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Topology optimization for incremental elastoplasticity

We discuss a topology optimization problem for an elastoplastic medium. The distribution of material in a region is optimized with respect to a given target functional taking into account compliance. The incremental elastoplastic problem serves as state constraint. We prove that the topology optimization problem admits a solution. First-order optimality conditions are obtained by considering a regularized problem and passing to the limit.


Introduction
Topology Optimization is concerned with determining the optimal shape of a mechanical piece against a number of criteria. Besides mechanical performance, these criteria may include weight, manufacturing costs, topological, and geometrical features. The distribution of material within an a-priori given region Ω ⊂ R n is the control parameter of the process. Given the portion E ⊂ Ω to be filled with material, one determines the mechanical response of the body and minimizes a target functional depending on E and such response. This very general setting applies to a number of different shape design problems, from mechanical engineering, to aerospace and automotive, to architectural engineering, to biomechanics [7].
On the more theoretical side, the linear elastic setting has been considered in a number of contributions. In [10] the existence of optimal shapes is tackled by introducing a penalization of interfaces between solid E and void Ω \ E and considering an additional phase-field regularization. The actual position of the body is hence modeled as a level set of a scalar order parameter z ∈ [0, 1] and gradients of this parameter are penalized. By removing such penalization, a solution of the original sharp-interface limit is recovered by means of a Γ-convergence argument [16]. Existence under additional stress constraints is discussed in [13] and the extension to hyperelasticity is presented in [36]. More recently, first-order optimality conditions have been obtained in the linear elastic setting [8,9,15] for both the sharp-interface model and its phase-field approximation, also for graded materials.
Inelastic effects such as large or cyclical stresses, permanent deformations, damage, and fracture appear ubiquitously in applications. Topology optimization in the inelastic setting is for instance paramount to the bending, punching, and machining of steel sheets, which are tasks of the utmost applicative relevance. Correspondingly, topology optimization in inelastic settings has already attracted strong attention in the engineering community, see the pioneering [42] and [5,26,29,31,34,47] among others. On the contrary, the mathematical theory seems to be less developed, with no rigorous existence and optimality result available to date.
We intend to fill this gap in this paper, by investigating a topology optimization problem in the setting of incremental, linearized elastoplasticity with hardening. Referring to Section 2 for all necessary assumptions and details, let us anticipate here that we aim at investigating the sharp-interface problem The order parameter z : Ω → {0, 1} determines the actual position of the solid, to be identified with the set {z = 1}. The smooth nonnegative function ℓ(·) represents the density of the body, in dependence of z . In particular, ℓ(0) = 0. We assume the whole container Ω to deform under the action of the body force ℓ(z)f and the boundary traction g . The elasticity tensor C, the linear hardening tensor H, and the yield stress d hence depend on z . The fields u : Ω → R n and p : Ω → R n×n are the displacement and the plastic strain, respectively, and ε(u) denotes the symmetrized gradient. Given the solid {z = 1}, the minimization of the incremental elastoplastic state functional E gives a unique elastoplastic state (u, p). Note that E is nothing but the complementary elastic energy of the body, augmented by the linear dissipation term d(z)|p|, modeling the plastic work from a prior nonplasticized state. Then, the displacement component u enters into the definition of J , which ultimately depends just on z . Note that the functional J features the total-variation norm of z , which indeed corresponds to the perimeter of the solid {z = 1} in Ω. In particular, the boundedness of J implies that z ∈ BV (Ω; {0, 1}), the space of functions of bounded variation [4].
In addition to the sharp-interface problem (1.1), we study its phase-field approximation where, for δ > 0, the phase-field target functional J δ is defined as with order parameter z now assumed to belong to H 1 (Ω; [0, 1]) and referred to as phase-field.
We prove that problems (1.1) and (1.2) admit solutions, see Propositions 2.1 and 2.4, respectively. In addition, we prove that optimal phase fields z δ solving problem (1.2) converge, up to subsequences, to a solution to the sharp-interface problem (1.1), see Corollary 2.7. Indeed, this relies on a general Γ-convergence result as δ → 0, see Theorem 2.6.
The phase-field problem (1.2) is then further regularized by replacing the dissipation term d(z)|p| by d(z)h γ (p) in E , where h γ is a smooth approximation of the norm, namely h γ (p) → |p| as γ → +∞. The ensuing regularized phase-field problem admits solutions z γ δ . For fixed δ > 0, as γ → +∞ the sequence z γ δ converges, up to subsequences, to a solution of the phase-field problem (1.2), see Proposition 2.8.
In addition, owing to the smoothness of the corresponding control-to-state map for all γ , one can derive first-order optimality conditions for the regularized phase-field problem, see Theorem 3.1 and Corollary 3.4. Eventually, by letting γ → +∞ we derive first-order optimality conditions for the phase-field problem (1.2), see Theorem 4.1.
Before closing this introduction, let us mention that the literature on existence and optimality conditions for elastoplastic problem is rather scant and, to our knowledge, always focusing on force and traction control. The incremental elastoplastic problem is discussed in [17,25,23] whereas the quasistatic setting is addressed in the series [44,45,46], see also [41]. Some related results on elastoplasticity are in [11], whereas [1] and [20,19] deal with quasistatic adhesive contact and shape memory alloys, respectively. The reader is referred to [38,39] for some abstract optimal control theory for rate-independent systems.
As concerns our quest for first-order optimality conditions, our analysis follows some argument from [17]. Here, the optimal control problem in incremental elastoplasticity is discussed, with elastoplastic state (u, p) controlled via f and traction g and the coefficients C, H, and d are given constants. In particular, the incremental elastoplastic state functional E in [17] is linear with respect to the controls. Here instead the order parameter z appears nonlinearly in E . This generates some additional difficulty which we overcome in a series of technical lemmas in Section 3 below. This is the plan of the paper: we introduce our setting and state and prove existence of both sharp-interface problems and phase-field approximations in Section 2, where we also discuss Γconvergence. The γ -regularized problem is then discussed in Section 3, where the corresponding first-order optimality conditions are derived. Eventually, in Section 4 we pass to the limit as γ → +∞ and obtain first-order optimality conditions for the phase-field approximation.

Setting of the problem
Let Ω be an open bounded subset of R n with Lipschitz boundary ∂Ω containing the elastoplastic body under kinematic hardening. The body is identified by means of the order parameter z ∈ [0, 1] at each point. The region {z > 0} hence represents the body, whereas {z = 0} on the portion of Ω which is not occupied by the body. As already mentioned, we assume also the portion {z = 0} to deform elastoplastically, being extremely soft however. This would correspond to the situation where the elastoplastic body is immersed in a soft polymeric matrix filling the remainder of the domain Ω.
The elastic and plastic properties at a point x ∈ Ω hence depend on the value z at that point. We describe such properties via prescribing the elastic and hardening tensors C(z) : M n S → M n S and H(z) : M n D → M n D , where M n S and M n D denote the spaces of symmetric and deviatoric and symmetric square matrices of order n, respectively. In the sequel, we will also use the symbol M n for the space of square matrices of order n. As usual, we assume C and H to be positive definite, uniformly w.r.t. z ∈ [0, 1], namely, there exists 0 ≤ α C ≤ β C < +∞ and 0 ≤ α H ≤ β H < +∞ such that for every z ∈ [0, 1] and every E ∈ M n S , for every z ∈ [0, 1] and every Q ∈ M n D , (2.2) where the dot indicates the scalar product between matrices. For technical reasons that will be clear in the proofs of Theorems 3.1 and 4.1, we assume that C and H are elements of C 1 ([0, 1]; L(M n ; M n )), where L(M n ; M n ) denotes the space of linear and continuous functions from M n to M n . We further suppose that C(·) and H(·) can be extended constantly outside the interval [0, 1], that is, C(z) = C(0) (resp. H(z) = H(0)) for z < 0 and C(z) = C(1) (resp. H(z) = H(1)) for z > 1, keeping the C 1 -regularity on R. This last requirement allows us to drop the usual constraint z ∈ [0, 1] on the order parameter, as it can be renormalized a posteriori without changing the solution of the problem (see (2.12)-(2.13) for further details). Finally, we assume that the elasticity tensor C(·) can be split into a volumetric and a deviatoric part, i.e., there exists C D : R → L(M n D ; M n D ) and κ : R → R joining the same properties of C(·) and such that 2)). In this case, C mat is the elasticity tensor of the material under consideration, while C void is interpreted as a residual elasticity tensor when no material is present (z = 0). As it is natural to expect, the tensor C(z) degradates when the parameter z decreases. In order to maintain the coerciveness of the problem, the residual tensor C void has to be considered. The very same interpretation holds for H mat and H void . Assume to be given w ∈ H 1 (Ω; R n ), whose trace on the subset Γ D ⊂ ∂Ω represents a prescribed boundary displacement. Here, Γ D is assumed to be open in the relative tolopogy of ∂Ω and H n−1 (Γ D ) > 0, where the latter is the (n−1)-Hausdorff surface measure in R n . Let us define the set of admissible states A(w) as where Eu is the symmetric part of ∇u, and ε and p are usually referred to as the elastic and the plastic strain, respectively.
As for the plastic dissipation, for simplicity of exposition we consider here a von Mises model. Hence, given an order parameter z ∈ L ∞ (Ω; [0, 1]) and p ∈ L 2 (Ω; M n D ), the plastic dissipation associated to p reads As we did for the tensors C and H, we assume that d can be extended to a C 1 -function on the whole R in such a way that d(z) = d(0) for z < 0 and d(z) = d(1) for z > 1. We notice that in (2.5) the presence of material, expressed through the order parameter z , directly affects the plastic dissipation. In a typical situation, d is also assumed to be increasing in the interval [0, 1], which implies that the plastic yield stress is lower for z ∼ 0 and higher for z ∼ 1. Assumption (2.6), in agreement with (2.1)-(2.2), says plastic dissipation occurs also in {z = 0}, although to possibly very small extent.
Besides the prescribed boundary displacement at Γ D , the system is subjected to an applied body force f ∈ L 2 (Ω; R n ) and an applied boundary traction g ∈ L 2 (Γ N ; R n ). Here, Γ N ⊂ ∂Ω is open in the topology of ∂Ω and Γ N ∩ Γ D = Ø, where Γ N and Γ D are closures in ∂Ω. Furthermore, we assume that Ω ∪ Γ N is regular in the sense of Gröger [22,Definition 2], that is, for every x ∈ ∂Ω there exists an open neighborhood U x ⊆ R n of x and a bi-Lipschitz ) coincides with one of the following sets: where y i is the i-th component of y ∈ R n . This last assumption turns to be useful in the proof of Theorem 3.1.
Given a state (u, ε, p) ∈ A(w) and an order parameter z ∈ L ∞ (Ω), we define the energy functional As we did for C, H, and d, in (2.7) we assume that ℓ ∈ C 1 (R) is such that ℓ is constant out of the interval [0, 1] with ℓ(0) = 0, ℓ(1) = 1. At a static (or time discrete) level, the functional E represents the total energy of a linearly elastoplastic material occupying Ω with density ℓ(z) and subjected to a displacement u, an elastic strain ε, and a plastic strain p. In particular, we notice that because of our assumptions on the tensors C and H and on the functions d and ℓ, we have E(z, u, ε, p) = E(0 ∨ z ∧ 1, u, ε, p) for every z ∈ H 1 (Ω) ∩ L ∞ (Ω). Thus, the natural constraint z ∈ [0, 1] needs not to be explicitly imposed at this level, for it will follow directly from the minimization of a cost functional, as shown in (2.12)-(2.13) below. This avoids some technicalities in Sections 3 and 4.
We now turn to the topology optimization problem. The primal aim of topology optimization is to detect the optimal distribution of a material inside a known box, which is the open set Ω in our context. From a mathematical standpoint, such a problem can be recasted as an optimal control problem, in which the set E ⊆ Ω to be filled with the material serves as a control parameter, while the physical properties of the material act as a constraint through the equibrium and constitutive equations. While such a mathematical approach has been already presented in many papers in the setting of linear or nonlinear elasticity (see, e.g., [9,8,13,15]), topology optimization in presence of inelastic structures has been considered only at an engineering and computational level [26,31]. The aim of our work is to present and discuss a model for topology optimization which accounts for possible plastic behaviors. The target functional we want to minimize here reads where u ∈ H 1 (Ω; R n ), E ⊆ Ω is a set of finite perimeter in Ω, i.e., such that the characteristic function 1 E of E belongs to the space BV (Ω) of functions of bounded variations [4], and the symbol Per(E; Ω) denotes the perimeter of E in Ω. The constant 1/6 in front of the perimeter term in J is just intended to ease notations with respect to the phase-field approximation argument, see Theorem 2.6. In particular, the analysis is independent from the specific value of this constant, which can be changed with no difficulty. With an abuse of notation, we write where △ stands for the symmetric difference of sets. Let us briefly comment on the cost functional J . The first two integrals appearing in (2.8) represent the compliance and measure the ability of the body to resist to the applied loads f and g . The perimeter functional, instead, penalizes sets E with large boundary, limiting the onset of a very fine microstructure. We notice that the functional J could be modified to take into account, for instance, accumulation of plastic strain p or a cost of production depending on the size of E . We stay with the specific form above for the sake of definiteness.
The sharp-interface optimization problem (1.1) is therefore specified as The forward problem (2.10) represents the elastoplastic equilibrium condition, and is formulated at a static, incremental level. Under this constraint, we look for the optimal shape E ⊂ Ω by minimizing J in (2.9).
We now briefly discuss the existence of solutions to (2.9)-(2.10).
Proof. The existence of a solution follows from an application of the Direct Method. Indeed,

From (2.1)-(2.2) and the previous inequality we deduce that
By the lower semicontinuity of E we have that the triple (u, ε, p) minimizes E(z, ·, ·, ·) in A(w). Finally, the lower semicontinuity of J implies that E is a solution of (2.9)-(2.10).
Remark 2.2. We remark that the forward problem (2.10) admits a unique solution for every control E ∈ BV (Ω), since E(1 E , ·, ·, ·) is strictly convex in the state variable. The same holds for the two phase-field optimization problem we consider below, namely (2.13) and (2.21).
In what follows we focus on a phase-field approximation of the optimization problem (2.9)-(2.10). Precisely, for δ > 0 we consider the target functional The first two integrals in (2.11) are an approximation of the compliance appearing in (2.8). The last integral, instead, is a typical Modica-Mortola term [32,33], which, according to the value of δ , forces z to take the values 0 or 1 and penalizes the transition between material ({z = 1}) and void ({z = 0}) regions. In particular, such a term Γ-converges to the perimeter functional Per(·; Ω)/6.
The phase-field optimization problem (1.2) is hence rewritten as Remark 2.3. We remark that the usual constraint z ∈ H 1 (Ω; [0, 1]) has not been explicitly imposed in (2.12). However, if z ∈ H 1 (Ω) ∩ L ∞ (Ω) is a solution of (2.12)-(2.13), then also z = 0 ∨ z ∧ 1 is a solution, with the same state configuration. This is due to the assumptions on C, H, d, and ℓ above, which are supposed to be continuously and constantly extended outside the interval [0, 1].
In the next proposition we prove the existence of solutions for (2.12)-(2.13) and show the Γ-convergence of the phase-field problem to the sharp-interface one as δ → 0.
Proof. The existence of solutions follows from the same argument of Proposition 2.1. We only have to notice that, by Remark 2.3, we may assume, without loss of generality, that a minimizing sequence In order to give a compact statement of the Γ-convergence result, we introduce G δ : In a similar way we may define G : BV (Ω; [0, 1]) × H 1 (Ω; R n ) → R ∪ {+∞} by replacing J δ with J and (2.13) with (2.10).
Remark 2.5. Notice that in the definition (2.14) of G δ we have imposed z ∈ H 1 (Ω; [0, 1]). As pointed out in Remark 2.3, such a requirement is not necessary when we look for minimizers of (2.12)-(2.13), as it can be imposed a posteriori on a solution. However, such a constraint has to be considered to state a precise Γ-convergence result, as the space of sets of finite perimeter is described by characteristic functions taking values in [0, 1] only.
Theorem 2.6 (Phase-field approximation of the sharp-interface model).
Proof. We start by proving the Γ-liminf inequality. Let δ k → 0, E ∈ BV (Ω), and z k ∈ the corresponding state variables, solutions of (2.10) and (2.13), respectively. Following the steps of the proof of Proposition 2.1, we get that . Furthermore, from the Γ-convergence of the Modica-Mortola term to the perimeter functional [32,33] we deduce As for the Γ-limsup inequality, for every E ∈ BV (Ω) we consider the recovery sequence z k ∈ H 1 (Ω; [0, 1]) constructed in [32,33] and such that z k → 1 E in L 1 (Ω). Again, let This concludes the proof of the theorem.
Proof. The thesis follows by the Γ-convergence shown in Theorem 2.6.
In the rest of the paper we focus on the first-order optimality conditions for (2.12)-(2.13). In this respect, we notice that the presence of the positively 1-homogeneous plastic dissipation (2.5) prevents us from deducing suitable optimality conditions by operating directly on (2.12)-(2.13). For this reason, we introduce a regularization of the energy functional (2.7) depending on a parameter γ > 0. Namely, for every z ∈ L ∞ (Ω), every (u, ε, p) ∈ A(w), and every γ ∈ (0, +∞), we define (2.16) For later use, we set and notice that h γ ∈ C ∞ (M n D ) is convex and such that for every γ ∈ (0, +∞) and every Q, Q 1 , Q 2 ∈ M n D . We now consider the regularized optimization problem We notice that the minimization problem (2.20)-(2.21) is expected to be regular w.r.t. z (see Theorem 3.1) as the regularization of the plastic dissipation . In the following proposition we state the existence of solutions of (2.20)-(2.21) and the convergence as γ → +∞ to solutions of (2.12)-(2.13). Proposition 2.8 (Existence and convergence of regularized optimal controls). Let f ∈ L 2 (Ω; R n ), g ∈ L 2 (Γ N ; R n ), and w ∈ H 1 (Ω; R n ). Then, for every γ ∈ (0, +∞) there exists a solution z γ ∈ H 1 (Ω; [0, 1]) to (2.20)-(2.21).
Since in the next sections we are going to deduce the optimality conditions for (2.12)-(2.13) as limit of those for (2.20)-(2.21), these will be valid only for a subclass of solutions of (2.12)-(2.13). Clearly, if (2.12)-(2.13) admits a unique solution, in view of Proposition 2.8 it can be approximated by solutions of (2.20)-(2.21), and its first order optimality conditions follows from Theorems 3.1 and 4.1.

Optimality of the regularized problem
This section is devoted to the computation of the first-order optimality conditions for the regularized problem (2.20)-(2.21). In particular, we follow here the main lines of [17].
In order to state the main result of this section, we need some additional notation. For γ ∈ (0, +∞), z, ϕ ∈ L ∞ (Ω), and (u, ε, p), (v, η, q) ∈ H 1 (Ω; R n ) × L 2 (Ω; M n S ) × L 2 (Ω; M n D ), we set Theorem 3.1 (Differentiability of the control-to-state map). Let p ∈ (2, +∞), f ∈ L p (Ω; R n ), g ∈ L p (Γ N ; R n ), w ∈ W 1,p (Ω; R n ), and γ ∈ (0, +∞). Then, the control-to-state operator S γ : L ∞ (Ω) → A(w) defined as is Frechét differentiable. Its derivative in the direction ϕ ∈ L ∞ (Ω) is given by In order to prove the Frechét differentiability of the control-to-state operator S γ stated in Theorem 3.1 we first need to investigate the integrability and continuity properties of S γ (z) for z ∈ L ∞ (Ω). This is the subject of the following three lemmata. Lemma 3.5. For every z ∈ R and every γ ∈ (0, +∞) let F z,γ : M n D → M n D be the map defined as for every Q ∈ M n D . Then, there exist three constants C 1 , C 2 , C 3 > 0 independent of γ and z and a constant C γ > 0 (depending only on γ ) such that the following inequalities hold: for every z, z 1 , z 2 ∈ R and every Q 1 , Q 2 ∈ M n D . Moreover, F z,γ is invertible for every z ∈ R and every γ ∈ (0, +∞), and its inverse satisfies for every Q 1 , Q 2 ∈ M n D , for a positive constant C independent of γ and z . Proof. Inequality (3.6) follows from assumptions (2.1)-(2.2), from the Lipschitz continuity of d, and from inequality (2.19). In particular, C γ depends on γ because the Lipschitz constant in (2.19) degenerates with γ . Inequality (3.7) is a consequence of (2.1)-(2.2), of the convexity of h γ , and of the sign of d. The constant C 1 is independent of γ ∈ (0, +∞) as (2.1)-(2.2) are. From (3.6)-(3.7) we infer that F z,γ is invertible, with inverse F −1 z,γ : M n D → M n D satisfying (3.9). Let us now show (3.8). By definition of F z,γ , for every z 1 , z 2 ∈ R and every Q 1 , Q 2 ∈ M n D we have Inequality (3.8) can be deduced from (3.10) by taking into account the assumptions (2.1)-(2.2), the regularity of C, H, and d w.r.t. z , inequality (3.7), and the bound |∇ Q h γ (Q 2 )| ≤ 1.
Lemma 3.6. For every γ ∈ (0, +∞) and every z ∈ R let the map b z,γ : M n S → M n S be defined as for every E ∈ M n S , where Π M n D : M n → M n D denotes the projection operator on M n D . Then, there exist two positive constants c 1 , c 2 such that for every γ ∈ (0, +∞), every z ∈ R, and every E 1 , Proof. Property (3.12) is a direct consequence of (3.9). Let us prove (3.13), instead. Following the ideas of [17, Section 5], let us define for z ∈ R, E ∈ M n S , and Q ∈ M n D the auxiliary function .
Proof. The proof of (3.16) and (3.17) follows from an application of [24,Theorem 1.1]. To apply such result, we first have to recast the Euler-Lagrange equations associated to the minimization problem (2.21) in terms of the sole displacement variable u.
As for I 3 , we simply use the Lipschitz continuity of C(·) and inequality (3.16) to show that (3.27) In a similar way, since ℓ is Lipschitz continuous we obtain (3.28) Finally, inserting (3.24)-(3.28) into (3.23) we infer In order to conclude for (3.17), we notice that inequality (3.7) tested with for a.e. x ∈ Ω and integrated over Ω implies By the triangle inequality, we also estimate ε 1 − ε 2 q , and the proof of (3.17) is complete.
We are now ready to prove Theorem 3.1.
Let us now estimate I t,j , j = 1, . . . , 5. We write I t,1 as In a similar way, we have that As for I t,4 , since h γ ∈ C ∞ (M n D ), for every t > 0 there exists ξ t laying on the segment [p, p t ] such that Inserting the previous equalities in (3.30), choosing the test function (v, η, q) = (v t , η t , q t ) ∈ A(0), using (2.1)-(2.2), the Lipschitz continuity of C(·), H(·), d(·), and ∇ Q h γ , the convexity of h γ , and Lemma 3.7, we obtain the estimate for some positive constant C γ dependent on γ ∈ (0, +∞). In view of the regularity of C(·), H(·), d(·), and ℓ(·), we can continue in (3.31) with for some C γ > 0 and some ν ∈ (1, +∞). In order to conclude for (3.29) we are led to show that the last term on the right-hand side of (3.32) tends to 0 as t → 0. To do this, we explicitly write ∇ 2 Q h γ (Q) for Q ∈ M n D : Since p t → p in L 2 (Ω; M n D ) as t → 0, up to a subsequence we can assume that p t → p a.e. in Ω. Hence, ξ t → p and This, together with (3.32), concludes the proof of (3.29).
We conclude this section with the proof of Corollary 3.4.

Proof of Corollary 3.4. Given
is a solution of (3.2), which in turn is equivalent to the Euler-Lagrange equation (3.3).
The equality (3.4) also follows from Theorem 3.1. Indeed, for t ∈ R we consider z t γ := z γ + tϕ and denote with (u t γ , ε t γ , p t γ ) = S γ (z t γ ) the corresponding solution of the forward problem (2.21). By optimality of z γ we have that J δ (z γ , u γ ) ≤ J δ (z t γ , u t γ ). Equality (3.4) is therefore obtained by dividing the previous expression by t and passing to the limit as t → 0, by taking into account the differentiability of the control-to-state operator S γ from Theorem 3.1.

Optimality conditions for γ → +∞
This section is devoted to the computation of the optimality conditions for (2.12)-(2.13). Such conditions are obtained by passing to the limit in (3.3) and (3.4). As the control-to-state operator for (2.12)-(2.13) is not differentiable anymore and quantities such as ∇ Q h γ and ∇ 2 Q h γ appearing in formulas (3.1)-(3.2) degenerate as γ → +∞, the limit passage is not completely trivial and deserves some further analysis. This is the content of the following theorem.
Let us show the first equality in (4.5). To do this, we set J γ : = {|p γ | ≤ 1 √ γ } and estimate π ϕ γ · p γ 2 1 as follows: We now show that I γ,1 and I γ,2 tend to 0 separately. For I γ,1 , by Hölder and Cauchy inequalities and by the continuity of d we have that there exists C > 0 independent of γ such that In order to conclude that I γ,1 → 0, we write explicitly |π ϕ γ · p ϕ γ |: Therefore, we can continue in (4.11) with which implies that I γ,1 → 0 as γ → +∞, as π ϕ γ and p ϕ γ are bounded in L 2 (Ω; M n D ). As for I γ,2 , instead, by the Cauchy inequality we have Thus, also I γ,2 tends to 0 as γ → +∞.
We now check with the third inequality in (4.5). Since |ρ| ≤ d(z), the function q →ˆΩ |q| d(z) − |ρ| dx is lower semicontinuous w.r.t. the weak topology of L 2 (Ω; M n D ). Hence, Since p ϕ γ is bounded in L 2 (Ω; M n D ), d(z γ )∇ Q h γ (p γ ) → ρ in L 2 (Ω; M 2 D ), and z γ ⇀ z weakly in H 1 (Ω; [0, 1]), the previous inequality reduces to where in the second line we have used the explicit expression of ∇ Q h γ (p γ ). Arguing as in (4.12) we deduce that the second integral on the right-hand side of (4.13) tends to 0 as γ → +∞.
As for the integral on J γ , instead, by the Cauchy inequality and the continuity of d there exists a positive constant C independent of γ such that In view of (2.6) and of the boundedness of π ϕ γ and p ϕ γ are bounded in L 2 (Ω; M n D ) we get Jγ |π ϕ γ · p ϕ γ | dx =ˆJ γ d(z γ ) we deduce that (4.15) lim Combining (4.13)-(4.15) we get that Ω |p ϕ | d(z) − |ρ| dx = 0 which implies |p ϕ | = 0 a.e. in {|ρ| ≤ d(z)}. This concludes the proof of the theorem.