Nonlinear Capacitary Problems for a General Distribution of Fibers

We determine the eﬀective electric properties of a composite with high contrast. The energy density is given locally in terms of a convex function of the gradient of the potential. The permittivity may take very large values in a fairly general distribution of parallel ﬁbers of tiny cross sections. For a critical size of the cross sections, we show that a concentration of electric energy may arise in a small region of space surrounding the ﬁbers. This extra contribution is caused by the discrepancy between the behaviors of the potential in the matrix and in the ﬁbers and is characterized by the density of the cross sections of the ﬁbers with respect to the cross section of the body in terms of some suitable notion of capacity. Our results extend those established in [7] in the periodic case for the p -Laplacian to a general nonlinear framework and a non-periodic distribution of ﬁbers.


Introduction and setting out of the problem
Composites comprising traces of materials with extreme physical properties have been investigated by several authors over the past decades in various contexts, such as diffusion equations [7,11,16,26,28], fluid mechanics [12], electromagnetic theory [9], linearized elasticity [6,8]. The common feature of this body of work is the emergence of a concentration of energy in a small region of space surrounding the strong components. This extra contribution is characterized by a local density of the geometric perturbations in terms of an appropriate capacity depending on the type of equations.
In this paper, we determine the effective electric properties of an electrified composite whereby a set of extremely thin fibers with very large permittivities is embedded in a matrix with permittivity of order 1. This study may as well concern various steady-state situations in Physics like heat diffusion for instance. It is interesting to refer to Electricity where capacity has a specific meaning. A similar problem has been studied by one of the authors with G. Bouchitté [7] in the periodic quasilinear case for fibers of circular cross section. In what follows, we investigate the non periodic case and consider a more general non linear framework and also fibers with arbitrarily shaped cross sections. This is worthwhile because fibers stem from draw plates and therefore are likely to display anisotropic behaviors governed by general convex functions. Let us notice that in the linear case, M. Briane and J. Casado-Díaz [15] obtained nonlocal effects with fibers the cross section of which is merely a bounded connected open subset of R 2 . Dropping the assumption of periodicity is a challenging task which may lead to quite different effective problems when composites with high contrast are considered. In our specific study, the effective problem turns out to show the same general features as in the periodic case, provided the fibers are not too closely spaced (see (1.6)).
We turn now to a more detailed introduction of the paper.
O is a bounded smooth open subset of R 2 . We consider the boundary value problem in Electrostatics (1.1) The solution u ε to (P ε ) describes the electric potential of an electrified fibered composite insulator, where the distributions of body and surface charges are denoted by q b and q s . The non periodic set T rε occupied by the fibers is defined in terms of a bounded domain S ⊂ R 2 with a Lipschitz boundary, of two small positive parameter ε, r ε such that 0 < r ε < < ε < < 1, and of a finite family (ω j ε ) j∈Jε (J ε ⊂ N) of points in " O. We set an approximation of the number of fibers included in the parallelepiped Y z ε × (0, L) containing x. The assumption 0 ≤ n ε (x) ≤ N in O, N ∈ N, n ε n weak star in L ∞ (O), (1.5) ensures that the fibers do not concentrate in some lower dimensional subset of O (see Remark 3.1 (ii)). We also suppose that min j,j ∈Jε,j =j for some sequence of positive reals (R ε ) satisfying (1.7) The hypothesis (1.6) guarantees that each fiber is separated by a sufficient distance from the other fibers and from the lateral boundary of O (the constant "5 √ 2" in (1.6) is chosen in order to get (6.37)). The periodic case corresponds to Ω ε = {εz, z ∈ I ε } and n ε given by n ε (x) = 1 if x ∈ z∈Iε Y z ε × (0, L), n ε (x) = 0 otherwise. With no loss of generality, we assume that where D denotes the open unit ball of R 2 . The density of electric energy is given in terms of a sequence of positive reals (λ ε ) and of two strictly convex functions f , g satisfying a growth condition of order p ∈ (1, +∞) of the type a|ξ| p ≤ f (ξ), g(ξ) ≤ b|ξ| p ∀ξ ∈ R 3 , (a, b > 0). (1.9) We suppose that lim ε→0 λ ε r 2 ε |S| ε 2 =k ∈ (0, +∞], (1.10) thus the density of electric energy is assumed to take large values in the fibers. For simplicity, we suppose that (see Remark 3.1 (iii)) u 0 = 0, ifk = +∞. (1.11)

Notations
For any weakly differentiable function ϕ : R N → R (N ∈ {2, 3}), we set " ∇ϕ := (∂ 1 ϕ, ∂ 2 ϕ, 0). (2.1) We denote by f ∞,p the "p-recession" function of f , defined by For all couples (U, V ) of open subsets of R 2 such that U ⊂ V and for all α ∈ R, we set cap f (U, V ; α) = inf P f (U, V ; α), (2.5) The letter C denotes different constants whose precise values may vary. We employ the usual convention ∞.0 = 0. We denote the Lebesgue measure on R N by L N , the Hausdorff k-dimensional measure on R N by H k , the space of Radon measures on O by M(O), the space of Borel functions on O by L 0 (O), respectively.

Main result
We assume that the sequence (γ (p) ε (r ε )) defined by (1.7) is convergent and set The effective behavior depends on the order of magnitude of the parameter γ (p) . A critical case occurs when 0 < γ (p) < +∞. Then, a gap between the mean potential of the constituent parts of the composite may appear, giving rise to a concentration of electric energy stored in a thin region of space enveloping the fibers. The effective electric energy then takes the form of a sum of three terms like The function u stands for the weak limit in W 1,p (O) of the sequence (u ε ) of the solutions to (1.1), and v represents a local approximation of the effective potential in the fibers. More precisely the function nv, where n is defined by (1.5), is the weak- * limit in M(O) of the sequence of measures (u ε µ ε ), being µ ε the measure defined by The functional Φ f ibers accounts for the effective electric energy stored in the fibers and is given by where n and g hom : R → R are respectively defined by (1.5) and g hom (a) := min g(q) : q ∈ R 3 , q 3 = a . (3.5) The second term of Φ describes the last mentioned concentration of energy in terms of the gap between the effective potential in the fibers and in the matrix. We obtain The sequences (c f ε (S; ±1)) are assumed to be convergent if p = 2. A study of cap f (see Section 5) yields for some positive reals c f ∞,2 (±1) independent of S (see Remark 3.1 (iv)). We prove that the limiting problem in a variational sense associated with (1.1) is given by otherwise, (3.9) Assume in addition that γ (p) > 0 and let µ ε be the measure defined by (3.3). Then the sequence of measures (u ε µ ε ) weak * converges in M(O) to nvL 3 O , where n is defined by (1.5) and v is the unique element of V p , given by (3.9), such that F hom (u) = Φ(u, v).
and the effective energy is that of the matrix augmented by a permittivity term in the direction of the fibers. If 0 < γ (p) < +∞, the effective electric energy is not a local functional. This means that it can not be written as the integration over O of a density of electric energy of the form h(x, u(x), ∇u(x), ...). By introducing the additional state variable v, we can write the effective energy under the form of a local functional of the couple (u, v). This internal or hidden state variable is the limit of a suitable scaling of the electric potential in the sole fibers and accounts for the micro-structure. The total effective electric energy is that of a body totally filled up by the matrix material augmented by a term which is the infimal convolution of the last mentioned permittivity term supplied by the distribution of fibers and a bonding term depending on the gap of electric potentials in the matrix and in the fibers. This bonding term describes a concentration of electric energy in the matrix in the immediate vicinity of the fibers, which may occur only when p ≤ 2. It induces a total effective energy lower than Φ(u, u). The structure of Φ stems from the contribution of each term entering the decomposition: where, given (R ε ) satisfying (1.7), the set D Rε × (0, L) is the R ε -neighborhood of the fibers defined by (6.3). The set (D Rε × (0, L)) \ T rε is a small portion of the matrix surrounding the fibers where electric energy may concentrate due to the gap between the mean electric potentials in the fibers and in the matrix. This will provide a limit capacitary term associated with f ∞,p ( " ∇u) on R ε D \ r ε S. The contribution of O \ (D Rε × (0, L)) is obvious and the contribution of the fibers is classical (see [1,29]).
(ii) The extension of our results to the case when the sequence (n ε ) is not bounded in L ∞ (O) but only in L 1 (O) and weak- * converges in M(O) to some measure µ is a challenging mathematical problem. The effective energy stored in the fibers is then likely to be simply deduced from (3.4) by substituting dµ for ndx. As regards the concentration of electric energy around the fibers, we expect it to take the form Φ cap (v − u) =´c f (S; v − u)dµ 0 for some suitable measure µ 0 absolutely continuous with respect to µ and satisfying µ 0 (E × (0, L)) = 0 for all sets E ⊂ " O such that cap f (E, " O; 1) = 0. Similar classes of measures arise in the study of Dirichlet problems on varying domains [19], [20], [21]. Computing this measure µ 0 , if possible in terms of µ, seems to be a big task.
(iii) The simplifying assumption (1.11) ensures that the effective electric energy stored in the fibers vanishes ifk = +∞. An alternative is to assume that u 0 takes the same values on the intersection of the opposite bases of O with Γ 0 .
(iv) If p = 2, the constants c f ∞,2 (±1) are simply defined by These constants can not be explicitely determined in terms of cap f ∞,2 . However, they can be calculated if f (.) = 1 2 |.| 2 (see (5.20, 5.36)): c (iv) The phenomenon observed in the critical case does not appear in dimension 2 whenever the sequence of conductivities (λ ε ) is supposed to be uniformly bounded from below. Indeed, M. Briane and J. Casado-Díaz showed [13,14] that in that case the nature of the problem is preserved through the homogenization process.
4 Conjecture for the case of a random distribution of fibers In this section, we indicate a possible generalization of the periodic model to the case of parallel fibers randomly distributed in accordance with a stationary point process. In the model under consideration, the cross sections are not uniformly (i.e., periodically) distributed but their distribution is periodic in law i.e., the probability of presence of the sections is invariant under a suitable group (τ z ) z∈Z 2 defined below. In the stochastic homogenization framework, the distribution of the sections is then said to be statistically homogeneous. We are going to give some precisions on this model. Let us first define the discrete dynamical system (Ω, P P P , (τ z ) z∈Z 2 ) that models the distribution of the sections of the fibers. Given d > 0, we set and denote by Σ the trace of the Borel σ-algebra of (R 2 ) ∞ on Ω. We equip Ω with the group (τ z ) z∈Z 2 defined by where ω − z must be understood as (ω i − z) i∈N , and we denote by F the σ-algebra made up of all the events of Σ which are invariant under the group (τ z ) z∈Z 2 . We assume the existence of a probability measure P P P on (Ω, Σ) for which (τ z ) z∈Z 2 is a measure preserving transformation, i.e., P P P #τ z = P P P for all z ∈ Z 2 , where P P P #τ z denotes the pushforward of the probability measure P P P by the map τ z . For any measurable function X : Ω → R, we denote by E E E F X its conditional expectation given F, i.e., the unique F-measurable function satisfyingˆE E E E F X dP P P =ˆE X dP P P for every E ∈ F.
Note that E E E F X is τ z -invariant (hence periodic) and that under the additional ergodic hypothesis which asserts that F is trivial, that is made up of events with probability measure 0 or 1, E E E F X is constant and nothing but the expectation E E E(X) :=´Ω X dP P P . Note also that the following asymptotic independance hypothesis lim |z|→+∞ P P P (E ∩ τ z E ) = P P P (E)P P P (E ), is a stronger but more intuitive condition yielding ergodicity.
The random set of fibers is defined by We will denote by (P ε (ω)) the problem associated with the random functional F ε (ω, .). Consider the random function In all likelyhood, the conditional expectation E E E F n 0 (ω) is the only additional corrector of the limit energy obtained in the periodic case. More precisely let us denote by Φ(ω, .) the random functional: Moreover, belonging to V p such that F hom (ω, u) = Φ(ω, u(ω), v(ω)). Furthermore, under the ergodic hypothesis (for instance under condition (4.2)), there holds E E E F n 0 (ω) = E E En 0 so that the functionals Φ, F hom and the functions u and v are deterministic.
We hope to treat the mathematical analysis in a forthcoming paper.

Study of the capacity cap f
Given a strictly convex function f : R 3 → R satisfying a growth condition of order p ∈ (1, +∞) of the type (1.9), our main objective in this section is to analyze the behavior with respect to certain small subsets of R 2 of the mapping cap f defined by (2.5). A similar study has already been performed in the setting of linear elasticity in [6,Section 3]. In [22], G. Dal Maso and I.V. Skrypnik have studied the capacity for monotone operators which are closely related to the ones considered in our paper. Also, their study has been extended to pseudo-monotone operators by J. Casado Díaz in [17]. Further results concerning capacities and many references on this subject may be found for instance in [2,25,27,30,32].
In what follows, the letter U denotes a non-empty bounded connected Lipschitz open subset of R 2 and V an open subset of R 2 such that U ⊂ V . The proof of the following Lemma is similar to that of [6, Lemma 1]: Lemma 5.1. The problem (2.5) has a minimizing sequence in D(V ).
If p < 2, we denote by K p (V ) the set of functions ψ ∈ L p * (V ) (p * := 2p 2−p ) for which all the partial derivatives ∂ 1 ψ, ∂ 2 ψ (in the sense of distributions) belong to L p (V ). It is easy to check that, equiped with the norm , which will be denoted by K p 0 (V ), is also a reflexive Banach space. Gagliardo-Nirenberg-Sobolev inequality (see for instance [10, Theorem 9.9]), namelŷ holds true whatever the choice of the open set V , with a constant C depending only on p (we can take for instance C = p 2−p but this constant is not optimal, see [10, footenote, p. 278]), unlike Poincaré inequality in W 1,p 0 (V ), which may fail to hold when V is unbounded. The space K p 0 (V ) coincides with W 1,p 0 (V ) if V is bounded and may be strictly larger otherwise. There holds K p 0 (R 2 ) = K p (R 2 ). The next lemma marks a noteworthy difference between the case p < 2 and the case 2 Lemma 5.2. (i) Assume that p < 2, and let α ∈ R. Then the problem has a unique solution and Moreover, the solution ψ to (5.4) satisfies, for a.e. x ∈ V , (ii) Assume that 2 ≤ p and that V is bounded in one direction, and let α ∈ R. Then the problem (2.5) has a unique solution.
Proof. (i) By (2.5) and Lemma 5.1, we have By repeating the argument of the proof of [6, Lemma 2], we find that Since f is convex and satisfies the growth condition (1.9), the functional ψ →´V f (∂ 1 ψ, ∂ 2 ψ, 0)dx is continuous on K p 0 (V ). We deduce that By (1.9) and Gagliardo-Nirenberg-Sobolev inequality there holds, for all ψ ∈ D(V ) (extending ψ to R 2 by setting By Lemma 5.1, there exists a minimizing sequence (ψ n ) in D(V ) to Problem (2.5). By (5.10), the sequence (ψ n ) is bounded in the reflexive Banach space K p 0 (V ), hence weakly converges in K p 0 (V ), up to a subsequence, to some ψ. As each function ψ n belongs to the convex strongly closed (hence weakly Taking (5.8) into account we infer that ψ is a solution to (5.4). The uniqueness of this solution follows from the strict convexity of f . The "markovian" property (5.5) results from the last mentioned uniqueness, and from the fact that for any ψ ∈ K α (U, V ), the function defined by ψ : Then we repeat the argument of the case p < 2, substituting (5.11) for (5.10), The next Lemma, whose proof is straightforward, states that the map (f, U, V, α) → cap f (U, V ; α) is convex with respect to α, decreasing with respect to V and increasing with respect to U and f .
In addition, if f 1 , f 2 : R 3 → R are two strictly convex functions satisfying a growth condition of order p ∈ (1, +∞) of the type (1.9) and if f 1 ≤ f 2 in R 3 , then In the following lemma, we investigate the continuity properties of cap f (U, V ; α) with respect to U and V .
(ii) Assume that p < 2, and let ψ n be the unique solution to P f (iii) Assume that 2 ≤ p and that V is bounded in one direction, and let ψ n be the solution to P f (U, V n ; α) (see (2.5) and Lemma 5.2 (ii)), extended to V in the same way. Then (ψ n ) converges weakly in By the arbitrary choice of t, Assertion (5.16) is proved. If 0 ∈ V , we can assume without loss of generality that D ⊂ V . Since the sequence (nD) is increasing, we deduce from (5.16) that lim n→+∞ cap f (U, nD; α) = cap f U, +∞ n=1 nD; α = cap f (U, R 2 ; α). Taking (5.12) into account, we then easily infer that lim λ→0 cap f U, 1 . By passing to the limit as λ → 0 in the third term of the last double inequality, we obtain (5.17).
(ii) If p < 2 and ψ n is the solution to P f K p 0 (U, V n ; α), then by (5.10) we have It then follows from (5.16) that Therefore the sequence (ψ n ) is bounded in K p (V ) and converges weakly in K p (V ), up to a subsequence, to some function ψ. As each function ψ n (extended by 0 to V ) belongs to the weakly closed subset On the other hand, by (5.16) and by the weak lower semi-continuity in (iii) Same argument as in the proof of Lemma 5.2 (ii). (iv) Let ψ n be the unique solution to P f Lemma 5.2). By (5.10) and (5.13), there holds hence (ψ n ) is bounded in K p (V ) and converges weakly, up to a subsequence, to some function ψ. Since each ψ n belongs to (5.3)). We deduce from (5.13) and from the weak lower semi-continuity in In the next Lemma, we investigate the case when f is positively homogeneous of degree p. Next properties (5.19) are easily deduced from Lemma 5.4 and from the change of variable formula. Formula (5.20) is deduced from the explicit computation performed in [7, p. 432] of the radial solution to the problem associated with cap 1 p |·| p r ε D, R ε D; α . Lemma 5.5. Assume that f is positively homogeneous of degree p and let λ > 0, α ∈ R. Then Proof. Let us fix t > 0. By Lemma 5.1, there exists ψ ∈ D(V ) such that ψ = α in λU and The function ϕ(y) := ψ(λy) belongs to D 1 λ V and satisfies ϕ = α in U and (∂ 1 ϕ, ∂ 2 ϕ, 0)(y) = λ(∂ 1 ψ, ∂ 2 ψ, 0)(λy). By applying the change of variables formula, taking the positive homogeneousness of degree p of f into account, we obtain By the arbitrary choice of t, we deduce that cap f (λU, V ; α) ≥ λ 2−p cap f U, 1 λ V . The inverse inequality can be proved in a similar way. The first line of (5.19) is established. The second line of (5.19) is obtained in a analogous manner, by setting ϕ(y) := 1 |α| ψ(y).
The next Lemma illustrates the contrasting behavior of the capacity cap f in the case p < 2 and in the case 2 ≤ p.
The case p = 2 is appreciably more involved. (i) There holds for some positive constants C 1 , C 2 .
(i) If p ≥ 2 and α ∈ R \ {0}, then the infimum for problem P f U, R 2 ; α (see (2.5)) is not achieved. Otherwise, should ψ ∈ W 1,p 0 (R 2 ) be a minimum, then by (1.9) and the second line of (5.21), |∇ψ| p L p (R 2 ;R 2 ) ≤ C´R 2 f (ψ)dx = Ccap f (U, R 2 ; α) = 0, hence ψ = 0, in contradiction with the fact that ψ = α in U . This lack of solution is similar to Stokes' paradox in fluid Mechanics [31]. (ii) If V = R 2 , weighted Sobolev spaces provide an interesting alternative approach to the questions of existence of a solution to P f U, R 2 ; α (see [3] for more details on this subject ). Indeed, it can be shown that where W 1,p µp (R 2 ) is the weighted Sobolev space defined by being µ p the measure on R 2 given by The property cap f (U, R 2 ; α) = 0 if p ≥ 2, stated in Lemma 5.6, can be recovered from the fact that if α = 0, then the constant function ψ = α belongs to W 1,p µp (R 2 ) if and only if p ≥ 2.

Technical preliminaries and a priori estimates
The proof of Theorem 3.1 rests on an extensive investigation into the asymptotic behavior of the sequence of the solutions to (P ε ) and, more generally, of sequences (u ε ) satisfying sup ε>0 F ε (u ε ) < +∞. (6.1) A commonly used method consists in introducing auxiliary sequences designed to characterize the comportment of the diverse constituents of the composite. The delicate step lies in the analysis of the behavior of the fibers. An interesting approach consists in investigating the sequence (u ε µ ε ), where µ ε denotes the measure with support included in the fibers defined by (3.3). To that aim, given a sequence (R ε ) satisfying (1.7), we introduce the operators . Rε , . rε , . ε defined on L p ((0, L); W 1,p (O)) by setting and analogously for D j rε . The series of estimates stated below will take a crucial part in the proof of Theorem 3.1 (the proof of Lemma 6.1 is situated at the end of Section 6). Lemma 6.1. There exists a constant C such that for all ϕ ∈ L p ((0, L); W 1,p ( " O)), where " ∇ and γ ii) If f ε µ ε f µ, then lim inf ε→0ˆj (f ε ) dµ ε ≥ˆj(f ) dµ for all convex and lower semi-continuous functions j on R satisfying a growth condition of order p. In addition Proof. The proof of this lemma is given in [7] with j = 1 p | · | p but the duality argument can be extended to any convex lower semi-continuous function satisfying a growth condition of order p. Assertion (6.5) results from the fact that if f + ε µ ε gµ and f ε µ ε f µ, then g ≥ f + µ−a.e., which can be easily checked by using positive continuous test functions (notice that in general, g = f + ).
By (6.21), the function u ε 1,ε takes constant values on each set S j rε ×{x 3 }, whereas the function u ε ε , defined by (6.2), may take up to four different values on S j rε × {x 3 } if S j rε ∩ ∂Y z ε = ∅ and j ∈ J z ε . For each z ∈ I ε , we denote by Z z ε the union of the cells Y z ε whose adherence has a non empty intersection with Y z ε . The set Z z ε is the subset of ε(z + [−1, 2[ 2 ) defined by Let us fix z ∈ I ε . For each j ∈ J z ε and for a.e. x 3 ∈ (0, L), we have Noticing that #A z ≤ 9, we infer (6.30) The next to last inequality in (6.30) is deduced from a change of variables in Poincaré-Wirtinger inequalitý . Noticing that by (1.4) and (1.5) there holds J z ε ≤ N , we deduce from (6.28), and (6.30) that By (6.29), each set Y z ε is included in at most 9 distinct sets Z z ε , therefore by (6.6) we have (6.32) The estimate (6.26) follows from (6.31) and (6.32).

Proof of the main result
The demonstration of Theorem 3.1 is based on the Γ-convergence method (for precise details about this method, we refer the reader to [4,5,18]). The "lowerbound" and the "upperbound" stated respectively in Proposition 7.1 and Proposition 7.2, indicate in particular that the sequence of functionals (F ε ) Γconverges with respect to the strong topology of L p (O) to the functional F hom defined by (3.9). The proof of Theorem 3.1 is deduced in the following manner from the two last mentioned propositions and from the a priori estimates established in Proposition 6.1:

Proof of Theorem 3.1.
We will only prove Theorem (3.1) in the most interesting case Let (u ε ) be the sequence of the solutions to (1.1). By (1.11), and since u 0 is continuous on O (see , we deduce that (u ε ) satisfies (6.1). Therefore, we can apply Proposition 6.1 and, after possibly extracting a subsequence, assume that (u ε ) converges weakly in W 1,p (O) to some u, and that the sequence (u ε µ ε ) weak * converges in M(O) to vL 3 O for some v ∈ V p . We just have to prove that (u, v) is the solution to (3.8). To that aim, we first apply Proposition 7.1, to get (7.2) By Proposition 7.2, there exists a sequence (ϕ ε ) such that, Since u ε is the solution to (1.1), there holds We infer from (7.2), (7.3), (7.4) and from the weak continuity on W 1,p (O) of the linear form ϕ → hence (u, v) is the unique solution to (P hom ) (the uniqueness results from the strict convexity of f and g).

Lower bound
Proof. We can suppose that lim inf ε→0 F ε (u ε ) < +∞, otherwise there is nothing to prove. Accordingly, after possibly extracting a subsequence, we can assume that (6.1) is verified and that the estimates (6.6) and the convergences (6.7) established in Proposition 6.1 take place. We choose a suitable sequence (R ε ) of positive reals satisfying (1.7) (the choice of (R ε ) will be made more precise in Lemma 8.2), and establish (see below) that where g hom and c f are defined by (3.5) and (3.6). Collecting (7.6), (7.7), (7.8) we obtain (7.5) which, joined with (6.7), achieves the proof of Proposition 7.1.
Proof of (7.8). If γ (p) = +∞ (in particular if p > 2), there is nothing to prove because then, by Proposition 6.1, v = u. From now on, we assume that 0 < γ (p) < +∞ (hence p ≤ 2). First, we show (see Lemma 8.1) that there exists an approximation (Ê u ε ) of u ε piecewise constant in x 3 satisfying Next, we fix a positive real δ satisfying 1 < δ < 2, (7.10) and define the set S −r δ ε rε by setting (U, α) = (S rε , r δ ε ) in for all Lipschitz domain S satisfying (7.16). Fixing an increasing sequence (S n ) n∈N of Lipschitz domains such that S n ⊂ S and n∈N S n = S, substituting S n for S in (7.18) and passing to the limit as n → +∞, thanks to (5.13), (5.18) and to the Monotone Convergence Theorem, we get (7.8).