_1(\Omega)u_\Omega$ is constant in $\Omega$: the step c) of
Weinberger and Willms's argument applies and $\Omega$ must be a disk.

If the second hypothesis (6.13) is verified, then every point of $\partial\Omega$ belongs to $\mathcal{R}_u \sqcup \mathcal{A}_u$. Assume there exists some $p \in \mathcal{A}_u$, then $\Delta u_\Omega + \Lambda_1(\Omega)u_\Omega$ must change sign in any neighbourhood of $p$. Indeed, the blow-up sequence $(u_\Omega)_{p,r}$ converges in $H^2(\mathbb{D}_1)$ to a homogeneous solution of type II, denoted $\hat{u}$, and $\Delta \hat{u} = \lim_{r\to 0} (\Delta u_{p,r} + r^2\Lambda u_{p,r})$ changes sign. This contradicts our hypothesis, so $\mathcal{A}_u$ must be empty and every point of $\partial\Omega$ is regular.

Thus $\partial\Omega \subset \mathcal{R}_{u_\Omega}$, $\Delta u_\Omega|_{\partial\Omega}$ extends continuously to $\partial\Omega$ with value $\pm 1$. Since $u_\Omega = 0$ on $\partial\Omega$ and $\Delta u_\Omega + \Lambda_1(\Omega)u_\Omega \ge 0$, then $\Delta u_\Omega + \Lambda_1(\Omega)u_\Omega = 1$ in $\partial\Omega$ and harmonic, so it is constant in $\Omega$. We conclude with step c) of Weinberger and Willms's argument described above: $\Omega$ is necessarily a disk.

<figure><img src="image_25.png" /></figure>

There are several obstacles to obtain a general proof of the optimality of the disk from this.
First is proving the hypothesis (6.12) would hold for a general minimizer ; either the simple con-
nectedness should be attained as a consequence of the optimality, from some unknown geometric
argument, or it could be considered as a constraint (in an attempt to prove the optimality of the
disk at least among simply connected sets). If it is considered as a constraint, then our main
Theorem 8 does not apply since we frequently use non-simply connected competitors (for instance
as early as in the proof of Lemma 11 and Lemma 14).

The density assumption is not obvious either ; although relying on heuristics in some parts,
the optimal set for the drag of Richardson in [31] shows that angular points may occur in optimal
shapes. Even assuming that we work among simply connected shapes, some geometric argument
specific to the buckling problem would be needed to prove that angular points do not occur for
minimizers. The same can be said for the assumption $\Delta u_{\Omega} + \Lambda_{1}(\Omega)u_{\Omega} \ge 0$ in (6.13).

Finally, even assuming that $\partial\Omega$ is connected, one would need to understand the behaviour of $\Delta u_\Omega|_{\partial\Omega}$ near junction point (here $\Delta u_\Omega|_{\partial\Omega}$ is expected to **not** change sign) and explosion points (which are expected to not exist).

**Acknowledgements:** This work was supported by the project ANR STOIQUES (ANR-24-CE40-2216) financed by the French Agence Nationale de la Recherche (ANR).

References

[1] H. W. Alt and L. A. Caffarelli. Existence and regularity for a minimum problem with free boundary. J. Reine Angew. Math., 325:105–144, 1981.

[2] S. Ashbaugh and Dorin Bucur. On the isoperimetric inequality for the buckling of a clamped plate. Z. Angew. Math. Phys., 54(5):756–770, 2003.

[3] Maitine Bergounioux and Yannick Privat. Shape optimization with stokes constraints over the set of axisymmetric domains. SIAM Journal on Control and Optimization, 51(1):599–628, 2013.

[4] J.-M. Bourot. On the numerical computation of the optimum profile in Stokes flow. J. Fluid Mech., 65:513–515, 1974.
[5] Dorin Bucur, Jimmy Lamboley, Mickaël Nahon, and Raphaël Prunier. Sharp quantitative stability of the dirichlet spectrum near the ball. *Communications on Pure and Applied Mathematics*, 2025.

[6] Giuseppe Buttazzo, Francesco Paolo Maiale, Dario Mazzoleni, Giorgio Tortone, and Bozhidar Velichkov. Regularity of the optimal sets for a class of integral shape functionals. *Archive for Rational Mechanics and Analysis*, 248(3):44, 2024.

[7] Luis A. Caffarelli, Henrik Shahgholian, and Karen Yeressian. A minimization problem with free boundary related to a cooperative system. *Duke Mathematical Journal*, 167(10):1825–1882, 2018.

[8] Constantin Carathéodory. Über die gegenseitige beziehung der ränder bei der konformen abbildung des inneren einer jordanschen kurve auf einen kreis. *Mathematische Annalen*, 73(2):305–320, 1913.

[9] Andrea Cianchi. Continuity properties of functions from Orlicz-Sobolev spaces and embedding theorems. *Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser.*, 23(3):575–608, 1996.

[10] Guy David and Tatiana Toro. Regularity of almost minimizers with free boundary. *Calculus of Variations and Partial Differential Equations*, 54:455–524, 2015.

[11] Serena Dipierro, Aram Karakhanyan, and Enrico Valdinoci. A free boundary problem driven by the biharmonic operator. *Pure Appl. Anal.*, 2(4):875–942, 2020.

[12] Charles Louis Fefferman and Elias M. Stein. $H^p$ spaces of several variables. *Acta Math.*, 129:137–193, 1972.

[13] Hans-Christoph Grunau and Marius Müller. A biharmonic analogue of the Alt-Caffarelli problem. *Math. Ann.*, 390(4):5259–5297, 2024.

[14] Hans-Christoph Grunau and Marius Müller. Global minimizers for a two-sided biharmonic Alt-Caffarelli problem. in preparation.

[15] Qing Han and Fanghua Lin. Elliptic partial differential equations, volume 1 of Courant Lect. Notes Math. New York, NY: Courant Institute of Mathematical Sciences; Providence, RI: American Mathematical Society (AMS), 2nd ed. edition, 2011.

[16] Fritz John and Louis Nirenberg. On functions of bounded mean oscillation. *Communications on pure and applied Mathematics*, 14(3):415–426, 1961.

[17] Bernhard Kawohl, Olivier Pironneau, Luc Tartar, Jean-Paul Zolésio, and Bernd Kawohl. Some nonconvex shape optimization problems. *Optimal Shape Design: Lectures given at the joint CIM/CIME Summer School held in Tróia, Portugal, June 1–6, 1998*, pages 7–46, 2000.

[18] D. Kinderlehrer and Louis Nirenberg. Regularity in free boundary problems. *Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser.*, 4:373–391, 1977.

[19] D. Kinderlehrer, Louis Nirenberg, and J. Spruck. Regularity in elliptic free boundary problems. I. *J. Anal. Math.*, 34:86–119, 1978.

[20] D. Kinderlehrer, Louis Nirenberg, and J. Spruck. Regularity in elliptic free boundary problems. II: Equations of higher order. *Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser.*, 6:637–683, 1979.
[21] Dennis Kriventsov and Fanghua Lin. Regularity for shape optimizers: the nondegenerate case. *Communications on Pure and Applied Mathematics*, 71(8):1535-1596, 2018.

[22] Jimmy Lamboley. About Hölder-regularity of the convex shape minimizing $\lambda_2$. *Appl. Anal.*, 90(1-2):263-278, 2011.

[23] Francesco Paolo Maiale, Giorgio Tortone, Bozhidar Velichkov, et al. Epsilon-regularity for the solutions of a free boundary system. *Rev. Mat. Iberoam*, 39(5):1947-1972, 2023.

[24] Dario Mazzoleni, Susanna Terracini, and Bozhidar Velichkov. Regularity of the optimal sets for some spectral functionals. *Geom. Funct. Anal.*, 27(2):373-426, 2017.

[25] Dario Mazzoleni, Susanna Terracini, and Bozhidar Velichkov. Regularity of the free boundary for the vectorial bernoulli problem. *Analysis & PDE*, 13(3):741-764, 2020.

[26] Olivier Pironneau. On optimum profiles in stokes flow. *Journal of Fluid Mechanics*, 59(1):117-128, 1973.

[27] George Pólya and Gábor Szegö. Isoperimetric inequalities in mathematical physics. (am-27). 1951.

[28] Christian Pommerenke. Boundary behaviour of conformal maps, volume 299 of *Grundlehren Math. Wiss*. Berlin: Springer-Verlag, 1992.

[29] Yannick Privat. *Quelques problèmes d'optimisation de formes en sciences du vivant*. PhD thesis, Université Henri Poincaré-Nancy 1, 2008.

[30] Ernst R Reifenberg. An epiperimetric inequality related to the analyticity of minimal surfaces. *Annals of Mathematics*, 80(1):1-14, 1964.

[31] S Richardson. Optimum profiles in two-dimensional stokes flow. *Proceedings of the Royal Society of London. Series A: Mathematical and Physical Sciences*, 450(1940):603-622, 1995.

[32] Luca Spolaor and Bozhidar Velichkov. An epiperimetric inequality for the regularity of some free boundary problems: The 2-dimensional case. *Communications on Pure and Applied Mathematics*, 72(2):375-421, 2019.

[33] Kathrin Stollenwerk. Optimal shape of a domain which minimizes the first buckling eigenvalue. *Calculus of Variations and Partial Differential Equations*, 55(1):5, 2016.

[34] Gábro Szegö. On membranes and plates. *Proceedings of the National Academy of Sciences*, 36(3):210-216, 1950.

[35] Giorgio Tortone and Bozhidar Velichkov. Vectorial Bernoulli Problems and Free Boundary Systems. *arXiv preprint arXiv:2510.09875*, 2025.

[36] Bozhidar Velichkov. Regularity of the one-phase free boundaries. Springer Nature, 2023.

[37] Stefan Warschawski. Über das randverhalten der ableitung der abbildungsfunktion bei konformer abbildung. *Mathematische Zeitschrift*, 35(1):321-456, 1932.

[38] Georg S. Weiss. Partial regularity for weak solutions of an elliptic free boundary problem. *Commun. Partial Differ. Equations*, 23(3-4):439-455, 1998.

[39] Georg S. Weiss. A homogeneity improvement approach to the obstacle problem. *Invent. Math.*, 138(1):23-50, 1999.

[40] Georg Sebastian Weiss. Partial regularity for a minimum problem with free boundary. *J. Geom. Anal.*, 9(2):317-326, 1999.
Affiliations

JIMMY LAMBOLEY

École Normale Supérieure, CNRS, PSL University, DMA and Sorbonne Université, Université Paris Cité, IMJ-
PRG, F-75005 Paris, France

*E-mail address: jimmy.lamboley@ens.fr*

MICKAËL NAHON

Univ. Grenoble Alpes, CNRS, Grenoble INP, LJK, 38000 Grenoble, France.

*E-mail address: mickael.nahon@univ-grenoble-alpes.fr*