On the Convergence of critical points of the Ambrosio-Tortorelli functional

This work is devoted to study the asymptotic behavior of critical points $\{(u_\varepsilon,v_\varepsilon)\}_{\varepsilon>0}$ of the Ambrosio-Tortorelli functional. Under a uniform energy bound assumption, the usual $\Gamma$-convergence theory ensures that $(u_\varepsilon,v_\varepsilon)$ converges in the $L^2$-sense to some $(u_*,1)$ as $\varepsilon\to 0$, where $u_*$ is a special function of bounded variation. Assuming further the Ambrosio-Tortorelli energy of $(u_\varepsilon,v_\varepsilon)$ to converge to the Mumford-Shah energy of $u_*$, the later is shown to be a critical point with respect to inner variations of the Mumford-Shah functional. As a byproduct, the second inner variation is also shown to pass to the limit. To establish these convergence results, interior ($\mathscr{C}^\infty$) regularity and boundary regularity for Dirichlet boundary conditions are first obtained for a fixed parameter $\varepsilon>0$. The asymptotic analysis is then performed by means of varifold theory in the spirit of scalar phase transition problems.


Introduction
Let Ω ⊂ R N be a bounded open set with Lipschitz boundary (N ≥ 1) and g ∈ H 1 2 (∂Ω) be a prescribed Dirichlet boundary data on ∂Ω. For infinitesimal parameters ε → 0 and η ε → 0 with 0 < η ε ≪ ε, we consider the Ambrosio-Tortorelli functional defined by (1. 2) The Mumford-Shah functional is well known as a theoretical tool to approach image segmentation [35,34,36]. It is also at the heart of the Francfort-Marigo model in fracture mechanics [14], and the numerical implementation of this model heavily relies on Ambrosio-Tortorelli type functionals [8]. The use of such phase-field approximation in numerics is usually justified through Γ-convergence theory.
In terms of the functionals defined above, it states that AT ε Γ-converges in the [L 2 (Ω)] 2 -topology as ε → 0 towards the Mumford-Shah functional (see e.g. the seminal paper [4]) As a consequence, the fundamental theorem of Γ-convergence ensures the convergence of global minimizers (u ε , v ε ) of AT ε to (u, 1) as ε → 0 where u ∈ SBV 2 (Ω) is a global minimizer of M S. This result is of course of importance, but it is somehow not fully satisfactory. Beyond the fact that the use of global minimizers in the models mentioned above remains under debate, this convergence result does not really provide a rigorous justification of the numerical simulations based on the Ambrosio-Tortorelli functional. One particular feature of AT ε is its lack of convexity due to the nonconvex coupling term v 2 |∇u| 2 with respect to the pair (u, v). This is a high obstacle to reach global minimizers through a numerical method. An idea employed in the context of image segmentation or fracture mechanics consists in performing an alternate minimization algorithm, see [9]. Each iteration of the scheme is well-posed since AT ε is continuous, coercive, and separately strictly convex. Letting the number of steps 1 going to infinity, the sequence of iterates turns out to converge to a critical point of the energy AT ε [8, Theorem 1], but this critical point might fail to be a global minimizer. Consequently, the original target of numerically approximating global minimizers of the Mumford-Shah functional might be lost. These issues motivate the question of convergence as ε → 0 of critical points of the Ambrosio-Tortorelli functional and it constitutes the main goal of this article, continuing a task initiated in [13,23] in dimension N = 1. In higher dimension, a fundamental issue in such an analysis is the regularity of critical points of AT ε . It is also of importance for numerics as the efficiency of the numerical methods crucially rests on it. Here, we fully resolve this last question showing smoothness of arbitrary critical points according to the smoothness of ∂Ω and the Dirichlet boundary data.
A critical point (u ε , v ε ) of the Ambrosio-Tortorelli functional is a weak (distributional) solution of the nonlinear elliptic system (u ε , v ε ) = (g, 1) on ∂Ω . (1.3) To be more precise, critical points of AT ε are defined as follows. AT ε (u + tφ, v + tψ) = 0 for all (φ, ψ) ∈ H 1 0 (Ω) × [H 1 0 (Ω) ∩ L ∞ (Ω)] , that isˆΩ (η ε + v 2 ε )∇u ε · ∇φ dx = 0 for all φ ∈ H 1 0 (Ω) , (1.4) and εˆΩ ∇v ε · ∇ψ dx +ˆΩ v ε − 1 4ε + v ε |∇u ε | 2 ψ dx = 0 for all ψ ∈ H 1 0 (Ω) ∩ L ∞ (Ω) . One may expect that critical points of AT ε with uniformly bounded energy converge along some subsequence ε → 0 to a limit satisfying some first order criticality conditions for M S. Unfortunately, the theory of Γ-convergence does not provide convergence of critical points towards critical points of the limiting functional. Even for local minimizers such a result usually fails. We refer to [22,Remark 4.5] and [10, Example 3.5.1] for counter-examples. However, it has been proved in some specific examples that critical points do converge to critical points, possibly under the assumption of convergence of critical values. This is the case for the Allen-Cahn (or Modica-Mortola) functional from phase transitions approximating the (N − 1)-dimensional area functional [37,21,43,44,19,16], the Ginzburg-Landau functional approximating the (N − 2)-dimensional area functional [2,6,28,42,38], and the Dirichlet energy of manifold valued stationary harmonic maps [27,29,30,31,32]. These functionals share many features with AT ε , and we shall take advantage of the existing theory to develop our asymptotic analysis of critical points of AT ε . In particular, we shall make an essential use of both outer and inner variations of the energy, a common approach in all these studies.
1.1. Outer and inner variations. Definition 1.1 is simply saying that the first outer variation of AT ε vanishes at (u ε , v ε ) ∈ A g (Ω) in any direction (φ, ψ). In case of a smooth functional like AT ε , outer variations coincide with Gâteaux differentials. For (u, v) ∈ A g (Ω) and (φ, ψ) ∈ H 1 0 (Ω)×[H 1 0 (Ω)∩L ∞ (Ω)] as before, we introduce the following notation for the first and second outer variations of AT ε (see Lemma A.1 for explicit formulas) AT ε (u + tφ, v + tψ) . (1.7) Concerning the Mumford-Shah functional, the notion of critical points requires some definition and notation. Before doing so, let us first comment on the functional M S in (1.2) we are considering. Contrary to AT ε , the admissible u's for M S are not required to agree with g on ∂Ω in the sense of traces. In turn, the additional term H N −1 (∂Ω ∩ {u = g}) in the expression of M S(u) penalizes "boundary jumps" where the inner trace of u (still denoted by u) differs from g. The expression u = g on ∂Ω is also intended in the sense of traces. In the sequel, we shall often use the following compact notation where J u = J u ∪ (∂Ω ∩ {u = g}), so that J u = Jû withû := u1 Ω + G1 R N \Ω ∈ SBV 2 (R N ) , and G ∈ H 1 (R N ) is an arbitrary extension of g.
Unlike AT ε , the Mumford-Shah functional is not smooth and outer variations must be accordingly defined (see e.g. [1,Section 7.4]). Given u, φ ∈ SBV 2 (Ω) such that J φ ⊂ J u , the first and second outer variations of M S at u in the direction φ are respectively defined and given by In this definition, the requirement J φ ⊂ J u ensures the differentiability at t = 0 of the function t → M S(u + tφ) since H N −1 ( J u+tφ ) remains constantly equal to H N −1 ( J u ) . As a consequence, these differentials provide only information on the "regular part" of the function u, and not on the jump set J u . Note also that the second order condition d 2 M S(u)[φ] ≥ 0 is obviously satisfied at any u, φ as above. On the other hand, the condition J φ ⊂ J u also implies that the direction φ must agree with g on ∂Ω ∩ {u = g}, in agreement with the notion of Dirichlet boundary condition. It is clear that outer variations are not sufficient to define a notion of critical point for M S since admissible perturbations leave the "singular part" H N −1 ( J u ) unchanged. The way to complement outer variations is to consider inner variations, i.e., variations under domain deformations. In doing so (up to the boundary), we shall assume that ∂Ω is at least of class C 2 .
Given a vector field X ∈ C 1 c (R N ; R N ) satisfying X ·ν Ω = 0 on ∂Ω (here ν Ω denotes the outward unit normal field on ∂Ω), we consider its flow map Φ : R × R N → R N , i.e., for every x ∈ R N , t → Φ(t, x) is defined as the unique solution of the system of ODE's    dΦ dt (t, x) = X(Φ(t, x)) , (1.8) According to the standard Cauchy-Lipschitz theory, Φ ∈ C 1 (R × R N ; R N ) is well-defined, and {Φ t } t∈R with Φ t := Φ(t, ·) is a one-parameter group of C 1 -diffeomorphisms of R N into itself satisfying Φ 0 = id. Then the requirement X · ν Ω = 0 on ∂Ω implies that Φ t (∂Ω) = ∂Ω for every t ∈ R. Hence (the restriction of) Φ t is a C 1 -diffeomorphism of ∂Ω into itself, and a C 1 -diffeomorphism of Ω into itself.
Setting {Φ t } t∈R to be the integral flow of X and , the first and second inner variations of M S at u are defined by It can be checked that, provided ∂Ω, g, and G are smooth enough, the above derivatives exist and they can be explicitly computed (see Lemma A.4). Analogously, we define inner variations of the Ambrosio-Tortorelli functional.
We define the first and second inner variations of AT ε at (u, v) by Once again, the limits in (1.9) exist whenever ∂Ω, g, and G are sufficiently smooth, and one can compute them explicitly (see Lemmas A.2 & A.3).
We emphasize that we are considering in Definitions 1.2 & 1.3 deformations up to the boundary. Compare to the usual deformations involving compactly supported perturbations in Ω of the original maps, it requires the additional test function G. This is of fundamental importance for the M S functional to recover information at the boundary since the Dirichlet boundary condition is implemented in the functional as a penalization. Of course, the type of deformations we are using includes as a particular case the usual ones defined only through a vector field X compactly supported in Ω, see Remark A.2.
1.2. First order criticality conditions for M S. In view of the discussion above, the non smooth character of M S forces the appropriate notion of critical point to involve both outer and inner variations. In other words, a critical point of the Mumford-Shah functional is a critical point with respect to both outer and inner variations, a property obviously satisfied by global (and even local) minimizers.
a bounded open set with boundary of class at least C 2 and g ∈ C 2 (∂Ω). and for all X ∈ C 1 c (R N ; R N ) and G ∈ C 2 (R N ) satisfying X · ν Ω = 0 and G = g on ∂Ω. From these criticality conditions, one can derive a set of Euler-Lagrange equations which can be written in a strong form if the smoothness of u * and J u * allows it. Specializing first the condition (1.10) to φ ∈ C ∞ c (Ω) yields div(∇u * ) = 0 in D ′ (Ω) . (1.12) Then, if J u * is regular enough, one can choose test functions φ in (1.10) with a non trivial jump set but smooth up to J u * from both sides. It leads to the following homogeneous Neumann condition see [1, formula (7.42)]. In other words, allowing test functions φ in (1.10) with J φ ⊂ J u * (and not only in φ ∈ C ∞ c (Ω)) provides the weak formulation of (1.13) which complements (1.12). Computing δM S(u * )[X, G] (see formula (A.19)) and using equation (1.12), the stationarity condition (1.11) appears to be independent of the test function G and it reduces tô Here div Ju * X = tr ((Id − ν u * ⊗ ν u * )DX) is the tangential divergence of X on the countably H N −1rectifiable set J u * with ν u * the approximate unit normal to that set. The boundary term in the right hand side of (1.14) is interpreted in the sense of duality by (1.12), and ∇ τ g denotes the tangential derivative of g. If J u * and u * are regular enough, then (1.14) provides the coupling equation where H u * denotes the scalar mean curvature of J u * with respect to the normal ν u * and |∇u * | 2 ± the (accordingly oriented) jump of |∇u * | 2 across J u * (see [1, Chapter 7, Section 7.4]).
In the one-dimensional case N = 1, if Ω = (0, L) for some L > 0, we can see that if u ∈ SBV 2 (0, L) satisfies conditions (1.12)-(1.13), then u is either piecewise constant with a finite number of jumps or u is a globally affine function (with no jump). Indeed, the very definition of SBV 2 (0, L) shows that u has a finite number of jumps. Then, condition (1.12) implies that u is affine in between to consecutive jump points, and (1.13) implies that the slope of all affine functions must be zero. However, condition (1.14) does not play any role because it only implies that |u ′ | is constant in (0, L), where u ′ is the approximate derivative of u. From this, we just deduce that u is a piecewise affine function with equal slopes in absolute value, and it is not sufficient by itself to prove that u is piecewise constant. It indicates that the use of SBV 2 -test functions in (1.10) can not be relaxed to a class of smooth functions (in any dimension).
1.3. Main results. As already mentioned, the main purpose of this article is to investigate the asymptotic behavior of critical points of the Ambrosio-Tortorelli functional as ε → 0. In view of the Γ-convergence result, one may expect that critical points converge to critical points, possibly under the assumption of convergence of energies. Without fully resolving this question, our analysis provides the first answer in this direction in arbitrary dimensions showing that a limit of critical points of AT ε must at least be a critical point of M S with respect to inner variations, i.e., a stationary point of M S. If a critical point (u ε , v ε ) of AT ε is smooth enough, then it is easy to see that it is also stationary, i.e., δAT ε (u ε , v ε ) = 0 (see Lemma A.2). Hence, if regularity of critical points AT ε holds, proving the convergence of the first inner variations implies the announced stationarity of the limit. This is the path we have followed, and the regularity issue is the object of our first main theorem.
is a critical point of AT ε , then (u ε , v ε ) ∈ [C ∞ (Ω)] 2 and the following regularity up to the boundary holds.
We emphasize that the regularity in Theorem 1.1 is highly non trivial since the second equation in (1.3) is of the form ∆v = f with f ∈ L 1 and standard linear elliptic theory does not directly apply. Instead, we shall rely on arguments borrowed from the regularity theory for harmonic maps into a manifold, or more generally for variational nonlinear elliptic systems, see e.g. [17]. The key issue is to prove Hölder continuity of v ε , that we achieve by proving that it belongs to a suitable Morrey-Campanato space. We treat interior and boundary regularity in a similar way through a reflection argument of independent interest originally devised in [40].
In our second main theorem, we show that, under the assumption of convergence of energies, limits (up to a subsequence) of critical points of AT ε are critical points of M S for the inner variations. Theorem 1.2. Assume that Ω ⊂ R N is a bounded open set of class C 2,1 and g ∈ C 2,α (∂Ω) for some α ∈ (0, 1). Let {(u ε , v ε )} ε>0 ⊂ A g (Ω) be a family of critical points of the Ambrosio-Tortorelli functional. Then, the following properties hold: (ii) If, further, the energy convergence (1.17) for all vector field X ∈ C 1 c (R N ; R N ) with X · ν Ω = 0 on ∂Ω. Remark 1.2. At this stage, it is still open whether or not u * is a critical point of M S as we do not know if the outer variation dM S(u * ) also vanishes on arbitrary functions φ ∈ SBV 2 (Ω) satisfying J φ ⊂ J u * (and not only on C ∞ c (Ω)). In other words, the weak form of the homogeneous Neumann condition (1.13) on J u * remains to be established. This is the only missing ingredient to obtain that u * is a critical point of M S.
An assumption of convergence of energies similar to (1.16) has been used in [33,24,25,26] to prove that critical points of the Allen-Cahn functional (from phase transitions) converge towards critical points of the perimeter functional, hence to minimal surfaces. The analysis without this assumption has been first carried out in [21], and it shows that critical points converge (in the sense of inner variations) towards integer multiplicity stationary varifolds, a measure theoretic generalization of minimal surfaces allowing for multiplicities. Interfaces with multiplicities do appear as limits of critical points of the Allen-Cahn energy and cannot be excluded, see e.g. [21,Section 6.3]. In our context, a similar phenomenon may appear, so that assumption (1.16) is probably necessary.
In [24,25,26], convergence of energies is also used to pass to the limit in the second inner variation. Following the same path, (1.16) allows us to pass to the limit in the second inner variation of AT ε . It shows that the second inner variations of AT ε do not converge to the second inner variation of M S, but to the second inner variation plus a residual additional term. As a byproduct, it follows that limits of stable critical points of AT ε satisfy an "augmented" second order minimality condition. Second order minimality criteria for M S has been addressed in [11,7]. We also note that the convergence of the second inner variation for the Allen-Cahn functional without the assumption of convergence of energies has been studied in [15], see also [20]. Convergence of second inner variations is our third and last main result. Theorem 1.3. Assume that Ω ⊂ R N is a bounded open set of class C 3,1 and g ∈ C 3,α (∂Ω) for some α ∈ (0, 1). Let {(u ε , v ε )} ε>0 ⊂ A g (Ω) be a family of critical points of the Ambrosio-Tortorelli functional and u * ∈ SBV 2 (Ω) ∩ L ∞ (Ω) be as in Theorem 1.2, satisfying the convergence of energy (1.16). Then, , then u * satisfies the second order inequality for all X ∈ C 2 c (R N ; R N ) and all G ∈ C 3 (R N ) with X · ν Ω = 0 and G = g on ∂Ω.
In the one-dimensional case, the asymptotic analysis as ε → 0 of critical points of the Ambrosio-Tortorelli functional has already been carried out in [13,23] for different sets of boundary conditions. In [13], a homogeneous Neumann boundary condition is assumed for the phase field variable v. The authors proved that if {(u ε , v ε )} ε>0 is a family of critical points of the Ambrosio-Tortorelli functional satisfying (1.15), then, up to a subsequence, (u ε , v ε ) → (u, 1) in [L 2 (Ω)] 2 with u ∈ SBV 2 (Ω) that is either globally affine or piecewise constant with a finite number of jumps, see Remark 1.1. This result is extended in [23] to the Ambrosio-Tortorelli functional with a fidelity term. Note that our present analysis also applies in the presence of a fidelity term, but we do not consider this case here in order not to add useless difficulties. In a short note [5], we have also carried out the 1D analysis in our setting, i.e., with the Dirichlet boundary condition on the v variable. In this case, we have established a convergence result for critical points without assuming the convergence of the energy (1.16), but proving (1.16) as a consequence of the energy bound (1.15). It allows us to exhibit non-minimizing critical points of AT ε satisfying our energy convergence assumption (1.16) (see [5, 1.4. Ideas of the convergence proof. The proof of Theorem 1.2 relies on the classical compactness argument and the lower bound inequality for the Ambrosio-Tortorelli functional. Indeed, the energy bound for a family {(u ε , v ε )} ε>0 ⊂ A g (Ω) of critical points for AT ε implies the L 2 (Ω)-convergence (up to a subsequence) of u ε to a limit u * ∈ SBV 2 (Ω), together with a Γ-liminf inequality M S(u * ) ≤ lim inf ε AT ε (u ε , v ε ). Our energy convergence assumption (1.16) leads to the equipartition of phase field energy, as well as the convergence of the bulk energy. Then, as in [21], we associate an (N − 1)-varifold V ε to the phase field variable v ε , which converges (again up to a subsequence) to a limiting varifold V * . The energy convergence (1.16) allows us to identify the mass of V * , that is V * = H N −1 J u * . Next, we use the equations satisfied by (u ε , v ε ) in their conservative form to pass to the limit, and find an equation satisfied by u * and V * . The idea is then to employ a blow-up argument similar to [2] to identify (the first moment of) V * , and show that it is the rectifiable varifold associated to J u * with multiplicity one.
To prove Theorem 1.3, we argue as in [24,25,26]. We observe that the convergence V ε ⇀ V * in the sense of varifolds and the identification of V * implies the convergence of quadratic terms ε∇v ε ⊗ ∇v ε ⇀ 1 2 ν u * ⊗ ν u * H N −1 J u * in the sense of measures. This information is precisely what is needed to pass to the limit in the second inner variation of AT ε , and we infer from a stability condition on (u ε , v ε ) ∈ A g (Ω) a stability condition on the limit u * ∈ SBV 2 (Ω).
The paper is organised as follows. Section 2 collects several notation that will be used throughout the paper. In Section 3, we study the regularity theory for critical points of the Ambrosio-Tortorelli functional proving first smoothness in the interior of the domain, and then smoothness at the boundary. In Section 4, we prove compactness of a family {(u ε , v ε )} ε>0 satisfying a uniform energy bound sup ε AT ε (u ε , v ε ) < ∞. The regularity result allows one to derive the conservative form of the equations satisfied by these critical points which itself provides bounds on the normal traces of u ε and v ε on ∂Ω. Then, in Section 5, we improve the previous results by assuming the energy convergence AT ε (u ε , v ε ) → M S(u * ). From this assumption we obtain equipartition of the phase field part of the energy. Then, we employ a reformulation in terms of varifolds to pass to the limit in the inner variational equations satisfied by critical points of AT ε to prove that the weak limit u * of u ε is a stationary point of the Mumford-Shah energy. The asymptotic behavior of the second inner variations is performed in Section 6.

Notation and preliminaries
2.1. Measures. The Lebesgue measure in R N is denoted by L N , and the k-dimensional Hausdorff measure by H k . We will sometime write ω k for the L k -measure of the k-dimensional unit ball in R k .
If X ⊂ R N is a locally compact set and Y an Euclidean space, we denote by M(X; Y ) the space of Yvalued bounded Radon measures in X endowed with the norm µ = |µ|(X), where |µ| is the variation of the measure µ. If Y = R, we simply write M(X) instead of M(X; R). By Riesz representation theorem, M(X; Y ) can be identified with the topological dual of C 0 (X; Y ), the space of continuous functions f : X → Y such that {|f | ≥ ε} is compact for all ε > 0. The weak* topology of M(X; Y ) is defined using this duality.
2.2. Functional spaces. We use standard notation for Lebesgue, Sobolev and Hölder spaces. Given a bounded open set Ω ⊂ R N , the space of functions of bounded variation is defined by We shall also consider the subspace SBV (Ω) of special functions of bounded variation made of functions u ∈ BV (Ω) whose distributional derivative can be decomposed as In the previous expression, ∇u is the Radon-Nikodým derivative of Du with respect to L N , 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 H N −1 -rectifiable subset of Ω oriented by the (approximate) normal direction of jump ν u : J u → S N −1 , and u ± are the one-sided approximate limits of u on J u according to ν u . Finally we define

2.3.
Varifolds. Let us recall several basic ingredients of the theory of varifolds (see [41] for a detailed description). We denote by G N −1 the Grassmannian manifold of all (N − 1)-dimensional linear subspaces of R N . The set G N −1 is as usual identified with the set of all orthogonal projection matrices onto (N − 1)-dimensional linear subspaces of R N , i.e., N × N symmetric matrices A such that A 2 = A and tr(A) = N − 1, in other words, matrices of the form A = Id − e ⊗ e for some e ∈ S N −1 . A We say that an (N − 1)-varifold is stationary in U if δV (ϕ) = 0 for all ϕ in C 1 c (U ; R N ). We recall that such a varifold satisfies the monotonicity formula [41, paragraph 40]).
2.4. Tangential divergence. Let Γ be a countably H N −1 -rectifiable set and let T x Γ its approximate tangent space defined for H N −1 -a.e. x ∈ Γ. We consider an orthonormal basis

Regularity theory for critical points of the Ambrosio-Tortorelli energy
In this section, we investigate interior and boundary regularity properties of critical points of the Ambrosio-Tortorelli functional AT ε for a parameter ε > 0 which is kept fixed.
3.1. Interior regularity. We first establish interior regularity following ideas used by T. Rivière in [39] to prove the regularity of harmonic maps with values into a revolution torus.
Proof. For simplicity, we drop the subscript ε in (u ε , v ε ) and write instead (u, v). We also assume N ≥ 2 since in the case N = 1, the regularity of (u, v) solution of (1.4)-(1.5) is elementary.
, the matrix field (η ε + v 2 )Id has bounded measurable coefficients and it satisfies in Ω in the sense of quadratic forms. It is therefore uniformly elliptic and the De Giorgi-Nash-Moser regularity theorem applies to equation (3.1). It provides the existence of α ∈ (0, 1) such that u ∈ C 0,α loc (Ω) together with the estimate: for every open subset ω such that ω ⊂ Ω (see e.g. Theorem 8.13 and Eq. (8.18) in [17]). Now we claim that the function v belongs to C 0,α loc (Ω). Before proving this claim, we complete the proof of the theorem. Assuming the claim to be true, we can use the Schauder estimates (see e.g. [17,Theorem 5.19]) to derive from equation (3.1) that u ∈ C 1,α loc (Ω). On the other hand, by (1.5), v weakly solves Since the right-hand-side of (3.3) belongs to C 0,α loc (Ω), it follows from standard Schauder estimates that v ∈ C 2,α loc (Ω). By a classical bootstrap, it now follows from equations (3.1) and (3.3) that both u and v are of class C ∞ in Ω.
Hence, it only remains to show the claim v ∈ C 0,α loc (Ω). To this purpose, we fix an arbitrary ball B 2R (x 0 ) ⊂ Ω, and we aim to prove that v ∈ C 0,α To show that v 2 ∈ C 0,α loc (B R (x 0 )), the Morrey-Campanato Theorem (see e.g. [17,Theorem 5.7]) ensures that it suffices to prove the following Morrey type estimate: Let y ∈ B R (x 0 ) and r ∈ (0, R) arbitrary. We denote by w ∈ v 2 + H 1 0 (B r (y)) the harmonic extension of v 2 in the ball B r (y), i.e., the unique (weak) solution of . Moreover, |∇w| 2 being subharmonic in B r (y), we get that for every ̺ < r, Recalling w also minimizes the Dirichlet integral among all functions agreeing with v 2 on ∂B r (y), we infer thatˆB In view of (3.2), we have thus proved that for every y ∈ B R (x 0 ) and 0 < ̺ ≤ r < R, ). By using a classical iteration lemma (see e.g. [17, Lemma 5.13]), we infer that for every y ∈ B R (x 0 ) and 0 < ̺ < R, for a constant C α depending only on α and N . Hence v 2 satisfies the Morrey estimate (3.6), and thus v 2 ∈ C 0,α loc (B R (x 0 )). In turn, v = v 1 + v 2 ∈ C 0,α loc (B R (x 0 )) and the proof of the claim is complete. 3.2. Maximum principle and boundary regularity. We first show a (standard) maximum principle which stipulates that v ε takes values between 0 and 1, and that u ε is bounded whenever the boundary condition g is.
Proof. For a generic function f ∈ L 1 (Ω), we set f + := (f + |f |)/2 and f − := (|f | − f )/2. For simplicity, we drop the subscript ε in (u ε , v ε ) and write instead (u, v). Since in Ω, and thus u L ∞ (Ω) ≤ M . Next we study the boundary regularity of a critical point (u ε , v ε ) of the Ambrosio-Tortorelli energy. Our strategy is to use a local reflexion argument to extend (u ε , v ε ) across the boundary. The extension will then satisfy a modified system of PDEs for which we can apply an interior regularity result (similar to that of Theorem 3.1). The reflexion argument originates in [40] and follows the arguments in [12]. Note that Lemma 3.1 and Theorem 3.2 together with a standard covering argument completes the proof of Theorem 1.1.
Proof. We start by describing the reflexion method that we use to extend functions across ∂Ω in a neighborhood of the point x 0 . We assume that x 0 ∈ ∂Ω and R > 0 are fixed, and that the assumption of the theorem is satisfied. Since ∂Ω ∩ B 4R (x 0 ) is (at least) of class C 2,1 , we can find a small δ 0 ∈ (0, R/2) such that the nearest point projection on ∂Ω ∩ B 4R (x 0 ), denoted by π Ω , is well-defined and (at least) of class C 1,1 in a tubular neighborhood of size 2δ

Next we consider the bounded open set
The mapping σ Ω being involutive, we have Differentiating the relation σ Ω (σ Ω (x)) = x yields Dσ Ω (x)Dσ Ω (σ Ω (x)) = Id, and thus where p x is the orthogonal projection from R N onto the tangent space T x (∂Ω) to ∂Ω at x, i.e., (Dσ Ω (x)) T is the reflexion matrix across the hyperplane T x (∂Ω). In particular, In view of (3.10), j and A are Lipschitz continuous in Ω and A is uniformly elliptic, i.e., there exist two constants 0 < λ Ω ≤ Λ Ω such that With these geometrical preliminaries, we are now ready to provide the extension of (u ε , v ε ) to Ω. We define for x ∈ Ω, and By the chain rule in Sobolev spaces and the fact that the traces of these functions coincide on both side of ∂Ω ∩ Ω, each one of them belongs to H 1 ( Ω). In addition, v ε and v ε also belong to L ∞ ( Ω) since v ε ∈ L ∞ (Ω). We finally set Now we show that these extensions satisfy suitable equations in the domain Ω.
Step 2: proof of (3.14). We proceed as above, starting with On the other hand, Since ϕ a ∈ H 1 0 ( Ω ∩ Ω), we can apply the second equation in (1.3) to deduce that Summing up (3.17), (3.18), and (3.19), and using that σ Ω is an involution leads to Changing variables in the two last integrals, we obtain and (3.14) follows.
We now provide a general regularity result generalizing the argument used in the proof of the interior regularity.
Proof. Throughout the proof, we fix an exponent α ∈ (0, γ/2) and we set β := γ − 2α > 0. We also denote by K an upper bound for z L ∞ (BR) , and by M an upper bound for (3.20). Then C > 0 shall stand for a constant (which may vary from line to line) depending only on N , α, γ, λ, Λ, K, M , and the Lipschitz constant of A. Let us fix x 0 ∈ B R/2 and ̺ ∈ (0, R/2] arbitrary, and consider w ∈ Recalling that Moreover, according to the maximum principle, w L ∞ (B̺(x0)) ≤ z L ∞ (B̺(x0)) ≤ K. First, we infer from the triangle inequality, We start by estimating the first term in the right-hand-side of (3.25) using (3.24), and the fact that A is Lipschitz continuous and uniformly elliptic. It yieldŝ To estimate the second term in the right-hand-side of (3.25), we make use of equation (3.21) to writê Using assumption (3.20) on f , we infer that On the other hand, Equation (3.23) satisfied by w implies that We now choose ̺ = ̺ k = 2 −(k+1) R for k ∈ N, and we obtain Next, we observe that if (θ k ) k∈N , (σ k ) k∈N , and (y k ) k∈N are real sequences such that θ k ∈ (1, ∞), θ := ∞ k=0 θ k < ∞, σ k ∈ (0, ∞), σ := ∞ k=0 σ k < ∞, and satisfying y k+1 ≤ θ k y k + σ k for all k ∈ N, then y k ≤ θ(y 0 + σ). Applying this principle with for all k ∈ N (we have also used the elementary estimates θ ≤ e C √ R and σ ≤ CR β/2 ). Since for all ̺ ∈ (0, R/2], there exists a unique k ∈ N such that ̺ k+1 < ̺ ≤ ̺ k and 1 Finally, by ellipticity of A and the arbitrariness of x 0 , we conclude that (3.22) holds with By Morrey's Theorem (see e.g. [17,Theorem 5.7]), it then follows that v ∈ C 0,α (B R/2 ).
We are now ready to prove the boundary regularity result in Theorem 3.2.
We first improve the regularity of u which satisfies (3.13). We aim to apply the De-Giorgi-Nash-Moser Theorem to infer that u is locally Hölder continuous in Ω and that a suitable Morrey estimate holds for ∇ u. Since equation (3.13) is linear with respect to u, we first observe that x ∈ Ω and all ξ ∈ R N , and h : The function f is a Carathéodory function, and since A is uniformly elliptic and the functions v and h are essentially bounded, we can find positive constants c 1 , c 2 , and c 3 such that x ∈ Ω and all ξ ∈ R N .
If ∂Ω∩B 4R (x 0 ) is of class C k,α and g ∈ C k,α (∂Ω∩B 4R (x 0 )) with k ≥ 3, one can iterate the preceding argument using elliptic boundary regularity to conclude that u and v belong to C k,α (Ω ∩ B θR (x 0 )) for some θ > 0 small enough.

1)
and The regularity of solutions established in Theorem 1.1 allows us to prove that critical points of the Ambrosio-Tortorelli functional satisfies a Noether type conservation law, which is the starting point of their asymptotic analysis.
where X τ := X − (X · ν Ω )ν Ω is the tangential part of X, and ∇ τ g is the tangential gradient of g on ∂Ω.
Proof. Let us fix an arbitrary X ∈ C 1 c (R N ; R N ). By Theorem 1.1, (u ε , v ε ) ∈ [C 2,α (Ω)] 2 and (1.3) is satisfied in the classical sense. Multiplying the first equation of (1.3) by X · ∇u ε (which belongs to C 1 (Ω)) and by integration by parts, a stantard computation yields Similarly, multiplying the second equation in (1.3) by X ·∇v ε (which belongs to C 1 (Ω)) and performing similar integration by parts leads to since v ε = 1 on ∂Ω. Then the conclusion follows summing up (4.4) and (4.5).
Remark 4.1. The fact that critical points (u ε , v ε ) enjoy the higher regularity [C 2,α (Ω)] 2 allows one to obtain a strong form of the conservative equations for (u ε , v ε ). In particular, some information on the boundary are recovered since the vector field X does not need to be tangential on ∂Ω. This additional information will be instrumental in Section 5 to characterize the boundary term occurring in the first inner variation of the Mumford-Shah functional.
Since ∇u * belongs to L 2 (Ω; R N ), and div(∇u * ) = 0, the normal trace ∇u * · ν Ω is well defined as an element of H − 1 2 (∂Ω). Recalling that v ε = 1 on ∂Ω, we get that We now improve this convergence into a weak convergence in L 2 (∂Ω). For that, let us consider a test function X ∈ C 1 c (R N ; R n ) such that X = ν on ∂Ω in relation (4.3). Using that the left-hand side of (4.3) is clearly controlled by the Ambrosio-Tortorelli energy (see (1.15)), we infer that On the one hand, we obtain that {∂ ν u ε } ε>0 is bounded in L 2 (∂Ω), hence ∂ ν u ε ⇀ ∇u * · ν Ω weakly in L 2 (∂Ω). On the other hand, there exists a nonnegative Radon measure λ * ∈ M + (∂Ω) such that Remark 4.2. Our choice of Dirichlet boundary conditions for both u and v in (1.3) allows one to obtain an ε-dependent boundary term which is nonnegative in the boundary integral involving X · ν Ω in (4.3). This sign information is essential to get a limit boundary term which is a measure λ * concentrated on ∂Ω. If we had chosen a Neumann condition for v and a Dirichlet condition for u as in [13], one would have obtained a more involved boundary term which would lead to a first order distribution on ∂Ω in the ε → 0 limit. It is not clear in this case how to perform the analysis in Section 5 (in particular Lemma 5.3).

Convergence of critical points
Our objective is to show that u * is a critical point of the Mumford-Shah functional. We now improve the convergence results established at the previous section by additionally assuming the convergence of the energy (1.16), i.e., AT ε (u ε , v ε ) → M S(u * ). Under this stronger assumption, we can improve the above established convergences and in particular obtain the equipartition of the phase-field energy.
The proof of Theorem 1.2 is based on (geometric) measure theoretic arguments. Let us define the where w ε := Φ(v ε ), and Φ(t) = t − t 2 /2 for t ∈ [0, 1]. By the coarea formula, this definition is equivalent to the definition of a varifold associated to a function in [21]. By standard compactness of bounded Radon measures, at the expense of extracting a further subsequence, there exists a varifold For H N −1 almost every x ∈ J u * , we set Owing to our various convergence results, we are now in position to pass to the limit in the inner variation equation (4.3). The limit expression is for now depending on the abstract limit varifold V * through its first moment A of V x , and the abstract boundary measure λ * introduced in Lemma 4.1.
Proof. Using the strong convergence (5.1) established before, it is easy to pass to the limit in the first integral and in the left hand side of (4.3). We get for all X ∈ C 1 c (R N ; R N ), According to Lemma 4.1 we can also pass to the limit in the boundary integrals in the right hand side of (4.3), we get that It remains to pass to the limit in the second integral in the left hand side of (4.3). Using the chain rule, we have ∇w ε = Φ ′ (v ε )∇v ε . The equipartition of energy (5.4) thus implies that Gathering (5.11), (5.12) and (5.13), we infer that (5.10) holds.
Let us now identify the first moment A of the measure V x . We first establish some algebraic properties of this matrix. where ρ denotes the spectral radius. Proof. To simplify notation, we set The matrix A ε is well-defined on the set Ω ∩ {∇w ε = 0}, it is a symmetric matrix corresponding to the orthogonal projection on {∇w ε } ⊥ . It satisfies A ε ≥ 0, tr(A ε ) = N −1, and ρ(A ε ) = 1 in Ω∩{∇w ε = 0}. For all ϕ ∈ C 0 (Ω), we havê which shows that tr(A) = (N − 1) H N −1 -a.e. on J u * . If further ϕ ≥ 0 and z ∈ R N , then As a consequence, for all z ∈ R N , we have Az · z ≥ 0 H N −1 -a.e. on J u * , from which we deduce that A is a nonnegative matrix H N −1 -a.e. on J u * . Since for all ϕ ∈ C 0 (Ω) we havê we deduce that (5.14) Using that the spectral radius ρ is a convex, continuous, and positively 1-homogeneous function on the set of symmetric matrices, it follows from Reshetnyak continuity Theorem (see [1,Theorem 2.39]) that for all ϕ ∈ C 0 (Ω), We now focus on the interior structure of the varifold V * . Proof.
Step 1: Let us show that for To this aim, we perform a blow-up argument on the first variation equation (5.10). Let x 0 ∈ J u * be such that (1) x 0 is a Lebesgue point of A with respect to H N −1 J u * ; (2) J u * admits an approximate tangent space at x 0 given by It turns out that H N −1 -almost every point x 0 ∈ J u * satisfies these properties. Indeed, (1) is a consequence of the Besicovitch differentiation theorem, (2) and (3) are consequences of the rectifiability of J u * (see Theorems 2.63 and 2.83 in [1]), while condition (4) is a consequence of (3) together with the fact that the measure |∇u * | 2 L N Ω is singular with respect to H N −1 J u * . Let x 0 ∈ J u * be such a point and let ̺ > 0 be such that . Taking φ x0,̺ as test vector field in (5.10) (note that φ x0,̺ = 0 in a neighbourhood of ∂Ω) yieldŝ Dividing this identity by ̺ N −2 yields We first show that the left hand side of the previous equality tends to zero as ̺ → 0. Indeed, thanks to our choice of x 0 , we have for some constant C > 0. For what concerns the right hand side, using first that x 0 is a Lebesgue point of A and (5.15), we get that Using next that J u * admits an approximate tangent space that we denote by T x0 at x 0 , we obtain that Hence,ˆT Let t ∈ (0, 1) be such that t < √ N −N , the measure ν : According to [2, Lemma 3.9] with β = s = N − 1, we get that the matrix A(x 0 ) has at most N − 1 nonzero eigenvalues. Recalling that tr(A(x 0 )) = N − 1 and that all eigenvalues of A(x 0 ) belong to [0, 1], this implies that A(x 0 ) has exactly N − 1 eigenvalues which are equal to 1, and one eigenvalue which is zero. Hence, there exists e ∈ S N −1 such that A = Id − e ⊗ e.
Next we focus on boundary points.
Proof. We perform again a blow-up argument, this time at boundary points. Let x 0 ∈ J u * ∩ ∂Ω be such that: (2) J u * admits an approximate tangent space at x 0 which coincides with the (usual) tangent space of ∂Ω at x 0 (this in particular implies that ν u * (x 0 ) = ±ν Ω (x 0 )); It turns out that H N −1 almost every point x 0 ∈ J u * ∩ ∂Ω satisfies these properties. Indeed, (1) is a consequence of the Besicovitch differentiation Theorem while (2)  can be obtained similarly replacing λ * by |∇u * · ν|H N −1 ∂Ω. We choose such a point x 0 ∈ ∂Ω ∩ J u * and we take ̺ > 0.
Step 1: We first prove that ν Ω (x 0 ) is an eigenvector of A(x 0 ). Consider first a test vector field φ of the form φ(x) is such that τ · ν Ω = 0 on ∂Ω. Plugging φ in (5.10) and using estimates similar to the proof of Lemma 5.3, we obtain Note that to get (5.17), the boundary term is cancelled thanks to the second property of (5). Let {τ 1 , . . . , τ N −1 } be an orthonormal basis of T x0 , and ν := ν Ω (x 0 ) be the outward unit normal to Ω at x 0 (i.e. ν is a normal vector to T x0 ). We choose the vector fieldτ in such a way thatτ (x 0 ) = τ i , and we decompose ∇ϕ along the orthonormal basis {τ 1 , . . . , τ N −1 , ν} of R N as ∇ϕ = From the arbitrariness of ϕ, it follows that ( 1] (recall that all eigenvalues of A(x 0 ) belong to [0, 1] by Lemma 5.2). Thus ν is an eigenvector of A(x 0 ), and by the spectral theorem, we can also assume without loss of generality that τ 1 , . . . , τ N −1 are also eigenvectors of A(x 0 ).
Step 2: We next show that A(x 0 ) is the projection matrix onto the tangent space to ∂Ω at x 0 .
Proof of Theorem 1.2. i) This point is a consequence of Lemma 4.1.
ii) Using that ν u * = ±ν Ω H N −1 -a.e. in ∂Ω ∩ J u * and gathering Lemmas 5.3 and 5.4 yields A = Id − ν u * ⊗ ν u * H N −1 -a.e. in J u * . Thus, according to (5.8) and (5.9), we get that . Specifying this identity to vector fields X ∈ C 1 c (R N ; R N ) satisfying X ·ν Ω = 0 on ∂Ω leads tô and (1.17) follows from the definition of the tangential divergence of X on the countably H N −1rectifiable set J u * .
The results of this section also give the following convergences that will be used in Section 6.

Passing to the limit in the second inner variation
The aim of this section is to complement Theorem 1.2 proving also the convergence of the second inner variation of AT ε . As a consequence, we shall deduce that if the limit u * comes from stable critical points of AT ε , then u * satisfies a certain stability condition for M S. Our analysis and result parallels completely the ones in [24,25,26] for the Allen-Cahn type energies arising in phase transitions problems.
Proof of Theorem 1.3. Assume that ∂Ω is of class C 3,1 and g ∈ C 3,α (∂Ω) for some α ∈ (0, 1). By Theorem 3.2, if (u ε , v ε ) is a critical points of the Ambrosio-Tortorelli functional then it belongs to [C 3,α (Ω)] 2 . To prove the convergence of the second inner variation, we use Lemma A.3 and formula (A.9). From Proposition 5.1, we know that On the other hand, Corollary 5.1 ensures that ε∇v ε ⊗ ∇v ε L N Ω * ⇀ 1 2 ν u * ⊗ ν u * H N −1 J u * weakly* in M(Ω; M N ×N ). Let X ∈ C 2 c (R N ; R N ) and G ∈ C 2 (R N ) be such that X · ν Ω = 0 and G = g on ∂Ω, and set Y := (DX)X. Observing that |DX T ∇v ε | 2 = (DX(DX) T ) : (∇v ε ⊗ ∇v ε ), we can pass to the limit in all the terms of δ 2 AT ε (u ε , v ε )[X, G] in (A.9) to find that Using the geometric formulas stated in the proof of Theorem 1.1 p. 1851-1852 in [24], we infer that According to the expression of the inner second variation of the Mumford-Shah energy stated in Lemma A.4, (6.1) and (6.2), we infer that Now assume that (u ε , v ε ) ∈ A g (Ω) is a stable critical point of AT ε , i.e.
Back to (6.3), it follows that Passing now to the limit in the second inner variation yields (1.19), and the proof of Theorem 1.3 is now complete. Remark 6.1. In [11,7], the authors explore second order minimality conditions for the Mumford-Shah functional in the case where the jump set is regular enough. Such conditions could be derived in our context, taking care of the Dirichlet boundary data and thus of the fact that the jump set can charge the boundary. We do not develop this point here and refer to [11,Theorem 3.6] where the authors provide another expression for δ 2 M S(u) defined for smooth vector fields X compactly supported in Ω (see Remark A.2). But we indicate that, as a consequence of Theorem 1.3, it can be seen that, if (u ε , v ε ) ∈ A g (Ω) is a stable critical point of AT ε such that, up to a subsequence, u ε → u * in L 2 (Ω) and (1.16) hold, then u * satisfies the second order minimality condition for the Mumford-Shah functional derived in [11,7]) provided J u * is sufficiently smooth. This follows by choosing X ∈ C ∞ c (R N ; R N ) of the form X = ϕν u * • Π Ju * in a neighborhood of J u * and satisfying ν u * · (DXν u * ) = 0 on J u * , where Π Ju * denotes the nearest point projection onto J u * and ϕ is an arbitrary smooth scalar function.

Appendix: First and second variations
In this appendix we derive explicit expressions for outer and inner variations of the Ambrosio-Tortorelli and Mumford-Shah functionals. First, we recall the expression of the first and second outer variations of AT ε are defined by (1.6) and (1.7).
The computations of inner variations rely on one-parameter groups of diffeomorphisms over Ω, or equivalently on their infinitesimal generators. More precisely, assuming that ∂Ω is of class C k+1 with k ≥ 1, and given a vector field X ∈ C k c (R N ; R N ) satisfying X · ν Ω = 0 on ∂Ω, we consider the integral flow {Φ t } t∈R of X defined through the resolution of the ODE (1.8) for every x ∈ R N . Then Φ 0 = id and the flow rule asserts that Φ t+s = Φ t • Φ s . Since X · ν Ω = 0 on ∂Ω, {Φ t } t∈R is a one-parameter group of C k -diffeomorphism from Ω into itself, and from ∂Ω into itself.
Given a (sufficiently smooth) boundary data g, we consider an arbitrary (smooth) extension G of g to Ω to define a one-parameter family of deformations {(u t , v t )} t∈R ⊂ A g (Ω) satisfying (u 0 , v 0 ) = (u, v) for a given pair (u, v) ∈ A g (Ω) by setting u t := u • Φ −t − G • Φ −t + G and v t := v • Φ −t . The first and second inner variations δAT ε and δ 2 AT ε of AT ε at (u, v) are then defined by (1.9).
Remark A.2. We emphasize that δAT ε (u, v)[X, G] and δ 2 AT ε (u, v)[X, G] depend on both the vector field X and the extension G of the boundary data g, because the family of deformations {(u t , v t )} t∈R depends on X and G. It allows one to perform inner variations of the energy up to the boundary. This type of deformations includes the more usual variation {(u • Φ −t , v • Φ −t } t∈R with X compactly supported in Ω. Indeed, in this case we may choose an extension G supported in a small neighborhood of ∂Ω in such a way that supp G ∩ supp X = ∅. Then G • Φ −t = G, and thus u t = u • Φ −t . If the pair (u, v) and ∂Ω are smooth enough, one can compute the first and second inner variations of AT ε at (u, v) using the Taylor expansion of (u t , v t ) with respect to the parameter t. One may for instance follow the general setting of [ (i) Then for every vector field X ∈ C 1 c (R N ; R N ) and every extension G ∈ C 2 (R N ) satisfying X · ν Ω = 0 and G = g on ∂Ω, (ii) If further ∂Ω is of class C 3 , g ∈ C 3 (∂Ω), and (u, v) ∈ A g (Ω) ∩ [C 3 (Ω)] 2 , then for every vector field X ∈ C 2 c (R N ; R N ) and every extension G ∈ C 3 (R N ) satisfying X · ν Ω = 0 and G = g on ∂Ω, Since (u, v) and G belong to C 2 (Ω), we can differentiate (u t , v t ) with respect to t and use (1.8) with the flow rule Φ t+s = Φ t • Φ s to find In particular, we have If ∂Ω is of class C 3 , g ∈ C 3 (∂Ω), and (u, v) ∈ A g (Ω) ∩ [C 3 (Ω)] 2 , then we can differentiate (u t ,v t ) with respect to t to obtain Hence,ü Next, elementary computations yield , so that the conclusion follows from (A.5)-(A.6) evaluating those derivatives at t = 0.
In case the pair (u, v) only belongs to the energy space A g (Ω), we can compute the first and second inner variations of AT ε by making the change variables y = Φ t (x) in the integrals defining AT ε (u t , v t ). Then one expands the result with respect to t using a Taylor expansion of Φ t . If X ∈ C 2 c (R N ; R N ), the second order Taylor expansion near t = 0 of the flow map Φ t induced by X is given by where Y ∈ C 1 c (R N ; R N ) denotes the vector field Y := (DX)X , DX being the Jacobian matrix of X (i.e., (DX) ij = ∂ j X i with i the row index and j the column index), and o(s) denotes a quantity satisfying o(s)/s → 0 as s → 0 uniformly with respect to x ∈ R N . Lemma A.3. Let Ω ⊂ R N be a bounded open set with boundary of class C 2 , g ∈ C 2 (∂Ω) and (u, v) ∈ A g (Ω).
If we take X ∈ C ∞ c (Ω; R N ) and G ∈ C 2 (R N ) an extension of g such that supp G ∩ supp X = ∅, and if we assume (u, v) ∈ A g (Ω) to be a critical of AT ε , then the expression of the second-inner variation (A.9) simplifies. Indeed the terms that contain Y = (DX)X disappear, since by regularity we have Proof of Lemma A. 3. For simplicity, we assume that ∂Ω is of class C 3 , g ∈ C 3 (∂Ω), and we observe that the computation of δAT ε below only requires C 2 regularity. We fix X ∈ C 2 c (R N ; R N ) and G ∈ C 3 (Ω) satisfying X ·ν = 0 and G = g on ∂Ω. We set u t := u • Φ −t and G t : We aim to compute the first and second derivatives at t = 0 of A, B, and C, starting with A. By the chain rule in Sobolev spaces, we have Using the change of variables x = Φ s (y) and (A.12)-(A.13) again, we obtain Consequently, and in particular, To compute B ′′ (0), we write B ′ (t) =: I(t) − II(t), and we set for simplicity H := X · ∇G. Since ∇H(Φ t ) = ∇H + t∇(X · ∇H) − t(DX) T ∇H + o(t) , we can change variables y = Φ t (x) and use again (A.11)-(A.12)-(A.13) to find I(t) = 2ˆΩ(η ε + v 2 )∇u · ∇H dx + 2tˆΩ(η ε + v 2 )∇u · ∇(X · ∇H) dx + 2tˆΩ(η ε + v 2 ) (∇u · ∇H)divX − (∇u ⊗ ∇H) : DX + (DX) T dx + o(t) .
Proof. The second inner variation of the part´Ω |∇u| 2 dx is computed exactly as in the proof of Lemma A.3, recalling that the chain rule still holds for the approximate gradient ∇(u•Φ −t ) = [DΦ −t ] T ∇u(Φ −t ). For the singular part of the energy, we use that H N −1 ( J ut ) = H N −1 (Jû t ) whereû t =û • Φ −t and u = u1 Ω + G1 R N \Ω . The second variation of such a functional is computed with the area formula as in [41,Chapter 2] together with the following geometric formulas τ i · ∂ τj X (τ j · ∂ τi X) + ((ν u ⊗ ν u ) : ∇X) 2 stated in the proof of Theorem 1.1 p. 1851-1852 in [24].