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# Curvature penalization of strongly anisotropic interfaces models and their phase-field approximation

Jean-François Babadjian, Blanche Buet, Michael Goldman

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Jean-François Babadjian, Blanche Buet, Michael Goldman. Curvature penalization of strongly anisotropic interfaces models and their phase-field approximation. 2025. hal-05311891

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Distributed under a Creative Commons Attribution 4.0 International License
CURVATURE PENALIZATION OF STRONGLY ANISOTROPIC INTERFACES
MODELS AND THEIR PHASE-FIELD APPROXIMATION

J.-F. BABADJIAN, B. BUET AND M. GOLDMAN

ABSTRACT. This paper studies the effect of anisotropy on sharp or diffuse interfaces models. When the surface tension is a convex function of the normal to the interface, the anisotropy is said to be weak. This usually ensures the lower semicontinuity of the associated energy. If, however, the surface tension depends on the normal in a nonconvex way, this so-called strong anisotropy may lead to instabilities related to the lack of lower semicontinuity of the functional. We investigate the regularizing effects of adding a higher order term of Willmore type to the energy. We consider two types of problems. The first one is an anisotropic nonconvex generalization of the perimeter, and the second one is an anisotropic nonconvex Mumford-Shah functional. In both cases, lower semicontinuity properties of the energies with respect to a natural mode of convergence are established, as well as $\Gamma$-convergence type results by means of a phase field approximation. In comparison with related results for curvature dependent energies, one of the original aspects of our work is that, in the context of free discontinuity problems, we are able to consider singular structures such as crack-tips or multiple junctions.

CONTENTS

<table><tr><td>1. Introduction</td><td>1</td></tr><tr><td>2. Preliminaries</td><td>6</td></tr><tr><td>2.1. Radon measures</td><td>6</td></tr><tr><td>2.2. Rectifiable sets</td><td>7</td></tr><tr><td>2.3. Sets of finite perimeter</td><td>7</td></tr><tr><td>2.4. Varifolds</td><td>7</td></tr><tr><td>2.5. Curves</td><td>9</td></tr><tr><td>2.6. Functions of bounded variation</td><td>11</td></tr><tr><td>3. Strongly anisotropic perimeters with curvature</td><td>11</td></tr><tr><td>3.1. Lower semi-continuity</td><td>11</td></tr><tr><td>3.2. Phase field approximation</td><td>13</td></tr><tr><td>4. Strongly anisotropic Mumford-Shah functional with curvature</td><td>18</td></tr><tr><td>4.1. Lower semicontinuity</td><td>22</td></tr><tr><td>4.2. Phase field approximation</td><td>23</td></tr><tr><td>5. Appendix: the Gauss-Bonnet formula for varifolds</td><td>36</td></tr><tr><td>5.1. Lusin property of the Gauss map</td><td>40</td></tr><tr><td>5.2. Some technical results</td><td>45</td></tr><tr><td>Acknowledgments</td><td>47</td></tr><tr><td>References</td><td>47</td></tr></table>

1. INTRODUCTION

Anisotropic sharp interface models involve a surface tension density depending on the orientation of the unknown interface. It thus contains preferable directions, leading to crystalline surfaces due to the formation of facets, corner or wrinkling. When the surface tension is a convex function of the normal to the interface, the anisotropy is said to be weak, and the associated variational problem or the underlying PDE is in general well-behaved. If however, the anisotropy is strong, which means that the surface tension is a non convex function of the normal, the problem becomes unstable since several directions might be forbidden. From a mathematical standpoint, it corresponds to a loss of ellipticity of the underlying set of PDEs, or a lack of lower semicontinuity of the associated energy. It naturally arises in many applications such as phase transitions or epitaxial
growth [23, 19, 51, 39]. In the context of fracture mechanics, tearing experiments on brittle thin sheets with strongly anisotropic surface energy lead to observed crack path which happen to be inconsistent with those analytically predicted (see e.g. [27] and references therein).

From a variational point of view, which is the one adopted in the present work, a generic and formal way to formulate strongly anisotropic interfacial problems consists in minimizing an energy functional of the form

$$ \Gamma \mapsto \mathcal{E}(\Gamma) := \int_{\Gamma} \varphi(\nu_{\Gamma}) d\mathcal{H}^{d-1}, $$

among all possible $(d-1)$-dimensional hypersurfaces $\Gamma \subset \Omega$ (say without boundary) subjected to suitable external constraints. In the previous expression, $\Omega \subset \mathbf{R}^d$ is a bounded open set with $d=2$, $3$, $\mathcal{H}^{d-1}$ stands for the $(d-1)$-dimensional Hausdorff measure, $\nu_{\Gamma}$ is a normal vector to $\Gamma$ (which is well defined provided $\Gamma$ is smooth enough) and $\varphi: \mathbf{R}^d \to \mathbf{R}^+$ is a continuous, 1-homogeneous, even but not necessarily convex surface tension. Minimizing this kind of interfacial energies might lead to the formation of microstructures due to fast oscillations of minimizing sequences $\{\Gamma_n\}_{n \in \mathbb{N}}$. Indeed, it might be energetically convenient to allow the normals $\nu_{\Gamma_n}$ to highly oscillate between finitely many fixed directions, in such a way that the sequence of sets $\{\Gamma_n\}_{n \in \mathbb{N}}$ 'weakly converges' to a limit (generalized) set $\Gamma$ whose energy is strictly lower than the value of the infimum. This is a typical behavior of variational problems lacking lower semicontinuity.

A possible remedy consists in relaxing the original energy $\mathcal{E}$, i.e., replacing it by its lower semicontinuous envelope $\overline{\mathcal{E}}$ with respect to a natural topology for which the energy is coercive. In the formal framework described above, it consists in replacing $\varphi$ by its convex envelope $\varphi^{**}$ (see e.g. [28] when $\Gamma$ is the boundary of a set) so that

$$ \overline{\mathcal{E}}(\Gamma) = \int_{\Gamma} \varphi^{**}(\nu_{\Gamma}) d\mathcal{H}^{d-1}. $$

The so-called relaxed problem now becomes well-posed in a suitable mathematical framework, in the sense that minimizing sequences for $\mathcal{E}$ converges to (generalized) solutions of $\overline{\mathcal{E}}$ and $\inf \mathcal{E} = \min \overline{\mathcal{E}}$. From a mathematical point of view, the relaxation procedure is perfectly satisfactory because high oscillations of minimizing sequences are ruled out. However, it has the drawback of modifying the underlying physics by suppressing the possible nucleation of new facets orientated within the nonconvex region.

Another possibility is to add a higher order term of curvature type (the so-called Willmore energy). This penalizes spatial oscillations and regularizes the problem. It leads to a regularized energy functional as in [23]

$$ \Gamma \mapsto \mathcal{E}^{(\lambda)}(\Gamma) := \int_{\Gamma} \varphi(\nu_{\Gamma}) d\mathcal{H}^{d-1} + \lambda \int_{\Gamma} |H_{\Gamma}|^2 d\mathcal{H}^{d-1}, $$

where $\lambda > 0$ is a (fixed) weight parameter, and $H_{\Gamma} = (\mathrm{div}^{\Gamma} \nu_{\Gamma}) \nu_{\Gamma}$ is the mean curvature vector of $\Gamma$. Although not obvious, it is expected that the presence of the higher order term $H_{\Gamma}$ improves the compactness properties of minimizing sequences $\{\Gamma_n\}_{n \in \mathbb{N}}$ for $\mathcal{E}^{(\lambda)}$, so that $\Gamma_n$ should now converge in a stronger way to some (generalized) set $\Gamma$, making it possible to pass to the limit in the nonconvex term, without appealing to any type of relaxation. Let us point out that the behavior of (volume-constrained) minimizers for small $\lambda$ has been investigated in [50, 11, 34, 35]. See also [22] for the application of this idea in a different context.

The first objective of the present work is to investigate the lower semicontinuity properties of this type of problems when $\Gamma = \partial E$ is the boundary of a set $E$. Set

$$ \mathcal{F}(E) := \mathcal{E}^{(1)}(\partial^* E) = \int_{\partial^* E} \varphi(\nu_E) d\mathcal{H}^{d-1} + \int_{\partial^* E} |H_E|^2 d\mathcal{H}^{d-1}. \quad (1.1) $$

Postponing precise definitions to the next sections, our first main result is the following lower semi-continuity result, see Theorem 1.1.

**Theorem 1.1.** Let $\{E_n\}_{n \in \mathbb{N}}$ be a sequence of sets of finite perimeter in $\Omega$ converging in $L^1$ to a set $E$ of finite perimeter in $\Omega$. If $V_E$ has bounded first variation in $L^2_{\mu_E}(\Omega; \mathbf{R}^d)$, then

$$ \liminf_{n \to \infty} \mathcal{F}(E_n) \geq \mathcal{F}(E). $$
**Remark 1.2.** Let us point out that this result, just like Theorem 1.3 and Theorem 1.4, comes with a compactness statement which is transparent from the proof. However since we need to assume a priori some regularity of the limit set we do not include it.

Another important question for such sharp interface variational problems is the approximation by means of phase field models. This has a long history, going back to [33] where a so-called Modica-Mortola (or Allen-Cahn) approximation is proposed to approximate the classical perimeter functional in the sense of $\Gamma$-convergence. Regarding the variational approximation of the sum of the isotropic perimeter and Willmore functionals, a phase field approximation result has been established in [43, 38] answering a long standing conjecture of De Giorgi (see also [28] in the case of weakly anisotropic surface energies). Following [51, 28], we investigate here the case of strongly anisotropic surface energies. Let $\varphi_{\varepsilon}$ be a suitable smooth approximation of $\varphi$, see (3.6). We introduce the phase field energy functional $\mathcal{F}_{\varepsilon}: H^2(\Omega) \to \mathbf{R}$ by

$$ \mathcal{F}_{\varepsilon}(v) = \frac{1}{c_0} \int_{\Omega} \varphi_{\varepsilon}(\nabla v) \left( \frac{\varepsilon}{2} |\nabla v|^2 + \frac{1}{\varepsilon} W(v) \right) dx + \frac{1}{c_0} \int_{\Omega} \frac{1}{\varepsilon} \left( -\varepsilon \Delta v + \frac{1}{\varepsilon} W'(v) \right)^2 dx. \quad (1.2) $$

Here $W(z) = (1-z^2)^2$ is the standard double-well potential and $c_0 = \int_{-1}^1 \sqrt{2W(z)} dz$. In the absence of the second term, and if the surface tension $\varphi \equiv 1$, the first contribution of the energy corresponds exactly to the classical Modica-Mortola $\Gamma$-convergence approximation of the perimeter functional. It is established in [28] that the first term alone gives rise in the limit to the relaxed anisotropic perimeter

$$ \int_{\partial^* E} \varphi^{**}(\nu_E) d\mathcal{H}^{d-1}, $$

where $\varphi^{**}$ is the convex hull of $\varphi$. The second contribution in the energy functional $\mathcal{F}_{\varepsilon}$ is exactly the same as in [43]. It corresponds to a phase field approximation of the Willmore energy

$$ \int_{\partial^* E} |H_E|^2 d\mathcal{H}^{d-1}. $$

Again in [28], it is established that if $\varphi$ is convex, $\mathcal{F}_{\varepsilon}$ $\Gamma$-converges to $\mathcal{F}$ provided $\partial E$ is $\mathcal{C}^2$.

Let us point out that following [51], in the first term of $\mathcal{F}_{\varepsilon}$, the anisotropy is carried out by multiplying the Modica-Mortola energy density by $\varphi_{\varepsilon}(\nabla v)$. It is argued in [51] that this is crucial in order to have an interfacial layer independent of the orientation which is then compatible with the optimal profile of the curvature term. This is in contrast with other phase field models such as [41, 52]. See also [14] for a thorough review of phase field approximations of curvature energies.

Our second main result, Theorem 1.3, corresponds to a generalization of the result [28] to general anisotropies $\varphi$.

**Theorem 1.3.** Let $d=2$, or $d=3$, let $\Omega$ be a bounded open subset of $\mathbf{R}^d$ and let $E$ be a set of finite perimeter in $\Omega$ such that $V_E$ has bounded first variation in $L_{\mu_E}^2(\Omega; \mathbf{R}^d)$.

(i) Lower bound: If $\{v_{\varepsilon}\}_{\varepsilon>0} \subset H^2(\Omega)$ is such that $v_{\varepsilon} \to \mathbf{1}_E - \mathbf{1}_{\Omega \setminus E}$ in $L^1(\Omega)$, then

$$ \mathcal{F}(E) \leq \liminf_{\varepsilon \to 0} \mathcal{F}_{\varepsilon}(v_{\varepsilon}). $$

(ii) Upper bound: If additionally $\partial E$ is a $\mathcal{C}^2$-hypersurface, there exists a sequence $\{\bar{v}_{\varepsilon}\}_{\varepsilon>0}$ in $H^2(\Omega)$ such that $\bar{v}_{\varepsilon} \to \mathbf{1}_E - \mathbf{1}_{\Omega \setminus E}$ in $L^1(\Omega)$ and

$$ \mathcal{F}(E) = \lim_{\varepsilon \to 0} \mathcal{F}_{\varepsilon}(\bar{v}_{\varepsilon}). $$

For both Theorem 1.1 and Theorem 1.3, we strongly rely on the existing results on lower semi-continuity and phase field approximations [48, 43] which are naturally expressed in the language of varifolds. With these results at hand, the proofs of both theorems are quite short (and might be seen as warm ups for the free-discontinuity counterparts, see below). The main observation we are making and which seems to have remained unnoticed so far, see e.g. [28], is that varifold convergence together with rectifiability of the limit imply convergence of the anisotropic perimeter term. This is the analog of the property that for sequences of functions, strong convergence is equivalent to the fact that the generated Young measures are Dirac masses, see [42]. The lower bound in Theorem 1.3 rests on a locality property for the mean curvature of varifolds in order to compare the curvature of $\partial E$ with the curvature of the limit varifold (which has natural semi-continuity properties), see [4, 48, 26]. Moreover, as can already be seen for $d=2$ in [7], the relaxation of $\mathcal{F}$ must be non-local and a good understanding of its structure for $d=3$ is still
missing. The restriction on the regularity on $\partial E$ in the upper bound of Theorem 1.3 comes from the difficulty of approximating varifolds in energy by smooth sets. This is a common limitation in this setting, see [43, 32]. See however [44] for some recent progress on this question.

The second type of sharp interface models with curvature penalization we are interested in, are motivated by applications to fracture mechanics involving brittle materials with strongly anisotropic surface energies, see [27].

In a two-dimensional antiplane setting ($d = 2$), the variational approach to fracture introduced in [20, 9] is based on the global minimization of a Mumford-Shah type functional

$$u \mapsto MS(u) := \int_{\Omega \setminus J_u} |\nabla u|^2 dx + \int_{J_u} \varphi(\nu_{J_u}) d\mathcal{H}^1, \quad (1.3)$$

where $u \in SBV^2(\Omega)$ is a scalar function corresponding to an antiplane displacement, and the crack is assimilated to the jump set $J_u$ of $u$. In the isotropic case, $\varphi \equiv 1$, we recover the classical Mumford-Shah energy. Under external loadings and/or boundary conditions, the direct method in the calculus of variations easily leads to the existence of minimizers for $MS$ in the case of weak anisotropy, i.e. if $\varphi$ is a convex function. In addition, a phase field approximation result due to Ambrosio-Tortorelli has been established in [5, 18].

If however, $\varphi$ is nonconvex, the problem might develop spatial oscillations as explained above. Therefore, the existence of minimizers for $MS$ and the variational approximation by means of an Ambrosio-Tortorelli type functional is not guaranteed anymore because of a lack of lower semi-continuity of $MS$. Once more, a possible remedy is to penalize spatial oscillations by introducing a curvature term inside the free energy. Contrary to the previously described problem, the crack $J_u$ has no reason to be a boundaryless 1-dimensional set so that one has to account both for possible endpoints (the so-called crack-tips) as well as multiple junctions in $J_u$. The following curvature dependent Mumford-Shah functionals has been introduced in [15, 12] in the context of image reconstruction (in the isotropic case),

$$\hat{\mathcal{G}}^{(\gamma)}(u, C, P) := \int_{\Omega \setminus C} |\nabla u|^2 dx + \int_C (\varphi(\nu_C) + |H_C|^2) d\mathcal{H}^1 + \gamma \mathcal{H}^0(P). \quad (1.4)$$

Here $\gamma > 0$ is a weight parameter, $C$ is a finite union of curves with endpoints $P$ and $u \in H^1(\Omega \setminus C)$. Notice in particular that this implies that if we consider $u \in SBV^2(\Omega)$, then $J_u \subset C$. This formulation is closer in spirit to the original strong formulation of the Mumford-Shah functional in [37] than its weak counterpart (1.3). Existence of minimizers for $\hat{\mathcal{G}}^{(\gamma)}$ (with an extra fidelity term) has been obtained in [15] and a phase field approximation has been proposed in [12]. However, as opposed to the phase field approximation in (1.2), the curvature was approximated by the more nonlinear term $\text{div}(\frac{\nabla v}{|\nabla v|})$. While this greatly simplifies the analysis, the question of an approximation through a phase field model more similar to (1.2) was raised in [12, Section 7]. One of the main objectives of this paper is to answer positively this question. Since we will build upon [48, 43], it is more natural for us to work in the SBV framework. We thus introduce the sharp interface energy,

$$\mathcal{G}^{(\gamma)}(u) := \int_{\Omega \setminus J_u} |\nabla u|^2 dx + \int_{J_u} (\varphi(\nu_{J_u}) + |H_{J_u}|^2) d\mathcal{H}^1 + \gamma \mathcal{H}^0(\mathcal{P}_{J_u}),$$

We will restrict ourselves to functions $u \in SBV^2(\Omega)$ with a jump set $J_u$ made of a finite family of curves. Here $\mathcal{P}_{J_u}$ denotes the finite set of endpoints of the curves in $J_u$ (counting both multiple junctions and true endpoints). We first prove in Theorem 4.4 below a lower semicontinuity result for $\mathcal{G}^{(\gamma)}$. To some extent it can be seen as an alternative proof, but under a more restrictive set of assumptions, of the results from [15, 12]. While we include it mostly for presentation purposes, as it illustrates in a simpler setting some of the ideas behind the proof of the lower bound for the phase field approximation, let us make the following observation. Since, in the notation of (1.4) we assume that $J_u = C$, such a lower semicontinuity property can only hold provided the coefficient $\gamma$ is small enough (as in the statement of Theorem 4.4). Here smallness depends only on the smoothness of $J_u \setminus \mathcal{P}_{J_u}$, the mutual distances between the points in $\mathcal{P}_{J_u}$ as well as the distance of $\mathcal{P}_{J_u}$ from $\partial\Omega$. Indeed, if two points in $\mathcal{P}_{J_u}$ are very close to each other, it can be energetically favorable to add a small jump set connecting these points (and thus paying the distance between them) rather than having two crack-tips (which come with a fixed cost of 2). In the language of
(1.4) this amounts to $C \neq J_u$. This is somewhat related to the relaxation phenomenon for the elastica studied in [7] and already mentioned above. Besides [48] the main ingredient for this lower semicontinuity results is Lemma 4.3 which states that if a varifold with curvature in $L^2$ contains a set with singular curvature, then either its length or its Willmore energy should be bounded from below. Interestingly, the proof relies mostly on the monotonicity formula even though the energy we consider is anisotropic.

We present now the phase field approximation of $G^{(\gamma)}$ that we consider. It is an anisotropic variant of the one proposed in [12, Section 7]. The idea is to have a first phase field variable $v$ as in the Ambrosio-Tortorelli model and a second phase field variable $w$ to approximate the point energy $H^0(P_{J_u})$. We first point out that as observed in [12], since the optimal profile of the classical Ambrosio-Tortorelli functional is not $H^2$, it is not suited for the approximation of curvature energies. Following [12], we thus consider a Modica-Mortola approximation with a penalization of the size of the set $\{v = -1\}$. In the language of (1.1) this amounts to require that $|E| \to 0$. Another possible solution would be to detect the jump set of $u$ through the derivative of the optimal profile for the Modica-Mortola energy as very recently suggested in [13]. Let us now discuss the point energy. For this we repeat the heuristics from [11, 12]. For every $\beta > 0$, let us consider the functional

$$ \mathcal{F}^{\beta}(E) = \int_{\partial E} (\beta^{-1} + \beta|H_E|^2)d\mathcal{H}^1. $$

By the Cauchy-Schwarz inequality and Gauss-Bonnet formula, we have

$$ \mathcal{F}^{\beta}(E) \geq 2 \int_{\partial E} |H_E| d\mathcal{H}^1 \geq 4\pi \quad (1.5) $$

with equality if and only if $E$ is a ball of radius $\beta$. We thus consider for $\beta_{\varepsilon} \to 0$ a phase field approximation of $\mathcal{F}^{\beta_{\varepsilon}}$ as in (1.2) and set

$$ G_{\varepsilon,\beta}(w) = \frac{1}{c_0\beta} \int_{\Omega} \left( \frac{\varepsilon}{2} |\nabla w|^2 + \frac{1}{\varepsilon} W(w) \right) dx + \frac{\beta}{c_0\varepsilon} \int_{\Omega} \left( -\varepsilon \Delta w + \frac{1}{\varepsilon} W'(w) \right)^2 dx. $$

For all $(u, v, w) \in H^1(\Omega) \times [H^2(\Omega)]^2$, we then introduce the following energy functional

$$ \begin{aligned} G_{\varepsilon}^{(\gamma)}(u, v, w) = & \frac{1}{4} \int_{\Omega} (1+v)^2 |\nabla u|^2 dx + \frac{1}{2c_0} \int_{\Omega} \varphi_{\varepsilon} (\nabla v) \left( \frac{\varepsilon}{2} |\nabla v|^2 + \frac{1}{\varepsilon} W(v) \right) dx \\ & + \frac{1}{8c_0} \int_{\Omega} \frac{1}{\varepsilon} \left( -\varepsilon \Delta v + \frac{1}{\varepsilon} W'(v) \right)^2 (1+w)^2 dx \\ & + \frac{\gamma}{4\pi} G_{\varepsilon, \beta_{\varepsilon}}(w) + \frac{1}{\eta_{\varepsilon}} \int_{\Omega} (1-v)^2 dx + \frac{1}{\eta_{\varepsilon}} \int_{\Omega} (1-w)^2 dx, \end{aligned} \quad (1.6) $$

where $\varepsilon > 0$, $\eta_{\varepsilon} > 0$ and $\beta_{\varepsilon} > 0$ are infinitesimal parameters which satisfy the following scaling laws as in [12]

$$ \lim_{\varepsilon \to 0} \frac{\varepsilon |\log \varepsilon|}{\beta_{\varepsilon}} = \lim_{\varepsilon \to 0} \frac{\beta_{\varepsilon}}{\eta_{\varepsilon}} = 0. $$

Notice that just like $\nabla u$ is not penalized in the regions where $v = -1$ (as in the classical Ambrosio-Tortorelli approximation), the curvature of $J_u$, which is approximated by the term on the second line of (1.6), is not penalized in the regions where $w = -1$. This is essential in order to approximate jump sets with crack tips or multiple junctions.

As in the case of the perimeter, we need to assume some regularity of the admissible jump sets $J_u$. We assume here that $J_u$ is the finite disjoint union of $C^2$ curves possibly intersecting at their endpoints $P_{J_u}$. At each of these end points, $J_u$ has an infinite curvature. We further assume that at multiple junctions the sum of the tangent vectors vanishes. This means that the generalized curvature (in the sense of varifolds) has a Dirac mass at all points of $P_{J_u}$. Unfortunately, the case of multiple junctions with zero curvature is more delicate and remains out of reach at the moment, see Remark 4.1. Our main result is then (see Theorem 1.4 and (2.11) for the definition of $\mathcal{A}(\Omega)$)

**Theorem 1.4.** Let $u \in \mathcal{A}(\Omega)$.

(i) **Lower bound.** There exists $\gamma_0 = \gamma_0(u) > 0$ such that for all $\gamma \in (0, \gamma_0)$ the following property is satisfied: for all $\{(u_{\varepsilon}, v_{\varepsilon}, w_{\varepsilon})\}_{\varepsilon>0} \subset H^1(\Omega) \times [H^2(\Omega)]^2$ such that $(u_{\varepsilon}, v_{\varepsilon}, w_{\varepsilon}) \to (u, 1, 1)$ in $[L^1(\Omega)]^3$ we have

$$ G^{(\gamma)}(u) \leq \liminf_{\varepsilon \to 0} G_{\varepsilon}^{(\gamma)}(u_{\varepsilon}, v_{\varepsilon}, w_{\varepsilon}). $$
(ii) **Upper bound.** There exists a sequence $\{(\bar{u}_\epsilon, \bar{v}_\epsilon, \bar{w}_\epsilon)\}_{\epsilon>0}$ in $H^1(\Omega) \times [H^2(\Omega)]^2$ such that $(\bar{u}_\epsilon, \bar{v}_\epsilon, \bar{w}_\epsilon) \to (u, 1, 1)$ in $[L^1(\Omega)]^3$,

$$G^{(\gamma)}(u) = \lim_{\epsilon \to 0} G_{\epsilon}^{(\gamma)}(\bar{u}_{\epsilon}, \bar{v}_{\epsilon}, \bar{w}_{\epsilon}).$$

The restriction on $\gamma$ in Theorem 1.4 has its root in the use of Lemma 4.3 just as in the derivation of Theorem 4.4, see the discussion above. The main idea in the proof of Theorem 1.4 is contained in Proposition 4.5. It states that if the phase field variable $w_\epsilon$ is not uniformly close to 1 (say in a ball $B$), then the phase field energy involving $w_\epsilon$ inside $B$

$$G_{\epsilon, \beta_{\epsilon}}(w_{\epsilon}, B) = \frac{1}{c_0 \beta_{\epsilon}} \int_{B} \left( \frac{\epsilon}{2} |\nabla w_{\epsilon}|^2 + \frac{1}{\epsilon} W(w_{\epsilon}) \right) dx + \frac{\beta_{\epsilon}}{c_0 \epsilon} \int_{B} \left( -\epsilon \Delta w_{\epsilon} + \frac{1}{\epsilon} W'(w_{\epsilon}) \right)^2 dx$$

must be bounded from below by $4\pi$. This is sharp as seen from (1.5). The difficulty actually lies in extending the geometric argument leading to (1.5) to the phase field energy $G_{\epsilon, \beta}$. The argument rests on a blow-up argument around a point $x_0 \in \Omega$ such that $w_\epsilon(x_0)$ is far away from 1. Using the results of [43], one shows that

$$\liminf_{\epsilon \to 0} G_{\epsilon, \beta_{\epsilon}}(w_{\epsilon}, B) \ge 2 \int_{\mathbb{R}^2} |H_V| d\mu_V, \quad (1.7)$$

for some 1-rectifiable integral varifold $V$ in $\mathbb{R}^2$, with square-integrable first variation. Using the monotonicity formula as in [40], it is not hard to bound the right-hand side from below by 4 which is unfortunately not sharp as already noticed in [40]. This is related to derivation of Li-Yau inequalities for varifolds through monotonicity formula which are sharp for $d=3$ but not for $d=2$, see [25, 40, 36]. Let us point out that in order to prove that the limit varifold $V$ is non-trivial, we use in Step 2 of the proof of Proposition 4.5 a variant of the argument from [40] but at the $\epsilon$-level. Going back to the optimal lower bound in (1.7) the question is thus to generalize the Gauss-Bonnet formula for 1-rectifiable integral varifolds in $\mathbb{R}^2$ with square-integrable first variation. Although such objects are expected to enjoy good geometric properties (see e.g. [2] where it is proved that 1-rectifiable, integral stationary varifolds are a locally finite sum of segments with constant multiplicities inside each of the segments), the generalization of the Gauss-Bonnet formula to varifolds seems to be a serious issue. In the 2014 preprint [30], a sharp lower bound on the mean curvature for $m$-rectifiable integral varifolds in $\mathbb{R}^n$ with $p$-integrable first variation is obtained under the critical assumption $2 \le p = m = n-1$ (see [29, Theorem 24.1]). Unfortunately, our case does not fit within this set of assumptions since $p=2$, $n=2$ and $m=1$. For that reason, we dedicate an Appendix in which we derive the Gauss-Bonnet formula (Theorem 5.1) needed for our purpose by closely following and adapting the arguments from [30] to our specific situation. The main point is to prove a Lusin type property of the Gauss map, see Proposition 5.3 and [45, 46] for related results. Since our goal is much more modest than in [30] (restricting in particular to 1-dimensional varifolds), our proofs are often shorter and the notation lighter than those from [30]. We hope that our Appendix could also help shed light on [30] which seems to have remain relatively unnoticed, see e.g. [40].

The paper is organized as follows. In Section 2, we introduce notation and the mathematical background needed for our subsequent analysis. Section 3 is devoted to the study of strongly anisotropic perimeters with curvature penalization. Theorem 1.1 provides a lower semicontinuity property of the functional $\mathcal{F}$, while Theorem 1.3 shows a $\Gamma$-convergence type approximation result by means of the phase field energy $\mathcal{F}_\epsilon$. In Section 4, we carry out the analysis of the strongly anisotropic Mumford-Shah energy with curvature penalization. Theorem 4.4 shows a lower semicontinuity result of $G^{(\gamma)}$ for small enough weight parameter $\gamma > 0$. Theorem 1.4 gives a $\Gamma$-convergence type results of $G^{(\gamma)}$ by means of the phase field energy $G_{\epsilon}^{(\gamma)}$. Eventually, the Appendix is devoted to show the generalized Gauss-Bonnet formula, Theorem 5.1, for varifolds.

## 2. PRELIMINARIES

2.1. **Radon measures.** The Lebesgue measure in $\mathbb{R}^d$ is denoted by $\mathcal{L}^d$, and the $k$-dimensional Hausdorff measure by $\mathcal{H}^k$. We will write $\omega_k$ for the $\mathcal{L}^k$-measure of the $k$-dimensional unit ball in $\mathbb{R}^k$.
Let $X$ be a locally compact metric space and $Y$ be an Euclidean space. We denote by $\mathcal{M}(X; Y)$ (or simply $\mathcal{M}(X)$ if $Y = \mathbf{R}$) the space of bounded Radon measures in $X$. By the Riesz representation Theorem, $\mathcal{M}(X; Y)$ can be identified with the topological dual of $\mathcal{C}_0(X; Y)$, the space of all continuous functions $f: X \to Y$ such that $\{|f| \ge \varepsilon\}$ is compact for all $\varepsilon > 0$. Similarly, the space of Radon measures $\mathcal{M}_{\text{loc}}(X; Y)$ (or simply $\mathcal{M}_{\text{loc}}(X)$ if $Y = \mathbf{R}$) is identified with the topological dual of the space $\mathcal{C}_c(X; Y)$ of all continuous functions $f: X \to Y$ compactly supported in $X$.

Let $\Omega \subset \mathbf{R}^d$ be an open set, $\mu \in \mathcal{M}_{\text{loc}}(\Omega)$ be a non negative Radon measure and $1 \le k \le d$. The $k$-dimensional density of $\mu$ at $x \in \Omega$ is defined by

$$ \Theta^k(\mu, x) := \lim_{\rho \to 0} \frac{\mu(B_\rho(x))}{\omega_k \rho^k} $$

provided the limit exists.

2.2. **Rectifiable sets.** A set $M \subset \mathbf{R}^d$ is countably $\mathcal{H}^k$-rectifiable if there exist countably many Lipschitz functions $f_i: \mathbf{R}^k \to \mathbf{R}^d$, $i \in \mathbf{N}$, such that

$$ \mathcal{H}^k \left( M \setminus \left( \bigcup_{i \in \mathbf{N}} f_i(\mathbf{R}^k) \right) \right) = 0. $$

If $\mathcal{H}^k(M) < \infty$, then $M$ admits an approximate tangent space, denoted by $T_x M$, at $\mathcal{H}^k$-a.e. $x \in M$, see [3, Theorem 2.83]. We denote by $\Theta^k(M, x) := \Theta^k(\mathcal{H}^k \llcorner M, x)$ the $k$-dimensional density of $M$ at $x$ and recall the Besicovitch-Mastrand-Mattila Theorem which states that $\Theta^k(M, \cdot) = 1 \mathcal{H}^k$-a.e. in $M$, while $\Theta^k(M, \cdot) = 0 \mathcal{H}^k$-a.e. in $\mathbf{R}^d \setminus M$, see [3, Theorems 2.56 and 2.63].

It is known that the previous definition of rectifiability is equivalent to require that $\mathcal{H}^k$-almost every $M$ can be covered by a countable union of $k$-dimensional $\mathcal{C}^1$-submanifolds. If one further assume the submanifolds to be of class $\mathcal{C}^2$, then the set $M$ is said to be $\mathcal{C}^2$-rectifiable.

2.3. **Sets of finite perimeter.** Let $\Omega \subset \mathbf{R}^d$ be an open set. A Lebesgue measurable set $E \subset \mathbf{R}^d$ is said to have finite perimeter in $\Omega$ if $D1_E \in \mathcal{M}(\Omega; \mathbf{R}^d)$. In that case, we have

$$ D1_E = \nu_E \mathcal{H}^{d-1} \llcorner \partial^* E, $$

where $\partial^* E$ is the reduced boundary of $E$ (a countably $\mathcal{H}^{d-1}$-rectifiable set), and $\nu_E$ is the approximate inner unit normal to $E$ (see [3, Theorem 3.59]). We denote by $P(E, \Omega) = |D1_E|(\Omega) = \mathcal{H}^{d-1}(\partial^* E \cap \Omega)$ the perimeter of $E$ in $\Omega$.

2.4. **Varifolds.** Let us recall several basic ingredients of the theory of (co-dimension 1) varifolds (see [49] for a detailed description). We denote by $\mathbf{G}_{d-1}$ the Grassmannian manifold of all $(d-1)$-dimensional linear subspaces of $\mathbf{R}^d$. The set $\mathbf{G}_{d-1}$ is identified with the set of all orthogonal projection matrices onto $(d-1)$-dimensional linear subspaces of $\mathbf{R}^d$, i.e., $d \times d$ symmetric matrices $A$ such that $A^2 = A$ and tr($A$) = $d-1$, in other words, matrices of the form $A = \text{Id} - e \otimes e$ for some $e \in \mathbf{S}^{d-1}$.

A $(d-1)$-varifold in $\Omega$ is a Radon measure on $\mathbf{G}_{d-1}(\Omega) := \Omega \times \mathbf{G}_{d-1}$. The class of $(d-1)$-varifolds in $\Omega$ is denoted by $\mathbf{V}_{d-1}(\Omega)$. The weight measure of $V \in \mathbf{V}_{d-1}(\Omega)$ is the nonnegative measure $\mu_V \in \mathcal{M}_{\text{loc}}(\Omega)$ defined by $\mu_V(B) = V(B \times \mathbf{G}_{d-1})$ for all Borel sets $B \subset \Omega$. We define the first variation of a $(d-1)$-varifold in $V$ in $\Omega$ as the first order distribution

$$ \delta V(\zeta) = \int_{\mathbf{G}_{d-1}(\Omega)} D\zeta(x) \cdot S dV(x, S) \quad \text{for } \zeta \in \mathcal{C}_c^1(\Omega; \mathbf{R}^d). $$

The varifold $V$ is said to have bounded first variation in $\Omega$, if the distribution $\delta V$ extends to a bounded Radon measure in $\Omega$. In that case, the Besicovitch differentiation Theorem shows that $\delta V$ splits into

$$ \delta V = -H_V \mu_V + \sigma, $$

where $H_V \in L_{\mu_V}^1(\Omega; \mathbf{R}^d)$ is the generalized mean curvature of $V$ and $\sigma \in \mathcal{M}(\Omega; \mathbf{R}^d)$ is singular with respect to $\mu_V$. We say that $V$ has bounded first variation in $L_{\mu_V}^p(\Omega; \mathbf{R}^d)$, $p \ge 1$, if $\sigma = 0$ and $H_V \in L_{\mu_V}^p(\Omega; \mathbf{R}^d)$.

Let $M \subset \Omega$ be a countably $\mathcal{H}^{d-1}$-rectifiable set, and $\theta: M \to \mathbf{R}^+$ be a locally $\mathcal{H}^{d-1}$-integrable (multiplicity) function. A particular class of varifolds is that of $(d-1)$-rectifiable varifolds $V =$
$\mathbf{v}(M, \theta)$ which can be written as

$$ \int_{G_{d-1}(\Omega)} \Phi(x, S) \, dV(x, S) := \int_{M} \Phi(x, T_{x}M)\theta(x) \, d\mathcal{H}^{d-1}(x) \quad \text{for } \Phi \in \mathcal{C}_{c}(\mathbf{G}_{d-1}(\Omega)). $$

In that case, the weight measure is given by $\mu_V = \theta\mathcal{H}^{d-1}\mathbf{L}M$. We will denote by $\nu_V$ an approximate unit normal to the rectifiable set $M$. Let us point out that since we mostly work in the framework of (unoriented) varifolds, all quantities involved will not depend on the choice of the normal. If further $\theta \in \mathbb{N}\mathcal{H}^{d-1}$-a.e. in $M$, then $V$ is called $(d-1)$-rectifiable integral varifold. We will sometimes denote by $V_M = \mathbf{v}(M, 1)$, $\mu_M = \mu_{V_M}$ and, if $V_M$ has bounded first variation in $\Omega$, by $H_M := H_{V_M}$ the generalized mean curvature of $V_M$.

If $M = \partial^*E$ for some set $E$ of finite perimeter in $\Omega$, we write $V_E = V_{\partial^*E}$, $\mu_E = \mu_{\partial^*E}$, and $H_E = H_{\partial^*E}$.

Let us recall the following monotonicity formula, see [40, Theorem 2.2], which will be used throughout the paper.

**Lemma 2.1.** Let $\Omega \subset \mathbf{R}^2$ be an open set and $V \in \mathbf{V}_1(\Omega)$ be a (non zero) 1-rectifiable integral varifold with bounded first variation in $L_{\mu_V}^p(\Omega; \mathbf{R}^2)$, with $p > 1$. Then, for every $x_0 \in \Omega$ and $r > 0$ with $\bar{B}_r(x_0) \subset \Omega$, $\Theta^1(\mu_V, x_0)$ exists and

$$ \Theta^1(\mu_V, x_0) \le \frac{\mu_V(B_r(x_0))}{2r} + \frac{1}{2} \int_{B_r(x_0)} |H_V| \, d\mu_V. \quad (2.1) $$

Moreover, if $\Omega = \mathbf{R}^2$ and $\mu_V(\mathbf{R}^2) < \infty$, then

$$ \frac{\mu_V(B_r(x_0))}{r} \le \int_{\mathbf{R}^2} |H_V| \, d\mu_V. \quad (2.2) $$

*Proof.* The fact that $\Theta^1(\mu_V, x_0)$ exists for any $x_0$ follows from [49, Corollary 17.8]. Formulas (2.1) and (2.2) can be inferred from the proof of [40, Theorem 2.2] (though not stated under this form). For the reader's convenience we include some details. By [40, (18)], for every $x_0 \in \Omega$ and $0 < r < R$ with $\bar{B}_r(x_0) \subset \Omega$,

$$ \frac{\mu_V(B_r(x_0))}{r} + \frac{1}{r} \int_{B_r(x_0)} H_V(x) \cdot (x - x_0) \, d\mu_V(x) \le \frac{\mu_V(B_R(x_0))}{R} + \int_{B_R(x_0)} H_V(x) \cdot \left( \frac{x - x_0}{R} - \frac{x - x_0}{|x - x_0|} \mathbf{1}_{B_R(x_0) \setminus B_r(x_0)} \right) \, d\mu_V(x). $$

On the one hand, keeping $R > 0$ fixed and letting $r \to 0$, since $H_V \in L_{\mu_V}^p(\Omega; \mathbf{R}^2)$ with $p > 1$ yields (2.1).

On the other hand, if $\Omega = \mathbf{R}^2$ and $\mu_V(\mathbf{R}^2) < \infty$, we can now keep $r > 0$ fixed and let $R \to \infty$. Since $H_V \in L_{\mu_V}^1(\mathbf{R}^2; \mathbf{R}^2)$, we obtain by dominated convergence

$$ \frac{\mu_V(B_r(x_0))}{r} \le -\frac{1}{r} \int_{B_r(x_0)} H_V(x) \cdot (x - x_0) \, d\mu_V(x) - \int_{\mathbf{R}^2 \setminus B_r(x_0)} H_V(x) \cdot \frac{x - x_0}{|x - x_0|} \, d\mu_V(x) \\ \le \int_{\mathbf{R}^2} |H_V| \, d\mu_V, $$

leading to (2.2). $\square$

Let us next state the following version of the Sobolev-Michael-Simon inequality (see [1, Theorems 7.1 and 7.3]).

**Theorem 2.2.** There exists a universal constant $c_1 > 0$ such that the following property is satisfied: let $\Omega \subset \mathbf{R}^2$ be an open set and $V \in \mathbf{V}_1(\mathbf{R}^2)$ be a 1-rectifiable integral varifold with bounded first variation in $L_{\mu_V}^1(\mathbf{R}^2; \mathbf{R}^2)$. Setting

$$ \gamma := \int_{\Omega} |H_V| \, d\mu_V, $$

then, for every compactly supported Lipschitz function $\zeta : \Omega \to \mathbf{R}$.

$$ (1 - c_1\gamma) \sup_{\text{Supp}(V)} |\zeta| \le c_1 \int_{\mathbf{G}_1(\Omega)} |S \cdot \nabla \zeta(x)| \, dV(x, S). $$
We finally recall the main result of [43] which will be at the basis of our forthcoming phase-field approximation analysis. Although not stated in this form, a careful reading of [43, Theorem 4.1 & Theorem 5.1] yields the following statement. Let us consider the standard double well potential $W: \mathbf{R} \to \mathbf{R}$ defined by

$$W(z) = (1 - z^2)^2 \quad \text{for } z \in \mathbf{R},$$

and set

$$c_0 = \int_{-1}^{1} \sqrt{2W(z)} dz = \frac{4\sqrt{2}}{3}. \qquad (2.3)$$

**Theorem 2.3.** Let $d = 2, 3$ and $\Omega$ be a bounded open subset of $\mathbf{R}^d$. Given $v_\epsilon \in H^2(\Omega)$, we define the measures

$$\mu_\epsilon = \frac{1}{c_0} \left( \frac{\epsilon}{2} |\nabla v_\epsilon|^2 + \frac{1}{\epsilon} W(v_\epsilon) \right) \mathcal{L}^d, \quad \alpha_\epsilon = \frac{1}{c_0 \epsilon} \left( -\epsilon \Delta v_\epsilon + \frac{1}{\epsilon} W'(v_\epsilon) \right)^2 \mathcal{L}^d,$$

the function

$$\xi_\epsilon = \frac{\epsilon}{2} |\nabla v_\epsilon|^2 - \frac{1}{\epsilon} W(v_\epsilon)$$

and the varifold $V_\epsilon$ as

$$\int_{\mathbf{G}_{d-1}(\Omega)} \Phi(x, S) dV_\epsilon(x, S) = \int_{\Omega} \Phi(x, \mathrm{Id} - \nu_\epsilon \otimes \nu_\epsilon) d\mu_\epsilon, \quad \Phi \in \mathcal{C}_c(\mathbf{G}_{d-1}(\Omega)).$$

where $\nu_\epsilon: \Omega \to \mathbf{S}^{d-1}$ are Borel extensions of $\frac{\nabla v_\epsilon}{|\nabla v_\epsilon|}$ on $\{\nabla v_\epsilon = 0\}$. Assume that

$$\sup_{\epsilon>0} \{\mu_\epsilon(\Omega) + \alpha_\epsilon(\Omega)\} < \infty.$$

Then, up to a subsequence, there exist $\alpha \in \mathcal{M}^+(\Omega)$ and $V \in \mathbf{V}_{d-1}(\Omega)$ such that $V_\epsilon \to V$ weakly* in $\mathcal{M}(\mathbf{G}_{d-1}(\Omega))$, $\mu_\epsilon \to \mu_V$, $\alpha_\epsilon \to \alpha$ weakly* in $\mathcal{M}(\Omega)$ and $\xi_\epsilon \to 0$ in $L_{\mathrm{loc}}^1(\Omega)$. Moreover, $V$ is a $(d-1)$-rectifiable integral varifold with bounded first variation in $L_{\mu_V}^2(\Omega; \mathbf{R}^d)$ and

$$|H_V|^2 \mu_V \le \alpha. \qquad (2.4)$$

**2.5. Curves.** Let $\Omega$ be an open subset of $\mathbf{R}^2$. We consider the class of 1-dimensional sets $\Gamma \subset \Omega$ which are made of a finite union of curves of class $\mathcal{C}^2$ without self-intersections and that intersect only at their endpoints. Specifically, there exist $\mathcal{C}^2$ mappings $\gamma_1, \dots, \gamma_N: [0, 1] \to \Omega$ such that $\dot{\gamma}_i \ne 0$ and, setting $\Gamma_i = \gamma_i([0, 1])$,

$$\Gamma = \bigcup_{i=1}^{N} \Gamma_i. \qquad (2.5)$$

We denote by $\mathcal{P}_\Gamma = \bigcup_{i=1}^N \{\gamma_i(0), \gamma_i(1)\}$ the set of all end points of the curves $\Gamma_i$. Given $p \in \mathcal{P}_\Gamma$, there is $I \subset \{1, \dots, N\}$ with $\#(I) \ge 1$ such that $p \in \bigcap_{i \in I} \Gamma_i$. Up to a change of orientation, we can assume that

$$p = \gamma_i(0) \quad \text{for } i \in I.$$

For $i \in I$, let $\dot{\gamma}_i(0)$ be the tangent vector to $\Gamma_i$ at $p$. We assume that (see Remark 4.1)$^1$

$$v_p := \sum_{i \in I} \dot{\gamma}_i(0) \ne 0. \qquad (2.6)$$

The class of all sets $\Gamma$ satisfying (2.5) and (2.6) is denoted by $\mathcal{C}(\Omega)$.

Let $V_\Gamma$ be the varifold associated to such a set $\Gamma \in \mathcal{C}(\Omega)$ (which is a rectifiable set). A classical computation shows that the first variation of $V_\Gamma$ is given by

$$\delta V_\Gamma(\zeta) = - \sum_{i=1}^{N} \int_{\Gamma_i} H_{\Gamma_i} \cdot \zeta \, d\mathcal{H}^1 - \sum_{p \in \mathcal{P}_\Gamma} \zeta(p) \cdot v_p \quad \text{for } \zeta \in \mathcal{C}_c^1(\Omega; \mathbf{R}^2),$$

where $H_{\Gamma_i}(x) = \frac{d}{dt} \left( \frac{\dot{\gamma}_i(t)}{|\dot{\gamma}_i(t)|} \right)$ is the classical curvature of $\Gamma_i$ at the point $x = \gamma_i(t) \in \gamma_i([0, 1])$. In other words, $V_\Gamma$ has bounded first variation which can be represented by the measure

$$\delta V_\Gamma = - \sum_{i=1}^{N} H_{\Gamma_i} \mathcal{H}^1 \llcorner \Gamma_i - \sum_{p \in \mathcal{P}_\Gamma} v_p \delta_p.$$

<footnote>$^1$If, for some $i \in I$, $\Gamma_i$ is a loop, i.e. $\gamma_i(0) = \gamma_i(1)$, then $\dot{\gamma}_i(0)$ should be replaced by $\dot{\gamma}_i(0) - \dot{\gamma}_i(1)$ in (2.6).</footnote>
Condition (2.6) can thus be interpreted as a nonzero curvature in the sense of varifolds at all intersection points of the curves $\Gamma_i$'s.

The following elementary lemma describes the tangent measures to $\mathcal{H}^1 \perp \Gamma$ for $\Gamma \in \mathcal{C}(\Omega)$. We omit the proof.

**Lemma 2.4.** Let $\Gamma \in \mathcal{C}(\Omega)$. For all $x_0 \in \Omega$ and $\varrho > 0$, we define the blow-up measure $\lambda_{x_0, \varrho}$ of $\mathcal{H}^1 \perp \Gamma$ by

$$ \int_{\mathbb{R}^2} \zeta d\lambda_{x_0, \varrho} := \frac{1}{\varrho} \int_{\Gamma} \zeta \left( \frac{y-x_0}{\varrho} \right) d\mathcal{H}^1(y) \quad \text{for } \zeta \in \mathcal{C}_c(\mathbb{R}^2) $$

and

$$ \Theta(\Gamma, x_0) := \lim_{\varrho \to 0} \frac{\mathcal{H}^1(\Gamma \cap B_{\varrho}(x_0))}{2\varrho}. $$

Then, $\lambda_{x_0, \varrho} \to \lambda_0$ weakly* in $\mathcal{M}_{\text{loc}}(\mathbb{R}^2)$ as $\varrho \to 0$, where

(i) if $x_0 \in \Omega \setminus \Gamma$, then $\lambda_0 = 0$ and $\Theta(\Gamma, x_0) = 0$;

(ii) if $x_0 \in \gamma_i([0, 1])$ for some (unique) $i \in \{1, \dots, N\}$, then $\lambda_0 = \mathcal{H}^1 \perp T$ where $T = \mathbf{R}\dot{\gamma}(t_0)$ is the tangent line to $\gamma_i([0, 1])$ at $x_0 = \gamma_i(t_0)$, and $\Theta(\Gamma, x_0) = 1$;

(iii) if $x_0 \in \mathcal{P}_\Gamma$, then $\lambda_0 = \sum_{i \in I} \mathcal{H}^1 \perp L_i$ and $L_i = \mathbf{R}^+ \dot{\gamma}_i(0)$ is the half-line tangent to $\Gamma_i$ at the point $x_0 = \gamma_i(0)$ and $\Theta(\Gamma, x_0) = \frac{\#(I)}{2}$.

The following result is a consequence of Taylor's expansion of the length function.

**Lemma 2.5.** For every $\Gamma \in \mathcal{C}(\Omega)$, there exists $\varrho_0 = \varrho_0(\Gamma) > 0$ such that for every $x_0 \in \mathcal{P}_\Gamma$ and $0 < \varrho \le \varrho_0$,

$$ B_{\varrho}(x_0) \cap \mathcal{P}_{\Gamma} = \{x_0\}, \quad \mathcal{H}^1(\Gamma \cap B_{\varrho}(x_0)) \le 2\varrho\Theta(\Gamma, x_0) + \varrho^2\Theta(\Gamma, x_0) \sup_{\Gamma \setminus \mathcal{P}_{\Gamma}} |H_{\Gamma}|. \quad (2.7) $$

*Proof.* Since both sides of (2.7) are additive in the number of curves meeting at $x_0$, it is enough to consider the case where $\Gamma$ is a single curve. We may further assume that $x_0 = 0$ and that $\gamma$ is an arc-length parametrization of $\Gamma$, i.e. setting $L = \mathcal{H}^1(\Gamma)$, $\gamma \in \mathcal{C}^2([0, L]; \mathbb{R}^2)$ satisfies $|\dot{\gamma}| = 1$ on $[0, L]$ and $\gamma(0) = 0$. We thus need to show the existence of $\varrho_0 > 0$ such that for every $\varrho \in (0, \varrho_0)$,

$$ \mathcal{H}^1(\Gamma \cap B_{\varrho}) \le \varrho + \varrho^2 \sup_{t \in (0, L)} |\dot{\gamma}(t)|. \quad (2.8) $$

Let $\varrho_0 > 0$ be small enough so that $\Gamma \cap B_{\varrho_0}$ has one connected component. In particular, for every $0 < \varrho < \varrho_0$, denoting by $\ell(\varrho) = \mathcal{H}^1(\Gamma \cap B_{\varrho})$, we have

$$ \Gamma \cap B_{\varrho} = \gamma([0, \ell(\varrho)]), \quad \Gamma \cap \partial B_{\varrho} = \{\gamma(\ell(\varrho))\}. $$

Moreover, since $\gamma \in \mathcal{C}^2([0, L]; \mathbb{R}^2)$, then $\ell$ is an increasing function of class $\mathcal{C}^2$ on $(0, \varrho_0)$. Deriving the equation $|\gamma(\ell(\varrho))|^2 = \varrho^2$ with respect to $\varrho$ yields

$$ \dot{\gamma}(\ell(\varrho)) \cdot \gamma(\ell(\varrho)) \dot{\ell}(\varrho) = \varrho \quad \text{for } \varrho \in (0, \varrho_0). \quad (2.9) $$

Recalling that $\gamma$ is an arclength parametrization of $\Gamma$ and that $\ell$ is increasing, we get

$$ \varrho \le |\gamma(\ell(\varrho))| |\dot{\gamma}(\ell(\varrho))| \ell(\varrho) = \varrho \dot{\ell}(\varrho), $$

hence $\dot{\ell}(\varrho) \ge 1$. Moreover, passing to the limit in (2.9) as $\varrho \to 0$ yields $|\dot{\gamma}(0)|^2 \dot{\ell}(0)^2 = 1$, hence $\dot{\ell}(0) = 1$. As a consequence, using Taylor expansion, we get that for $\varrho \in (0, \varrho_0)$,

$$ \ell(\varrho) = \ell(0) + \varrho \dot{\ell}(0) + \int_{0}^{\varrho} (\varrho - s) \ddot{\ell}(s) ds = \varrho + \int_{0}^{\varrho} (\varrho - s) \ddot{\ell}(s) ds. \quad (2.10) $$

In order to estimate $\ddot{\ell}$, we derive (2.9) which leads to

$$ 1 = \dot{\ell}^2 (1 + \gamma(\ell) \cdot \ddot{\gamma}(\ell)) + \gamma(\ell) \cdot \dot{\gamma}(\ell) \ddot{\ell} \quad \text{in } (0, \varrho_0). $$

Using (2.9) and recalling that $\dot{\ell} \ge 1$, we obtain for every $s \in (0, \varrho_0)$

$$ \begin{aligned} \frac{s}{\dot{\ell}(s)} \ddot{\ell}(s) &= \gamma(\ell(s)) \cdot \dot{\gamma}(\ell(s)) \ddot{\ell}(s) \le -\dot{\ell}(s)^2 \gamma(\ell(s)) \cdot \ddot{\gamma}(\ell(s)) \\ &\le \dot{\ell}(s)^2 |\gamma(\ell(s))| |\ddot{\gamma}(\ell(s))| = \dot{\ell}(s)^2 s |\ddot{\gamma}(\ell(s))|. \end{aligned} $$
Since $\dot{l}(0) = 1$, we can further reduce $\varrho_0$ in such a way that $\dot{l}(s)^3 \le 2$ for every $s \in (0, \varrho_0)$, which implies that

$$ \ddot{l}(s) \le 2|\ddot{\gamma}(l(s))| \quad \text{for } s \in (0, \varrho_0). $$

Inserting inside (2.10) yields

$$ \ell(\varrho) \le \varrho + \varrho^2 \sup_{s \in (0, \varrho_0)} |\ddot{\gamma}(l(s))|, $$

which completes the proof of (2.8), hence of the lemma.

□

**2.6. Functions of bounded variation.** Given an open set $\Omega \subset \mathbf{R}^d$, the space of functions of bounded variation is defined by

$$ BV(\Omega) = \{u \in L^1(\Omega) : Du \in \mathcal{M}(\Omega; \mathbf{R}^d)\}. $$

We shall also consider the subspace $SBV(\Omega)$ of special functions of bounded variation made of functions $u \in BV(\Omega)$ whose distributional derivative can be decomposed as $Du = \nabla u \mathcal{L}^d + (u^+ - u^-)\nu_u \mathcal{H}^{d-1} \mathbf{L} J_u$. In the previous expression, $\nabla u$ is the Radon-Nikodým derivative of $Du$ with respect to $\mathcal{L}^d$, and it is called the approximate gradient of $u$. The Borel set $J_u$ is the (approximate) jump set of $u$. It is a countably $\mathcal{H}^{d-1}$-rectifiable subset of $\Omega$ oriented by the (approximate) normal direction of jump $\nu_u : J_u \to \mathbf{S}^{d-1}$, and $u^\pm$ are the one-sided approximate limits of $u$ on $J_u$ according to $\nu_u$. Finally we define

$$ SBV^2(\Omega) = \{u \in SBV(\Omega) : \nabla u \in L^2(\Omega; \mathbf{R}^d) \text{ and } \mathcal{H}^{d-1}(J_u) < \infty\}. $$

In the sequel, we will be interested only in the two-dimensional case $d = 2$, and in the smaller class of functions

$$ A(\Omega) := \{u \in SBV^2(\Omega) : J_u \in \mathcal{C}(\Omega)\}. \qquad (2.11) $$

**3. STRONGLY ANISOTROPIC PERIMETERS WITH CURVATURE**

Let $d=2$ or $d=3$ and $\Omega \subset \mathbf{R}^d$ be bounded open set. Let us consider an even and continuous anisotropy function $\varphi : \mathbf{S}^{d-1} \to \mathbf{R}^+$ satisfying

$$ \frac{1}{C} \le \varphi(z) \le C \quad \text{for } z \in \mathbf{S}^{d-1}, \qquad (3.1) $$

for some $C \ge 1$. Since $\varphi$ is an even function, the function $\tilde{\varphi} : \mathbf{G}_{d-1} \to \mathbf{R}^+$ given, for every $S = \text{Id} - e \otimes e \in \mathbf{G}_{d-1}$ (with $e \in \mathbf{S}^{d-1}$), by

$$ \tilde{\varphi}(S) := \varphi(e) \qquad (3.2) $$

is well defined and continuous.

For every Lebesgue measurable set $E$ with finite perimeter in $\Omega$, we consider the following energy functional

$$ \mathcal{F}(E) = \begin{cases} \int_{\Omega \cap \partial^* E} (\varphi(\nu_E) + |H_E|^2) d\mathcal{H}^{d-1} & \text{if } V_E \text{ has bounded first variation in } L_{\mu_E}^2(\Omega; \mathbf{R}^d), \\ = \infty & \text{otherwise.} \end{cases} $$

**3.1. Lower semi-continuity.** We start with the question of lower semicontinuity for $\mathcal{F}$ with respect to the following (natural) convergence.

**Definition 3.1.** A sequence $\{E_n\}_{n \in \mathbb{N}}$ of Lebesgue measurable subsets of $\Omega$ converges to the Lebesgue measurable set $E \subset \Omega$ if $\mathbf{1}_{E_n} \to \mathbf{1}_E$ strongly in $L^1(\Omega)$, or equivalently, if $\mathcal{L}^d(E_n \Delta E) \to 0$.

We now address the

*Proof of Theorem 1.1.* There is no loss of generality to suppose that

$$ \liminf_{n \to \infty} \mathcal{F}(E_n) < \infty. $$

At the expense of extracting a subsequence, we can thus assume that the $\liminf$ above is a limit and that $\sup_n \mathcal{F}(E_n) < \infty$. In particular, for each $n \in \mathbb{N}$, denoting by $V_n := V_{E_n}$ the varifold associated to $E_n$ and by $\mu_n := \mathcal{H}^{d-1} \mathbf{L} \partial^* E_n$ the weight measure of $V_n$, then $\{V_n\}_{n \in \mathbb{N}}$ is a sequence of $(d-1)$-rectifiable integral varifolds in $\Omega$ with bounded first variation in $\Omega$ satisfying $\delta V_n = -H_{E_n} \mu_n$ for some $H_{E_n} \in L_{\mu_n}^2(\Omega; \mathbf{R}^d)$. In addition, according to (3.1),

$$ \sup_{n \in \mathbb{N}} \int_{\Omega \cap \partial^* E_n} (1 + |H_{E_n}|^2) d\mathcal{H}^{d-1} < \infty. $$
By Allard's integral compactness theorem, see e.g. [49, Theorem 42.7 & Remark 42.8], we infer that, up to a subsequence, $V_n \to V$ weakly* in $M(G_{d-1}(\Omega))$ for some $(d-1)$-rectifiable integral varifold $V \in V_{d-1}(\Omega)$. Thus, there exist a countably $\mathcal{H}^{d-1}$-rectifiable set $M$ and a positive integer valued $\mathcal{H}^{d-1}$-summable function $\theta$ such that for every $\Phi \in C_c(G_{d-1}(\Omega))$,

$$ \int_{G_{d-1}(\Omega)} \Phi(x, S) dV(x, S) = \int_{\Omega \cap M} \theta(x) \Phi(x, T_x M) d\mathcal{H}^{d-1}(x), $$

where $T_x M$ is the approximate tangent space to $M$ at $x$. Moreover, as $\mu_n \to \mu_V$ weakly* in $M(\Omega)$ and $\delta V_n = -H_{E_n} \mu_n \to \delta V$ weakly* in $M(\Omega; \mathbf{R}^d)$, using a standard lower semicontinuity result, see e.g. [3, Example 2.36], we get that $\delta V = -H_V \mu_V$ for some $H_V \in L_{\mu_V}^2(\Omega; \mathbf{R}^d)$ and

$$ \int_{\Omega} |H_{\mu_V}|^2 d\mu_V \le \liminf_{n \to \infty} \int_{\Omega} |H_{E_n}|^2 d\mu_n. $$

Since $\mathbf{1}_{E_n} \to \mathbf{1}_E$ strongly in $L^1(\Omega)$ and $\mu_n = \mathcal{H}^{d-1} \mathbf{L} \partial^* E_n = |D\mathbf{1}_{E_n}|$ we have $\sup_n |D\mathbf{1}_{E_n}|(\Omega) < \infty$ and consequently $D\mathbf{1}_{E_n} \to D\mathbf{1}_E$ weakly* in $M(\Omega; \mathbf{R}^d)$. Moreover, $\mu_n = |D\mathbf{1}_{E_n}| \to \mu_V$ weakly* in $M(\Omega)$ and therefore

$$ \mu_E = \mathcal{H}^{d-1} \mathbf{L} \partial^* E = |D\mathbf{1}_E| \le \mu_V. $$

Since in addition $V_E$ has bounded first variation in $L_{\mu_E}^2(\Omega; \mathbf{R}^d)$, according to [29, Theorem 1] (see also [47, Theorem 5.1] and [48, Theorem 3.1] in dimension $d=2$), we infer that $\partial^* E$ is $C^2$-rectifiable. Then [48, Corollary 4.3] implies that, $\mathcal{H}^{d-1}$-a.e. in $\partial^* E$, we have $H_V = H_E$, hence

$$ \int_{\Omega \cap \partial^* E} |H_E|^2 d\mathcal{H}^{d-1} \le \int_{\Omega} |H_V|^2 d\mu_V \le \liminf_{n \to \infty} \int_{\Omega \cap \partial^* E_n} |H_{E_n}|^2 d\mathcal{H}^{d-1}. \quad (3.3) $$

We now consider the anisotropic perimeter term. Notice first that by [3, Proposition 2.85] we have the locality of the approximate tangent spaces: $T_x M = T_x(\partial^* E)$ for $\mathcal{H}^{d-1}$-a.e. $x \in \partial^* E$. Let now $\zeta \in C_c(\Omega)$ be such that $0 \le \zeta \le 1$ in $\Omega$. Then, the function $(x, S) \in G_{d-1}(\Omega) \mapsto \zeta(x)\tilde{\varphi}(S)$ belongs to $C_c(G_{d-1}(\Omega))$ so that by varifold convergence,

$$ \begin{aligned} \liminf_{n \to \infty} \int_{\Omega \cap \partial^* E_n} \varphi(\nu_{E_n}) d\mathcal{H}^{d-1} &\ge \liminf_{n \to \infty} \int_{G_{d-1}(\Omega)} \zeta(x) \tilde{\varphi}(S) dV_n(x, S) \\ &= \int_{G_{d-1}(\Omega)} \zeta(x) \tilde{\varphi}(S) dV(x, S) = \int_{\Omega} \zeta(x) \tilde{\varphi}(T_x M) d\mu_V(x) \\ &\ge \int_{\Omega \cap \partial^* E} \zeta \varphi(\nu_E) d\mathcal{H}^{d-1}. \end{aligned} \quad (3.4) $$

Passing to the supremum with respect to all $\zeta$ as above and gathering (3.3) together with (3.4) yields

$$ \int_{\Omega \cap \partial^* E} (\varphi(\nu_E) + |H_E|^2) d\mathcal{H}^{d-1} \le \liminf_{n \to \infty} \int_{\Omega \cap \partial^* E_n} (\varphi(\nu_{E_n}) + |H_{E_n}|^2) d\mathcal{H}^{d-1}. $$

This proves the desired semicontinuity property of $\mathcal{F}$.

□

**Remark 3.2.** In dimension $d=2$, the following elementary argument avoiding geometric measure theory can be used (see e.g. [7] or [12, Lemma 2.1]).

We first recall that if $\gamma \in H^2([0, 1]; \mathbf{R}^2)$ is a closed curve and $\Gamma = \gamma([0, 1])$, denoting by $\kappa := |H|$ the mean curvature of $\Gamma$, we have by the Gauss-Bonnet Theorem, Cauchy-Schwarz and Young's inequalities

$$ 2\pi \le \int_{\Gamma} \kappa d\mathcal{H}^1 \le \mathcal{H}^1(\Gamma)^{\frac{1}{2}} \left( \int_{\Gamma} \kappa^2 d\mathcal{H}^1 \right)^{\frac{1}{2}} \le \frac{1}{2} \int_{\Gamma} (1 + \kappa^2) d\mathcal{H}^1 \le \frac{3(3-\kappa^2)}{2} \int_{\Gamma} (\varphi(\nu) + \kappa^2) d\mathcal{H}^1, $$

hence

$$ \int_{\Gamma} (\varphi(\nu) + \kappa^2) d\mathcal{H}^1 \ge \frac{4\pi}{C} \quad \text{and} \quad \int_{\Gamma} \kappa^2 d\mathcal{H}^1 \ge \frac{4\pi^2}{\mathcal{H}^1(\Gamma)}. \quad (3.5) $$

In order to have compactness and lower semicontinuity for the natural topology, we need to extend a bit the definition of $\mathcal{F}$ as follows. For $(E, \Gamma)$ such that $\partial E \subset \Gamma$ with $\Gamma = \bigcup_{i \in I} \Gamma^i$ and $\Gamma^i$ are closed loops which can only intersect tangentially, we set

$$ \mathcal{F}(E, \Gamma) = \sum_{i \in I} \int_{\Gamma^i} (\varphi(\nu) + \kappa^2) d\mathcal{H}^1. $$
Let now $(E_n, \Gamma_n)$ be such that $\sup_n \mathcal{F}(E_n, \Gamma_n) < \infty$. By the first inequality in (3.5),

$$ \#(I_n) \frac{4\pi}{C} \leq \sum_{i \in I_n} \int_{\Gamma_n^i} (\varphi(\nu) + \kappa^2) d\mathcal{H}^1 = \mathcal{F}(E_n, \Gamma_n), $$

so that $\#(I_n)$ is uniformly bounded with respect to $n$. By the second inequality in (3.5) we see that $\mathcal{H}^1(\Gamma_n^i)$ are bounded from above and below. Up to translations, we thus find that the parametrizations $\gamma_n^i$ are bounded in $H^2([0, 1]; \mathbf{R}^2)$ and thus converging weakly in that space (and also strongly in $H^1([0, 1]; \mathbf{R}^2)$) to some curves $\gamma^i$. We then find

$$ \liminf_{n \to \infty} \mathcal{F}(E_n, \Gamma_n) \geq \sum_{i \in I} \int_{\Gamma^i} (\varphi(\nu) + \kappa^2) d\mathcal{H}^1. $$

Moreover, the sequence $\{\mathbf{1}_{E_n}\}_{n \in \mathbb{N}}$ is bounded in $BV(\Omega)$ and thus, up to a further subsequence, $E_n$ converges to some set $E$. Since $\mathcal{H}^1 \perp \Gamma^n \geq \mathcal{H}^1 \perp \partial E_n$, by lower semicontinuity of the perimeter we find,

$$ \mathcal{H}^1 \perp \Gamma \geq \mathcal{H}^1 \perp \partial E. $$

This proves that $\partial E \subset \Gamma$ and concludes the proof of the compactness and lower semicontinuity.

3.2. **Phase field approximation.** Given $v \in H^2(\Omega)$, we consider the following phase-field approximation of $\mathcal{F}$ based on [51, 28]: for $\varepsilon > 0$ and $\sqrt{\varepsilon}r_\varepsilon \to 0$,

$$ \mathcal{F}_{\varepsilon}(v) = \frac{1}{c_0} \int_{\Omega} \varphi_{\varepsilon}(\nabla v) \left( \frac{\varepsilon}{2} |\nabla v|^2 + \frac{1}{\varepsilon} W(v) \right) dx + \frac{1}{c_0} \int_{\Omega} \frac{1}{\varepsilon} \left( -\varepsilon \Delta v + \frac{1}{\varepsilon} W'(v) \right)^2 dx, $$

where $\varphi_\varepsilon$ extends $\varphi$ from the sphere to $\mathbf{R}^d$ as proposed in [28, Section 2] to handle regions of $\Omega$ where $\nabla v = 0$ and $\varphi(\frac{\nabla v}{|\nabla v|})$ cannot be defined. More precisely, for $z \in \mathbf{R}^n$,

$$ \varphi_{\varepsilon}(z) = \bar{\varphi} \left( \frac{z}{\sqrt{r_{\varepsilon}^2 + |z|^2}} \right) \quad \text{and} \quad \bar{\varphi} = \xi\varphi + (1 - \xi) \frac{\min_{\mathbf{S}^{d-1}} \varphi}{4} \quad (3.6) $$

where $0 \le \xi \le 1$ is a cut-off function satisfying $\xi = 1$ in $\mathbf{R}^d \setminus B_{1/2}$, $\xi = 0$ in $B_{1/4}$ and $|\nabla \xi| \le 4$. As proved in [28], such definitions ensure that $\bar{\varphi}$ is Lipschitz continuous (with Lipschitz constant $L > 0$) with

$$ \frac{1}{4C} \leq \varphi_{\varepsilon}(z) \leq C \quad \text{for all } z \in \mathbf{R}^d, \quad (3.7) $$

hence

$$ \int_{\Omega} \varphi_{\varepsilon}(\nabla v) \left( \frac{\varepsilon}{2} |\nabla v|^2 + \frac{1}{\varepsilon} W(v) \right) dx \geq \frac{1}{4C} \int_{\Omega} \left( \frac{\varepsilon}{2} |\nabla v|^2 + \frac{1}{\varepsilon} W(v) \right) dx. \quad (3.8) $$

Furthermore,

$$ \left| \varphi_{\varepsilon}(z) - \varphi\left(\frac{z}{|z|}\right) \right| \leq \frac{L r_{\varepsilon}}{2|z|} \quad \text{for all } z \neq 0. \quad (3.9) $$

**Remark 3.3.** Let us point out that in [28] the stronger hypothesis $r_\varepsilon = O(\varepsilon)$ is assumed compared to ours $\sqrt{\varepsilon}r_\varepsilon = o(1)$. From the upper bound construction one could actually conjecture that the sharpest condition should be $\varepsilon r_\varepsilon = o(1)$.

3.2.1. *Proof Theorem 1.3: the lower bound inequality.* We first prove the following intermediate statement.

**Lemma 3.4.** Assume that $\sup_\varepsilon \mathcal{F}_\varepsilon(v_\varepsilon) < \infty$. Then, in the notation of Theorem 2.3, up to a subsequence, there exist $V \in \mathbf{V}_{d-1}(\Omega)$ such that $V_\varepsilon \to V$ weakly* in $M(\mathbf{G}_{d-1}(\Omega))$ and $\mu_\varepsilon \to \mu_V$. Moreover, $V$ is a $(d-1)$-rectifiable integral varifold with bounded first variation in $L^2_{\mu_V}(\Omega; \mathbf{R}^d)$ and

$$ \liminf_{\varepsilon \to 0} \mathcal{F}_\varepsilon(v_\varepsilon) \geq \int_{\Omega} (\varphi(\nu_V) + |H_V|^2) d\mu_V. \quad (3.10) $$

*Proof.* Let

$$ K = \sup_{\varepsilon>0} \mathcal{F}_{\varepsilon}(v_{\varepsilon}) < \infty. $$

We rewrite the energy as (recall the definitions of $\mu_\varepsilon$ and $\alpha_\varepsilon$ from Theorem 2.3)

$$ \mathcal{F}_{\varepsilon}(v_{\varepsilon}) = \int_{\Omega} \varphi_{\varepsilon}(\nabla v_{\varepsilon}) d\mu_{\varepsilon} + \alpha_{\varepsilon}(\Omega). $$
According to (3.8), we get that the sequence ${\mu_{\varepsilon}(\Omega) + \alpha_{\varepsilon}(\Omega)}_{\varepsilon>0}$ is bounded so that Theorem 2.3 applies. In particular, up to a subsequence, $V_{\varepsilon} \to V$, where $V$ is a $(d-1)$-rectifiable integral varifold, and there exist a countably $\mathcal{H}^{d-1}$-rectifiable set $M$ and a positive integer valued $\mathcal{H}^{d-1}$-summable function $\theta$ such that $\mu_V = \theta\mathcal{H}^{d-1} \perp M$ and

$$ \int_{G_{d-1}(\Omega)} \Phi(x, S) dV(x, S) = \int_M \theta(x) \Phi(x, T_x M) d\mathcal{H}^{d-1}(x), \quad \Phi \in C_c(G_{d-1}(\Omega)). $$

Moreover,

$$ \xi_{\varepsilon} = \frac{\varepsilon}{2} |\nabla v_{\varepsilon}|^2 - \frac{1}{\varepsilon} W(v_{\varepsilon}) \to 0 \quad \text{in } L_{\text{loc}}^{1}(\Omega). \qquad (3.11) $$

We infer from (3.8) that for all $\varepsilon > 0$,

$$ K \geq \int_{\Omega} \varphi_{\varepsilon}(\nabla v_{\varepsilon}) \left( \frac{\varepsilon}{2} |\nabla v_{\varepsilon}|^2 + \frac{1}{\varepsilon} W(v_{\varepsilon}) \right) dx \geq \frac{1}{4C\varepsilon} \int_{\Omega} W(v_{\varepsilon}) dx $$

hence, by Hölder inequality,

$$ \int_{\Omega} \sqrt{W(v_{\varepsilon})} dx \leq 2\sqrt{KC|\Omega|\varepsilon}. $$

Combining (3.9) and Young inequality we then get

$$ \begin{aligned} \int_{\Omega} \varphi_{\varepsilon}(\nabla v_{\varepsilon}) \left( \frac{\varepsilon}{2} |\nabla v_{\varepsilon}|^2 + \frac{1}{\varepsilon} W(v_{\varepsilon}) \right) dx &\geq \int_{\Omega} \varphi_{\varepsilon}(\nabla v_{\varepsilon}) |\nabla v_{\varepsilon}| \sqrt{2W(v_{\varepsilon})} dx \\ &\geq \int_{\{\nabla v_{\varepsilon} \neq 0\}} \varphi \left( \frac{\nabla v_{\varepsilon}}{|\nabla v_{\varepsilon}|} \right) |\nabla v_{\varepsilon}| \sqrt{2W(v_{\varepsilon})} dx - \frac{Lr_{\varepsilon}}{2} \int_{\Omega} \sqrt{2W(v_{\varepsilon})} dx, \end{aligned} $$

and thus

$$ \begin{aligned} \liminf_{\varepsilon \to 0} \int_{\Omega} \varphi_{\varepsilon}(\nabla v_{\varepsilon}) \left( \frac{\varepsilon}{2} |\nabla v_{\varepsilon}|^2 + \frac{1}{\varepsilon} W(v_{\varepsilon}) \right) dx &\geq \liminf_{\varepsilon \to 0} \int_{\{\nabla v_{\varepsilon} \neq 0\}} \varphi \left( \frac{\nabla v_{\varepsilon}}{|\nabla v_{\varepsilon}|} \right) |\nabla v_{\varepsilon}| \sqrt{2W(v_{\varepsilon})} dx. \quad (3.12) \end{aligned} $$

Note that $v_{\varepsilon} = \frac{\nabla v_{\varepsilon}}{|\nabla v_{\varepsilon}|}$ on $\{|\nabla v_{\varepsilon}| \neq 0\}$, while $\varphi(v_{\varepsilon})\mu_{\varepsilon} = \frac{\varphi(v_{\varepsilon})}{\varepsilon c_0} W(v_{\varepsilon})\mathcal{L}^d$ on $\{\nabla v_{\varepsilon} = 0\}$. Recalling (3.1), we get by (3.11) that for all $\zeta \in C_c(\Omega)$ with $0 \leq \zeta \leq 1$,

$$ \begin{aligned} & \left| \frac{1}{c_0} \int_{\{\nabla v_{\varepsilon} \neq 0\}} \varphi \left( \frac{\nabla v_{\varepsilon}}{|\nabla v_{\varepsilon}|} \right) |\nabla v_{\varepsilon}| \sqrt{2W(v_{\varepsilon})} \zeta \, dx - \int_{\Omega} \varphi(v_{\varepsilon}) \zeta \, d\mu_{\varepsilon} \right| \\ & \leq \frac{C}{c_0} \int_{\Omega} \left| |\nabla v_{\varepsilon}| \sqrt{2W(v_{\varepsilon})} - \left( \frac{\varepsilon}{2} |\nabla v_{\varepsilon}|^2 + \frac{1}{\varepsilon} W(v_{\varepsilon}) \right) \right| \zeta \, dx \\ & = \frac{C}{c_0} \int_{\Omega} \left( \sqrt{\frac{\varepsilon}{2}} |\nabla v_{\varepsilon}| - \sqrt{\frac{W(v_{\varepsilon})}{\varepsilon}} \right)^2 \zeta \, dx \leq \frac{C}{c_0} \int_{\text{Supp}(\zeta)} |\xi_{\varepsilon}| \, dx \to 0, \quad (3.13) \end{aligned} $$

where we used, in the last inequality, that for $a, b \geq 0$, $|a - b| \leq a + b$ and thus $(a - b)^2 \leq |a - b|(a + b) = |a^2 - b^2|$. Thanks to (3.2), (3.12), (3.13), and by convergence of varifolds, we have

$$ \begin{aligned} & \liminf_{\varepsilon \to 0} \frac{1}{c_0} \int_{\Omega} \varphi_{\varepsilon}(\nabla v_{\varepsilon}) \left( \frac{\varepsilon}{2} |\nabla v_{\varepsilon}|^2 + \frac{1}{\varepsilon} W(v_{\varepsilon}) \right) dx \\ & \geq \liminf_{\varepsilon \to 0} \int_{\Omega} \varphi(v_{\varepsilon}) \zeta \, d\mu_{\varepsilon} \geq \liminf_{\varepsilon \to 0} \int_{G_{d-1}(\Omega)} \zeta(x) \tilde{\varphi}(S) \, dV_{\varepsilon}(x, S) = \int_{G_{d-1}(\Omega)} \zeta(x) \tilde{\varphi}(S) \, dV(x, S). \end{aligned} $$

Passing to the supremum with respect to all $\zeta$ yields

$$ \liminf_{\varepsilon \to 0} \frac{1}{c_0} \int_{\Omega} \varphi_{\varepsilon}(\nabla v_{\varepsilon}) \left( \frac{\varepsilon}{2} |\nabla v_{\varepsilon}|^2 + \frac{1}{\varepsilon} W(v_{\varepsilon}) \right) dx \geq \int_{\Omega} \tilde{\varphi}(T_x M) d\mu_V = \int_{\Omega} \varphi(v_V) d\mu_V. $$

The corresponding lower bound on the curvature term is a direct consequence of (2.4). This concludes the proof of (3.10). □
We now complete the proof of the lower bound in Theorem 1.3. There is no loss of generality in assuming that

$$ \liminf_{\epsilon \to 0} \mathcal{F}_{\epsilon}(v_{\epsilon}) < \infty. $$

Thus, up to a subsequence (not relabeled), we can assume that the liminf is actually a limit. We may thus apply Lemma 3.4. By the lower bound estimate in the Modica-Mortola Theorem (see [33]), we have

$$ \mu_V = \theta \mathcal{H}^{d-1} \llcorner M \ge \mathcal{H}^{d-1} \llcorner \partial^* E. \quad (3.14) $$

By the locality of the approximate tangent spaces together with (3.14) yields

$$ \int_{\Omega} \varphi(\nu_V) d\mu_V \ge \int_{\Omega \cap \partial^* E} \varphi(\nu_E) d\mathcal{H}^{d-1}. \quad (3.15) $$

This shows the lower bound inequality for the anisotropic interfacial term. Since $V_E$ has bounded first variation in $L_{\mu_E}^2(\Omega; \mathbf{R}^d)$, according to [29, Theorem 1] (see also [47, Theorem 5.1] and [48, Theorem 3.1] in dimension $d=2$), we infer that $\partial^* E$ is $\mathcal{C}^2$-rectifiable. Thus, (3.14) and [48, Corollary 4.3] imply that $H_V = H_E \mathcal{H}^{d-1}$-a.e. in $\partial^* E$, hence by (3.14), we infer that

$$ \int_{\Omega \cap \partial^* E} |H_E|^2 d\mathcal{H}^{d-1} = \int_{\Omega \cap \partial^* E} |H_V|^2 d\mathcal{H}^{d-1} \le \int_{\Omega} |H_V|^2 d\mu_V. \quad (3.16) $$

Combining (3.10), (3.15) and (3.16) leads to

$$ \liminf_{\epsilon \to 0} \mathcal{F}_{\epsilon}(v_{\epsilon}) \ge \int_{\Omega \cap \partial^* E} (\varphi(\nu_E) + |H_E|^2) d\mathcal{H}^{d-1} = \mathcal{F}(E). $$

This corresponds to the desired lower bound.

□

**3.2.2.** *Proof of Theorem 1.3: the upper bound inequality.* Let $q: t \in \mathbf{R} \mapsto \tanh(t\sqrt{2})$ be the optimal profile, i.e. the unique solution of the ODE

$$ \begin{cases} -q'' + W'(q) = 0 & \text{in } \mathbf{R} \\ q(\pm\infty) = \pm 1, \end{cases} \quad (3.17) $$

which also satisfies

$$ q' = \sqrt{2W(q)} \quad \text{in } \mathbf{R}. \quad (3.18) $$

As in [16, Lemma 4.1] (see also [17]) we fix a cut-off function $\zeta \in C_c^\infty(\mathbf{R}; [0, 1])$ such that

$$ \zeta(t) = 1 \quad \text{for } t \in [-1, 1], \quad \zeta(t) = 0, \quad \text{for } |t| \ge 2. $$

Given $\lambda > 1$ to be fixed later, we introduce the scaling parameter

$$ \delta_{\epsilon} := \lambda \epsilon |\log \epsilon|. \quad (3.19) $$

and set

$$ q_{\epsilon}(t) = \zeta \left( \frac{2t}{\delta_{\epsilon}} \right) q \left( \frac{t}{\epsilon} \right) + (\text{sign } t) \left( 1 - \zeta \left( \frac{2t}{\delta_{\epsilon}} \right) \right), \quad t \in \mathbf{R}. \quad (3.20) $$

We first prove two elementary properties of $q_\epsilon$ which will be useful in the forthcoming recovery sequence construction.

**Lemma 3.5.** For every $\lambda > 1$, we have

$$ \lim_{\epsilon \to 0} \int_{-\delta_{\epsilon}}^{\delta_{\epsilon}} \frac{\epsilon}{2} |q'_{\epsilon}|^2 dt = \lim_{\epsilon \to 0} \int_{-\delta_{\epsilon}}^{\delta_{\epsilon}} \frac{1}{\epsilon} W(q_{\epsilon}) dt = \frac{c_0}{2}. \quad (3.21) $$

Moreover,

$$ \lim_{\epsilon \to 0} \int_{|t| \ge \delta_{\epsilon}/2} \left( \frac{\epsilon}{2} |q'_{\epsilon}|^2 + \frac{W(q_{\epsilon})}{\epsilon} \right) dt = 0. \quad (3.22) $$

*Proof.* Since

$$ q_{\epsilon}(t) = \zeta\left(\frac{2t}{\delta_{\epsilon}}\right)\left(q\left(\frac{t}{\epsilon}\right) - 1\right) + 1 \quad \text{in } \mathbf{R}^{+} $$

we get by (3.18)

$$ \begin{aligned} q'_{\epsilon}(t) &= \frac{2}{\delta_{\epsilon}} \zeta'\left(\frac{2t}{\delta_{\epsilon}}\right) \left(q\left(\frac{t}{\epsilon}\right) - 1\right) + \frac{1}{\epsilon} \zeta\left(\frac{2t}{\delta_{\epsilon}}\right) q'\left(\frac{t}{\epsilon}\right) \\ &= \left(q\left(\frac{t}{\epsilon}\right) - 1\right) \left[ \frac{2}{\delta_{\epsilon}} \zeta'\left(\frac{2t}{\delta_{\epsilon}}\right) - \frac{\sqrt{2}}{\epsilon} \zeta\left(\frac{2t}{\delta_{\epsilon}}\right) \left(1 + q\left(\frac{t}{\epsilon}\right)\right) \right]. \end{aligned} $$
Using that for $t \in [\delta_{\varepsilon}/2, \delta_{\varepsilon}]$,

$$ \left| 1 - q\left(\frac{t}{\varepsilon}\right) \right| \le 2e^{-\frac{\sqrt{2}}{2}\frac{\delta_{\varepsilon}}{\varepsilon}} = 2e^{-\frac{\sqrt{2}}{2}\lambda|\log \varepsilon|} \le \varepsilon^{\frac{\sqrt{2}}{2}\lambda}, \quad (3.23) $$

we obtain (because $\lambda > 1$)

$$ \sup_{\frac{\delta_{\varepsilon}}{2} \le t \le \delta_{\varepsilon}} \left( \frac{\varepsilon}{2} |q'_{\varepsilon}(t)|^2 + \frac{W(q_{\varepsilon}(t))}{\varepsilon} \right) \le C \left( \varepsilon \left( \frac{1}{\delta_{\varepsilon}^2} + \frac{1}{\varepsilon^2} \right) + \frac{1}{\varepsilon} \right) \varepsilon^{\lambda}, $$

for some constant $C > 0$ independent of $\varepsilon$. A similar estimate holds in $[-\delta_{\varepsilon}, -\frac{\delta_{\varepsilon}}{2}]$, and it leads to (3.22).

On the other hand, by (3.18)

$$ \begin{aligned} \int_{-\delta_{\varepsilon}/2}^{\delta_{\varepsilon}/2} \frac{\varepsilon}{2} |q'_{\varepsilon}(t)|^2 dt &= \int_{-\delta_{\varepsilon}/2}^{\delta_{\varepsilon}/2} \frac{W(q_{\varepsilon}(t))}{\varepsilon} dt \\ &= \frac{1}{2} \int_{-\delta_{\varepsilon}/2\varepsilon}^{\delta_{\varepsilon}/2\varepsilon} \sqrt{2W(q(s))} q'(s) ds. \end{aligned} \quad (3.24) $$

Since $\delta_{\varepsilon}/2\varepsilon = \lambda|\log \varepsilon|/2 \to \infty$, the right-hand side of (3.24) tends to $c_0/2$ (recall (2.3)) as $\varepsilon \to 0$. In combination with (3.22) this completes the proof of the lemma. $\square$

**Lemma 3.6.** There exists $\varepsilon_0 \in (0, 1)$ satisfying: for all $p > 1$, there exists $\lambda > 1$ such that

$$ \int_{-\delta_{\varepsilon}}^{\delta_{\varepsilon}} \left| -\varepsilon q''_{\varepsilon} + \frac{1}{\varepsilon} W'(q_{\varepsilon}) \right|^2 dt \le \varepsilon^p, \quad \forall \varepsilon \in (0, \varepsilon_0). $$

*Proof.* For all $t \in \mathbf{R}^+$, we compute thanks to (3.17)

$$ \begin{aligned} q''_{\varepsilon}(t) &= \frac{4}{\delta_{\varepsilon}^2} \zeta''\left(\frac{2t}{\delta_{\varepsilon}}\right) \left(q\left(\frac{t}{\varepsilon}\right) - 1\right) + \frac{4}{\delta_{\varepsilon}\varepsilon} \zeta'\left(\frac{2t}{\delta_{\varepsilon}}\right) q'\left(\frac{t}{\varepsilon}\right) + \frac{1}{\varepsilon^2} \zeta\left(\frac{2t}{\delta_{\varepsilon}}\right) q''\left(\frac{t}{\varepsilon}\right) \\ &= \frac{4}{\delta_{\varepsilon}^2} \zeta''\left(\frac{2t}{\delta_{\varepsilon}}\right) \left(q\left(\frac{t}{\varepsilon}\right) - 1\right) + \frac{4}{\delta_{\varepsilon}\varepsilon} \zeta'\left(\frac{2t}{\delta_{\varepsilon}}\right) q'\left(\frac{t}{\varepsilon}\right) + \frac{1}{\varepsilon^2} \zeta\left(\frac{2t}{\delta_{\varepsilon}}\right) W'\left(q\left(\frac{t}{\varepsilon}\right)\right). \end{aligned} $$

Using (3.18), we find

$$ \begin{aligned} -\varepsilon q''_{\varepsilon}(t) + \frac{1}{\varepsilon} W'(q_{\varepsilon}(t)) &= \frac{4\varepsilon}{\delta_{\varepsilon}^2} \zeta''\left(\frac{2t}{\delta_{\varepsilon}}\right) \left(1 - q\left(\frac{t}{\varepsilon}\right)\right) - \frac{4\sqrt{2}}{\delta_{\varepsilon}} \zeta'\left(\frac{2t}{\delta_{\varepsilon}}\right) \left(1 - q\left(\frac{t}{\varepsilon}\right)^2\right) \\ &\quad - \frac{1}{\varepsilon} \left[ \zeta\left(\frac{2t}{\delta_{\varepsilon}}\right) W'\left(q\left(\frac{t}{\varepsilon}\right)\right) - W'\left(\zeta\left(\frac{2t}{\delta_{\varepsilon}}\right) \left(q\left(\frac{t}{\varepsilon}\right) - 1\right) + 1\right) \right]. \end{aligned} $$

Using the explicit expression of $W'$, we now rewrite the term inside the brackets on the right-hand-side of the previous equality as

$$ \begin{aligned} & \zeta\left(\frac{2t}{\delta_{\varepsilon}}\right) W'\left(q\left(\frac{t}{\varepsilon}\right)\right) - W'\left(\zeta\left(\frac{2t}{\delta_{\varepsilon}}\right) \left(q\left(\frac{t}{\varepsilon}\right) - 1\right) + 1\right) \\ &= 4\zeta\left(\frac{2t}{\delta_{\varepsilon}}\right) \left(q\left(\frac{t}{\varepsilon}\right) - 1\right) \left\{ q\left(\frac{t}{\varepsilon}\right) \left(q\left(\frac{t}{\varepsilon}\right) + 1\right) \right. \\ & \qquad \left. - \left[ \zeta\left(\frac{2t}{\delta_{\varepsilon}}\right) \left(q\left(\frac{t}{\varepsilon}\right) - 1\right) + 1 \right] \left[ \zeta\left(\frac{2t}{\delta_{\varepsilon}}\right) \left(q\left(\frac{t}{\varepsilon}\right) - 1\right) + 2 \right] \right\} \end{aligned} $$

so that

$$ \begin{aligned} -\varepsilon q''_{\varepsilon}(t) + \frac{1}{\varepsilon} W'(q_{\varepsilon}(t)) &= \left(1 - q\left(\frac{t}{\varepsilon}\right)\right) \left\{ \frac{4\varepsilon}{\delta_{\varepsilon}^2} \zeta''\left(\frac{2t}{\delta_{\varepsilon}}\right) - \frac{4\sqrt{2}}{\delta_{\varepsilon}} \zeta'\left(\frac{2t}{\delta_{\varepsilon}}\right) \left(1 + q\left(\frac{t}{\varepsilon}\right)\right) \right. \\ & \qquad \left. + \frac{4}{\varepsilon} \zeta\left(\frac{2t}{\delta_{\varepsilon}}\right) \left[q\left(\frac{t}{\varepsilon}\right) \left(q\left(\frac{t}{\varepsilon}\right) + 1\right) - \left(\zeta\left(\frac{2t}{\delta_{\varepsilon}}\right) \left(q\left(\frac{t}{\varepsilon}\right) - 1\right) + 1\right) \left(\zeta\left(\frac{2t}{\delta_{\varepsilon}}\right) \left(q\left(\frac{t}{\varepsilon}\right) - 1\right) + 2\right)\right] \right\}. \end{aligned} $$

A similar computation holds in $\mathbf{R}^-$. In any case, we see that

$$ -\varepsilon q''_{\varepsilon}(t) + \frac{1}{\varepsilon} W'(q_{\varepsilon}(t)) = 0 \quad \text{if } |t| \le \delta_{\varepsilon}/2 \text{ or } |t| \ge \delta_{\varepsilon}, \quad (3.25) $$
while, recalling (3.23),

$$ \sup_{\frac{\delta_\varepsilon}{2} \le |t| \le \delta_\varepsilon} \left| -\varepsilon q_\varepsilon''(t) + \frac{1}{\varepsilon} W'(q_\varepsilon(t)) \right| \le C \varepsilon^{\lambda \sqrt{2}} \left( \frac{1}{\lambda^2 \varepsilon |\log \varepsilon|^2} + \frac{1}{\lambda \varepsilon |\log \varepsilon|} + \frac{1}{\varepsilon} \right), $$

for some constant $C > 0$ independent of $\varepsilon$ and $\lambda$. Hence, for all $p > 1$, there exists $\lambda > 1$ independent of $\varepsilon$ such that

$$ \sup_{\frac{\delta_\varepsilon}{2} \le |t| \le \delta_\varepsilon} \left| -\varepsilon q_\varepsilon''(t) + \frac{1}{\varepsilon} W'(q_\varepsilon(t)) \right| \le \varepsilon^{p/2} \quad (3.26) $$

and the conclusion follows from (3.25) and (3.26), after integration over the interval $[-\delta_\varepsilon, \delta_\varepsilon]$. $\square$

Let $\mathbf{sd} := \text{sdist}(\cdot, \partial E)$ be the signed distance to $\partial E$ with the convention that $\mathbf{sd} \le 0$ in $E$ and $\mathbf{sd} \ge 0$ in $\mathbb{R}^d \setminus E$. Since $\partial E$ is of class $C^2$, there exists $\delta > 0$ such that $\mathbf{sd}$ is of class $C^2$ in $\bar{U}_\delta$, where $U_\delta = \{x \in \mathbb{R}^d : \text{dist}(x, \partial E) < \delta\}$. We finally set

$$ \bar{v}_\varepsilon(x) = q_\varepsilon(\mathbf{sd}(x)) \quad \text{and} \quad \bar{\mu}_\varepsilon = \frac{1}{c_0} \left( \frac{\varepsilon}{2} |\nabla \bar{v}_\varepsilon|^2 + \frac{1}{\varepsilon} W(\bar{v}_\varepsilon) \right) \mathcal{L}^d $$

so that $v_\varepsilon$ is constant equal to $\pm 1$ outside $U_{\delta_\varepsilon}$ and $\text{Supp}(\bar{\mu}_\varepsilon) \subset \bar{U}_{\delta_\varepsilon}$.

In particular we have

$$ \| \bar{v}_\varepsilon - (\mathbf{1}_E - \mathbf{1}_{\Omega \setminus E}) \|_{L^1(\Omega)} \le C \mathcal{L}^d(U_{\delta_\varepsilon}) \le C \lambda \varepsilon |\log \varepsilon| $$

so that $\bar{v}_\varepsilon \to \mathbf{1}_E - \mathbf{1}_{\Omega \setminus E}$ in $L^1(\Omega)$.

We now prove the convergence of the energy. Since the convergence of the curvature type term

$$ \frac{1}{c_0} \int_{\Omega} \frac{1}{\varepsilon} \left( -\varepsilon \Delta \bar{v}_{\varepsilon} + \frac{1}{\varepsilon} W'(\bar{v}_{\varepsilon}) \right)^2 dx \to \int_{\Omega \cap \partial E} |H_E|^2 d\mathcal{H}^{d-1} $$

follows as in e.g. [8, 16], we do not include the details. See also **Step 6** of the proof of the upper bound construction for Theorem 1.4 where similar computations are done. We thus turn to the proof of

$$ \limsup_{\varepsilon \to 0} \int_{\Omega} \varphi_{\varepsilon}(\nabla \bar{v}_{\varepsilon}) d\bar{\mu}_{\varepsilon} \le \int_{\partial E} \varphi(\nu_{\varepsilon}) d\mathcal{H}^{d-1}. \quad (3.27) $$

For $t \in \mathbb{R}$, let $\Gamma_t = \{\mathbf{sd} = t\}$ so that $\nabla \mathbf{sd} = \nu_{\Gamma_t}$ on $\Gamma_t$ for $|t|$ small enough. Since $\nabla \bar{v}_\varepsilon = q'_\varepsilon(\mathbf{sd}) \nabla \mathbf{sd}$ and $|\nabla \mathbf{sd}| = 1$, we use the co-area formula to infer

$$ c_0 \int_{\Omega} \varphi_{\varepsilon}(\nabla \bar{v}_{\varepsilon}) d\bar{\mu}_{\varepsilon} = \int_{-\delta_{\varepsilon}}^{\delta_{\varepsilon}} \left( \frac{\varepsilon}{2} |q'_{\varepsilon}(t)|^2 + \frac{1}{\varepsilon} W(q_{\varepsilon}(t)) \right) \left( \int_{\Gamma_t} \varphi_{\varepsilon}(q'_{\varepsilon}(t)\nu_{\Gamma_t}) d\mathcal{H}^{d-1} \right) dt. $$

Since $\varphi_\varepsilon \le C$ and for $|t| \le \delta_\varepsilon$

$$ \mathcal{H}^{d-1}(\Gamma_t) \le (1 + C|t|)\mathcal{H}^{d-1}(\partial E) \le C\mathcal{H}^{d-1}(\partial E) \quad (3.28) $$

we have by (3.22)

$$ \limsup_{\varepsilon \to 0} \int_{|t| \ge \delta_\varepsilon/2} \left( \frac{\varepsilon}{2} |q'_\varepsilon(t)|^2 + \frac{1}{\varepsilon} W(q_\varepsilon(t)) \right) \left( \int_{\Gamma_t} \varphi_\varepsilon(q'_\varepsilon(t)\nu_{\Gamma_t}) d\mathcal{H}^{d-1} \right) dt \\ \le C \mathcal{H}^{d-1}(\partial E) \limsup_{\varepsilon \to 0} \int_{|t| \ge \delta_\varepsilon/2} \left( \frac{\varepsilon}{2} |q'_\varepsilon|^2 + \frac{1}{\varepsilon} W(q_\varepsilon) \right) dt = 0. $$

Using (3.9) and (3.28), we thus find

$$ c_0 \limsup_{\varepsilon \to 0} \int_{\Omega} \varphi_{\varepsilon}(\nabla \bar{v}_{\varepsilon}) d\bar{\mu}_{\varepsilon} = \limsup_{\varepsilon \to 0} \int_{-\frac{\delta_{\varepsilon}}{2}}^{\frac{\delta_{\varepsilon}}{2}} \left( \frac{\varepsilon}{2} |q'_{\varepsilon}(t)|^2 + \frac{1}{\varepsilon} W(q_{\varepsilon}(t)) \right) \left( \int_{\Gamma_t} \varphi_{\varepsilon}(q'_{\varepsilon}(t)\nu_{\Gamma_t}) d\mathcal{H}^{d-1} \right) dt \\ \le \limsup_{\varepsilon \to 0} \int_{-\frac{\delta_{\varepsilon}}{2}}^{\frac{\delta_{\varepsilon}}{2}} \left( \frac{\varepsilon}{2} |q'_{\varepsilon}(t)|^2 + \frac{1}{\varepsilon} W(q_{\varepsilon}(t)) \right) \left( \int_{\Gamma_t} \varphi(\nu_{\Gamma_t}) d\mathcal{H}^{d-1} \right) dt \\ + C L \mathcal{H}^{d-1}(\partial E) \limsup_{\varepsilon \to 0} r_{\varepsilon} \int_{-\frac{\delta_{\varepsilon}}{2}}^{\frac{\delta_{\varepsilon}}{2}} \left( \frac{\varepsilon}{2} |q'_{\varepsilon}|^2 + \frac{1}{\varepsilon} W(q_{\varepsilon}) \right) \frac{1}{|q'_{\varepsilon}|} dt. $$
For the first right-hand side term we have by (3.21)

$$
\begin{align*}
& \limsup_{\epsilon \to 0} \int_{-\frac{\delta_\epsilon}{2}}^{\frac{\delta_\epsilon}{2}} \left( \frac{\epsilon}{2} |q'_\epsilon(t)|^2 + \frac{1}{\epsilon} W(q_\epsilon(t)) \right) \left( \int_{\Gamma_t} \varphi(\nu_{\Gamma_t}) d\mathcal{H}^{d-1} \right) dt \\
& \le \limsup_{\epsilon \to 0} \left( \sup_{|t| \le \delta_\epsilon} \int_{\Gamma_t} \varphi(\nu_{\Gamma_t}) d\mathcal{H}^{d-1} \right) \int_{-\frac{\delta_\epsilon}{2}}^{\frac{\delta_\epsilon}{2}} \left( \frac{\epsilon}{2} |q'_\epsilon|^2 + \frac{1}{\epsilon} W(q_\epsilon) \right) dt = c_0 \int_{\partial E} \varphi(\nu_E) d\mathcal{H}^{d-1}.
\end{align*}
$$

For the second right-hand side term, since $|q'_\epsilon(t)| = \sqrt{2W(q_\epsilon(t))}/\epsilon$ for $|t| \le \delta_\epsilon/2$, we find

$$
r_{\epsilon} \int_{-\frac{\delta_{\epsilon}}{2}}^{\frac{\delta_{\epsilon}}{2}} \left( \frac{\epsilon}{2} |q'_{\epsilon}|^2 + \frac{1}{\epsilon} W(q_{\epsilon}) \right) \frac{1}{|q'_{\epsilon}|} dt = r_{\epsilon} \epsilon \int_{-\frac{\delta_{\epsilon}}{2}}^{\frac{\delta_{\epsilon}}{2}} |q'_{\epsilon}| dt \le 2\epsilon r_{\epsilon} \to 0.
$$

This concludes the proof of (3.27). □

4. STRONGLY ANISOTROPIC MUMFORD-SHAH FUNCTIONAL WITH CURVATURE

We extend the analysis of the previous section to the approximation of the Mumford-Shah
functional with curvature penalization. In this section we set the dimension $d = 2$ and consider
$\Omega \subset \mathbf{R}^2$ a bounded open set. For $\gamma > 0$ and $u \in \mathcal{A}(\Omega)$ we define the anisotropic Mumford-Shah
functional with curvature penalization by

$$
\mathcal{G}^{(\gamma)}(u) = \int_{\Omega} |\nabla u|^2 dx + \int_{J_u} (\varphi(\nu_u) + |H_{J_u}|^2) d\mathcal{H}^1 + \gamma \mathcal{H}^0(\mathcal{P}_{J_u}). \quad (4.1)
$$

Here $\varphi : \mathbf{S}^{d-1} \to \mathbf{R}^+$ is a continuous anisotropy function satisfying (3.1) as in Section 3. By definition (4.1) of the energy, if $\mathcal{G}^{(\gamma)}(u) < \infty$ then $u \in SBV^2(\Omega)$ and $J_u \in \mathcal{C}(\Omega)$ is a finite union of $\mathcal{C}^2$-curves whose intersection points $\mathcal{P}_{J_u}$ have a nonzero curvature in the sense of varifolds.

**Remark 4.1.** If the jump set $J_u$ is made of three line segments intersecting at a single point, say 0, with an angle $2\pi/3$ (the so-called triple junction configuration), and $u$ is constant inside each connected components of the complementary of $J_u$ (see Figure 1), it is known that the varifold $v(J_u, 1)$ associated to $J_u$ is stationary, i.e., it has zero curvature in the sense of varifolds. However, it is not clear whether the point 0 should be counted in the point energy (4.1) as a junction point, or if it should not as a point of zero generalized curvature.

<figure><img src="image_3.png" /><figcaption>FIGURE 1. The triple junction configuration</figcaption></figure>

In the context of our phase-field approximation, see Subsection 4.2 (with $\Omega = B_1$ and $\varphi \equiv 1$ to
simplify), the construction of the upper bound provided in Subsection 4.2.2 leads to the existence
of a recovery sequence $\{(\bar{u}_\epsilon, \bar{v}_\epsilon, \bar{w}_\epsilon)\}_{\epsilon>0}$ such that

$$
\mathcal{G}_{\epsilon}^{(\gamma)}(\bar{u}_{\epsilon}, \bar{v}_{\epsilon}, \bar{w}_{\epsilon}) \to 3 + \gamma.
$$

The question of deciding whether the point 0 should be counted or not in the point energy is related
to the possibility of improving the upper bound by constructing a sequence $\{(\hat{u}_\epsilon, \hat{v}_\epsilon, \hat{w}_\epsilon)\}_{\epsilon>0}$ such
that

$$
\mathcal{G}_{\epsilon}^{(\gamma)}(\hat{u}_{\epsilon}, \hat{v}_{\epsilon}, \hat{w}_{\epsilon}) \to 3.
$$

See [14, Section 2.3] for a related discussion.