A characterization of gradient Young-concentration measures generated by solutions of Dirichlet-type problems with large sources

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A characterization of gradient Young-concentration measures generated by solutions of Dirichlet-type

problems with large sources

Gisella Croce, Catherine Lacour, Gérard Michaille

To cite this version:

Gisella Croce, Catherine Lacour, Gérard Michaille. A characterization of gradient Young- concentration measures generated by solutions of Dirichlet-type problems with large sources.

ESAIM: Control, Optimisation and Calculus of Variations, EDP Sciences, 2008, pp.2008048.

�10.1051/cocv:2008048�. �hal-00369367�

DOI:10.1051/cocv:2008048 www.esaim-cocv.org

A CHARACTERIZATION OF GRADIENT YOUNG-CONCENTRATION MEASURES GENERATED BY SOLUTIONS OF DIRICHLET-TYPE PROBLEMS

WITH LARGE SOURCES^∗

Gisella Croce

, Catherine Lacour

and G´ erard Michaille

Abstract. We show how to capture the gradient concentration of the solutions of Dirichlet-type problems subjected to large sources of order √¹ε concentrated on anε-neighborhood of a hypersurface of the domain. To this end we define the gradient Young-concentration measures generated by sequences of finite energy and establish a very simple characterization of these measures.

Mathematics Subject Classification.49Q20, 28A33.

Received January 7, 2008.

Published online July 19, 2008.

1. Introduction

Let us consider the one-dimensional Dirichlet problem in (−1,1) with a second member measure

⎧⎪

⎨

⎪⎩

(σ_ε(x)u)= 1

√ε m k=0

a_kδ_t^ε_k u(−1) =u(1) = 0,

(1.1)

where ε > 0, m ∈ N, a_k ∈ R^∗, (t^ε_k)_k=0,...,m is a non decreasing family of numbers in [−^ε₂,^ε₂] with t^ε₀ = −^ε₂, t^ε_m= ^ε₂, lim_ε→0^tεk+1_ε^−tεk = _m¹ :=l fork= 0, . . . , m−1, andσ_εis given by

σ_ε(x) = 1

ε, ifx∈(−1,1)\(−^ε₂,^ε₂) 1, ifx∈(−^ε₂,^ε₂).

Keywords and phrases. Gradient Young measures, concentration measures, minimization problems, quasiconvexity.

∗The research of the first author was partially supported by ATER grant of Universit´e Montpellier II.

1 LMAH (Laboratoire de Math´ematiques Appliqu´ees du Havre), Universit´e du Havre, 25 rue Philippe Lebon, BP 540, 76058 Le Havre, France. gisella.croce@univ-lehavre.fr

2 I3M (Institut de Math´ematiques et de Mod´elisation de Montpellier), UMR-CNRS 5149, Universit´e Montpellier II, Case courrier 051, Place Eug`ene Bataillon, 34095 Montpellier Cedex 5, France. lacour@math.univ-montp2.fr

3 EMIAN, Universit´e de Nˆımes, France. micha@math.univ-montp2.fr

Article published by EDP Sciences c EDP Sciences, SMAI 2008

Let ¯u_εbe the solution of (1.1); then clearly ¯u_εis of the form

¯ u_ε(x) =

⎧⎪

⎨

⎪⎩

εc_ε, ifx∈(−1,−₂^ε)

c_ε+^√sk_ε, ifx∈(t^ε_k, t^ε_k+1), k= 0, . . . , m−1 ε(c_ε+^s√^mε), ifx∈(^ε₂,1),

where s_k := _k

i=0a_i and c_ε is a constant which can be computed from the boundary conditions. Setting

¯ v_ε = √^u¯ε

ε one can show that the measure ^½₍₋^ε

2,^ε₂)ε|v¯_ε|²dxweakly converges to l_m−1

k=0 (L+s_k)²δ₀, where L is the limit of √

εc_ε (see Sect. 4 for the computations). Consequently the gradient of the solution ¯u_ε presents a concentration phenomenon in L²((−1,1)). The knowledge of the limit measure l_m−1

k=0(L+s_k)²δ₀ is not completely satisfactory to describe the concentration phenomenon. Indeed ¯v_ε clearly oscillates nearx= 0 from the observation that not all the coefficientss_kare necessarily of the same sign. A natural question to ask is how to capture these oscillations at the limit. To this end, we compute the weak limit ¯μ= ¯μ_x⊗π¯ of the measure

¯

μ_ε:=δ¯vε

|¯v

ε|(x)⊗^½₍₋^ε

2,^ε₂)ε|¯v_ε|²dx∈M([−1,1]× {−1,1}) in the sense that

ε

1

−1

½(−^ε₂,^ε₂)ϕ(¯v_ε)θ(x) dx→ ¹

−1θ(x)

{−1,1}ϕ(ζ) d¯μ_x

d¯π

for all 2-homogeneous and continuous functions ϕonR and all functionsθ in C([−1,1]). We write ¯μ_x⊗π¯ to denote the slicing decomposition of the measure ¯μ with respect to its projection measure ¯π on [−1,1]. We will call ¯μ the gradient Young-concentration measure generated by(¯v_ε>0). Note that (¯μ_x)_x∈(−1,1) is a family of probability measures on the unit sphere S⁰ := {−1,1} of R. Why is ¯μ useful to capture the oscillations of ¯v_ε at the limit? It is easy to show that ¯π is nothing but the weak limit l_m−1

k=0(L+s_k)²δ₀ of the measure

½(−^ε₂,^ε₂) ε|¯v_ε|²dx. Moreover, we show that (see Sect. 4 for the details)

¯

μ_x=aδ₁+bδ₋₁ where, if we assume for instanceL≥0,

a=

k=0,...,m−1:sk≥0

(L+s_k)²

m−1

k=0

(L+s_k)²

, b= 1−a=

k=0,...,m−1:sk<0

(L+s_k)²

m−1

k=0

(L+s_k)²

(if L≤0, take the condition s_k >0 in the sum defining a, ands_k ≤0 in the sum definingb). One sees that whereas the measure ¯π provides the intensityl_m−1

k=0(L+s_k)² and the support{0}of the concentration, the coefficients a and b in [0,1] provide the desired additional information, i.e., the proportion of change of sign of ¯v_ε captured at the limit, with respect to the total mass of the limit measure ¯π.

In this article we want to generalize the previous notion of gradient Young-concentration measure when ¯u_ε is the solution of a Dirichlet problem on a domain Ω ofR^N, N >1, with a source of magnitude order ^√1_ε. We assume that a (N −1)-dimensional manifold Σ, possessing an unit normal vector ν at H^N−1 a.e. x, splits Ω into two subdomains and that the conductivity outside theε-neighborhood Σ_ε of Σ grows as ¹_ε. The simplest model is given by

¯

u_ε∈ argmin

1

ε _Ω\Σε|∇u|² dx+

Σε

|∇u|² dx− 1

√ε F_ε, u:u∈W₀^1,2(Ω)

(1.2)

whereF_ε∈H⁻¹(Ω); settingv:= √^uε, (1.2) is equivalent to

¯

v_ε∈ argmin

Ω\Σε

|∇v|² dx+ε

Σε

|∇v|² dx− Fε, v:v∈W₀^1,2(Ω)

.

By analogy with the one-dimensional case, the estimation

supε>0 Ω\Σε

|∇vε|² dx+ε

Σε

∂v_ε

∂ν ² dx

<+∞

implies that ^∂_∂ν^¯vε presents a concentration phenomenon on S ⊂⊂ Ω. To describe it, we will characterize the weak cluster points ¯μ of the sequenceδ∂¯vε

∂ν /|^∂¯_∂ν^vε|(x)⊗ε^½_Bε|^∂_∂ν^v¯ε|² dx in the set of non-negative bounded Borel measuresM⁺( ¯Ω×S⁰) on ¯Ω×S⁰, whereB_εdenotes theε-neighborhood ofS. We will call ¯μthe gradient Young- concentration measure generated by (¯v_ε)_ε>0. In contrast to the one-dimensional case, it is difficult to express the measure ¯μand we adopt the following strategy: under some additional hypotheses onFε, one can prove that a subsequence of ¯v_εstrongly converges in L²(Ω) to some function ¯v∈W^1,2(Ω\Σ); our objective is to provide an intrinsic characterization of ¯μ(i.e., independent of the sequence) with respect to the Sobolev function ¯v, in the spirit of [3,4]. Actually, we characterize the limits (v, μ) generated by sequences (v_ε)_ε satisfying the partial finite energy condition

supε>0 Ω\Σε

|∇v_ε|² dx+ε

Σε

∂v_ε

∂ν ² dx

<+∞.

We show that v and μ =μ_x⊗π with μ = (a(x)δ₁+b(x)δ₋₁)⊗¯π, are linked by the necessary and sufficient conditions:

v∈W^1,2(Ω\Σ), v= 0 on∂Ω;

spt(π)⊂S;¯ dπ

dH^N−1S(x)(a(x)c+b(x)d)≥ϕ([v](x))

(1.3)

forH^N−1_S a.e. xand for all (c, d)∈R⁺×R⁺ where ϕ(ξ) =

cξ², ifξ≥0 dξ², ifξ≤0.

In addition, in Example 3.2 we will exhibit a large class of non trivial pairs (v, μ) satisfying (1.3) and generated by sequences (¯v_ε)_ε>0 of solutions to Dirichlet problems with large sources concentrated on Σ_ε. Moreover the probability measureμ_xcan be completely expressed.

It is worth noticing that the internal energy functional of (1.2) or (1.4), (1.5) below, rewritten in terms of the rescaled functionv, is the type of functionals considered in [5,6] in order to model adhesive bounded joints through the computation of a variational limit problem. In this paper, it is not our purpose to describe a variational limit of (1.2), (1.4) or (1.5), even if this program would be interesting because of the additional difficulty due to the source. Among the physical motivations of (1.2), or of more general Dirichlet problems of the form

−div (σ_ε(x)∂f(∇u)) = ^√1_εF_εon Ω

u= 0 on∂Ω, (1.4)

ν

Ω

Σ ^S _B

ε

Figure 1. The physical configuration.

where σ_ε := ¹_ε^½_Ω\Σε+^½_Σε and f :R^N →Ris a positively 2-homogeneous convex function, one may mention various applications to:

– heat conduction or electrostatic problems subjected to large concentrated sources, with high conductivity outside the support of the sources;

– membrane problems with large concentrated exterior loading and high stiffness outside the support of the loading.

Note that we treat this study in a vectorial environment, that is, we consider the following problems defined overW₀^1,2(Ω,R^m), withm≥1 by

¯

u_ε∈ argmin

1

ε^p−1 _Ω\Σεf(∇u) dx+

Σε

g(∇u) dx− 1

ε^1−1/p Fε, u:u∈W₀^1,p(Ω,R^m)

, p >1 (1.5)

where f, g : R^N×m → R are two positively p-homogeneous and quasiconvex functions, and _ε1−1/p¹ F_ε is the exterior loading. To shorten the proofs, we only consider the casep= 2.

2. Towards the definition of gradient Young-concentration measures

LetN,mbe two positive integers. Throughout this paperL^N andH^N−1denote theN-dimensional Lebesgue measure and the (N−1)-dimensional Hausdorff measure on Ω, respectively. We will sometimes write dxfor the measureL^N and|B|for theN-dimensional Lebesgue measure or the (N−1)-dimensional Hausdorff measure of any Borel subsetBof Ω, if there is no ambiguity. In theN-dimensional Euclidean spaceR^N with respect to the orthonormal frame (O, e₁, . . . , e_N), we consider a (N−1)-hypersurface Σ which splits Ω into two subdomains Ω⁺and Ω⁻,i.e. Ω = Ω⁺∪Σ∪Ω⁻. To avoid certain technical complications, we will additionally assume that Σ is included in the hyperplane vect(e₁, . . . , e_N−1) generated by{e1, . . . , e_N−1}and orthogonal to the unit vector ν =e_N, but we point out that we could treat the problem in the more general case where Σ is the graph of a Lipschitz function. Such a general geometry leads to technical complications in the proofs, but the overall strategy remains the same. For any point xin Ω, we write ˆxfor its projection on vect(e₁, . . . , e_N−1), so that x= (ˆx, x_N). We set Σ_ε:={x+tν:x∈Σ, −₂^ε< t < ^ε₂}and, forS⊂⊂Σ,B_ε:={x+tν:x∈S, −^ε₂ < t < ^ε₂} (see Fig. 1).

We assume that (F_ε)_ε>0 is a bounded sequence in the topological dual of W₀^1,1(Ω,R^m), and we define on W₀^1,2(Ω,R^m) the family of functionals

F_ε(u) =1

ε _Ω\Σεf(∇u) dx+

Σε

g(∇u) dx− 1

√ε Fε, u

where f, g : R^N×m →R are two positively 2-homogeneous and quasiconvex functions satisfying the standard growth conditions: there exist two constants 0< α≤β such that

α|ζ|²≤f(ζ)≤β(1 +|ζ|²) ∀ζ∈R^N×m, (2.1) same forg. We consider the functionalG_εdefined inW₀^1,2(Ω,R^m) by

G_ε(v) =F_ε(√ εv) =

Ω\Σε

f(∇v) dx+ε

Σε

g(∇v) dx− Fε, v

and the problem ¯v_ε ∈ argmin_v∈W1,2

0 (Ω,R^m)G_ε. Note that, by using the direct methods of the Calculus of Variations, one can easily establish that argmin_v∈W1,2

0 (Ω,R^m)G_ε = ∅. We could also treat the problem with various other boundary conditions. Throughout this paperC will denote various constants independent of ε and we do not relabel the subsequences considered.

Lemma 2.1. Let (v_ε)_ε>0 be a sequence in W₀^1,2(Ω,R^m)satisfyingsup_ε>0G_ε(v_ε)<∞. Then supε>0

ε

Σε

|∇v_ε|² dx+

Ω\Σε

|∇v_ε|² dx

<+∞. (2.2)

Proof. Estimate (2.2) is a straightforward consequence of estimates (a), (b) and (c) below: there exists a non-negative constantCsuch that

(a)

Σε

ε|∇v_ε|² dx≤C

1 +

Ω

|∇v_ε|dx

; (b)

Ω\Σε

|∇v_ε|² dx≤C

1 +

Ω

|∇v_ε|dx

; (c) sup

ε>0 Ω

|∇vε| dx <+∞.

Proof of (a). Since sup_ε>0G_ε(v_ε)<+∞, using (2.1), and Poincar´e’s inequality, we have α

Σε

ε|∇v_ε|²dx≤

Σε

εg(∇v_ε) dx ≤ C+F_εv_εW^1,1

0 (Ω,R^m)

≤ C

1 +

Ω

|∇v_ε|dx

.

Proof of (b). Sinceα

Ω\Σε|∇v_ε|²dx≤

Ω\Σεf(∇v_ε) dx, reproducing the proof of (a), one has

Ω\Σε

|∇v_ε|² dx≤C

1 +

Ω

|∇v_ε|dx

.

Proof of (c). Using the Cauchy-Schwartz inequality and (a), one can show that

Σε

|∇vε| dx≤C

1 +

Ω

|∇vε|dx ¹

2

.

Using the Cauchy-Schwartz inequality and (b), one can show that

Ω\Σε

|∇v_ε|dx≤C

1 +

Ω

|∇v_ε|dx ¹

2

.

From the previous estimates one has

Ω

|∇vε|dx=

Σε

|∇vε|dx+

Ω\Σε

|∇vε|dx≤C

1 +

Ω

|∇vε| dx ¹₂

,

thus sup

ε>0 Ω

|∇vε| dx <+∞.

Obviously ¯v_ε ∈ argmin_v∈W1,2

0 (Ω,R^m)G_ε(v) satisfies sup

ε>0G_ε(¯v_ε)<+∞; consequently, thanks to Lemma2.1,

¯

v_ε satisfies (2.2), thus the weaker condition supε>0

ε

Σε

∂v_ε

∂ν

² dx+

Ω\Σε

|∇vε|² dx

<+∞.

We will consider this condition in the definition of gradient Young-concentration measures. Let us introduce the space

W_∂Ω^1,2(Ω\Σ,R^m) :=

v∈W^1,2(Ω⁺,R^m)∩W^1,2(Ω⁻,R^m) :v= 0 on∂Ω in the sense of traces . Then we have:

Lemma 2.2. Let (v_ε)_ε>0 be a sequence in W₀^1,2(Ω,R^m)such that supε>0

ε

Σε

∂v_ε

∂ν

² dx+

Ω\Σε

|∇vε|² dx

<+∞. (2.3)

Then there exist a not relabeled subsequence andv∈W_∂Ω^1,2(Ω\Σ,R^m)such that v_ε→v in L²(Ω,R^m).

Proof. For every functionw∈W₀^1,2(Ω,R^m) we define itsε-translateT_εwby T_εw(ˆx, x_N) =

w(ˆx, x_N +ε/2), ifx∈Ω⁺, w(ˆx, x_N −ε/2), ifx∈Ω⁻.

First step. We claim that there exist z ∈ W_∂Ω^1,2(Ω\Σ,R^m) and a subsequence v_ε such that T_εv_ε z in W_∂Ω^1,2(Ω\Σ,R^m) and strongly inL²(Ω,R^m). Indeed because of the boundary condition, by extending all the con- sidered functions by zero outside of Ω, one may assume that Ω⁺and Ω⁻are cubes; clearly∇T_εv_ε=T_ε∇v_εso that sup_ε

Ω\Σ|∇T_εv_ε|² dx= sup_ε

Ω\Σε|∇v_ε|² dx <+∞. Poincar´e’s inequality then yieldsT_εv_εW^1,2

∂Ω(Ω\Σ,R^m)≤C and the claim follows immediately. We denote byz⁺ and z⁻ the traces ofz considered as a Sobolev function on Ω⁺ and Ω⁻ respectively.

Second step. We claim that there existsv∈L²(Ω,R^m) such that we can extract from the previous subsequence (v_ε)_ε>0 a subsequence which weakly converges tov in L²(Ω,R^m). Indeed

Ω

v_ε² dx=

Ω\Σε

v_ε²dx+

Σε

v_ε² dx=

Ω⁺∪Ω⁻|Tεv_ε|² dx+

Σε

v_ε² dx. (2.4)

We are going to show that

Σεv_ε²dx→0 asε→0. An integration with respect tox_N gives v_ε(ˆx, x_N) =v_ε(ˆx, ε/2) +

xN

ε/2

∂v_ε

∂ν (ˆx, s)ds.

It is not difficult to show, setting Σ⁺_ε = Σ_ε∩Ω⁺, that this implies

Σ⁺ε

|vε|²≤Cε

S+ε/2ν|vε|²dˆx+ε

Σ⁺ε

∂v_ε

∂ν ²ds

.

The same calculation on Σ⁻_ε = Σ_ε∩Ω⁻ gives

Σ⁻ε

|v_ε|²≤Cε

S−ε/2ν|v_ε|²dˆx+ε

Σ⁻ε

∂v_ε

∂ν ²ds

.

Summing up, we obtain

Σε

|vε|² dx≤Cε

S+ε/2ν|vε|² dˆx+

S−ε/2ν|vε|² dˆx+ε

Σε

∂v_ε

∂ν ² dx

.

By using (2.3) and the fact that

ε→0lim S+ε/2ν|v_ε|²dˆx+

S−ε/2ν|v_ε|²dˆx

=

S|z⁺|² dˆx+

S|z⁻|²dˆx, we have

ε→0lim Σε

|v_ε|²dx= 0. (2.5)

Estimate (2.5) and the fact thatT_εv_ε converges tozin L²(Ω,R^m) imply

ε→0lim Ω

|v_ε|² dx=

Ω

|z|² dx (2.6)

from (2.4). In particular we deduce that v_εis bounded inL²(Ω,R) and the conclusion follows immediately.

Last step. To conclude, we establish thatv=z. Indeed, for everyϕ∈ C_c^∞(Ω,R^m) we have

Ω

v·ϕdx= lim

ε→0 Ω

v_ε·ϕdx= lim

ε→0 Ω

T_εv_ε·T_εϕdx=

Ω

z·ϕdx.

This completes the proof becausev_ε v in L²(Ω,R^m) and, from (2.6),v_εL²_(Ω,R)→ v_L²_(Ω,R). We denote by S^m−1 the unit sphere of R^m and by M⁺( ¯Ω×S^m−1) the set of non negative bounded Borel measures on ¯Ω×S^m−1. For every measureμ inM⁺( ¯Ω×S^m−1),μ=μ_x⊗πdenotes its slicing decomposition.

We recall thatμ_x is a probability measure onS^m−1 andπis the projection measure ofμon ¯Ω.

Recall that the 2-homogeneous extension ˜ϕ:R^m→Rof a continuous function ϕonS^m−1 is defined for all ζ∈R^mby

˜ ϕ(ζ) =

|ζ|²ϕ(_|ζ|^ζ ), ifζ= 0, 0, otherwise.

It is well known (see [3], p. 741) that the 2-homogeneous extension of a Lipschitz functionϕinS^m−1fulfills the following locally Lipschitz property: there existsc=c(ϕ) such that

∀ζ, ζ∈R^m, |ϕ(ζ)˜ −ϕ(ζ˜ )| ≤c|ζ−ζ|(|ζ|+|ζ|). (2.7)

To shorten notation we will not distinguishϕfrom its 2-homogeneous extension ˜ϕ.

Definition 2.3. We say that a pair (v, μ) belongs to the set E ⊂L²(Ω,R^m)×M⁺( ¯Ω×S^m−1) of elementary gradient Young-concentration measuresif and only if v∈W₀^1,2(Ω,R^m) andμ=δ∂v

∂ν/|^∂v_∂ν|(x)⊗ |_∂ν^∂v|² dx.

We introduce the convergence of a sequence ((v_ε, μ_ε))_ε>0 ofE to a pair (v, μ) ofL²(Ω,R^m)×M⁺( ¯Ω×S^m−1) in the sense of concentration measuresas follows:

(v_ε, μ_ε)(v, μ) ⇐⇒

v_ε→v strongly inL²(Ω,R^m) ε^½_Bεμ_ε μ^∗ inM⁺( ¯Ω×S^m−1).

Definition 2.4. Let B be the set of pairs (v, μ) of E such that (2.3) holds. The set YC of gradient Young- concentration measures is the sequential closure ofBfor the convergence in the sense of concentration measures defined above,i.e.

(v, μ)∈ YC ⇐⇒ ∃v_ε∈W₀^1,2(Ω,R^m) s.t.

⎧⎪

⎪⎪

⎪⎨

⎪⎪

⎪⎪

⎩

v_ε→vstrongly in L²(Ω,R^m), supε>0

ε

Σε

∂v_ε

∂ν

²dx+

Ω\Σε

|∇v_ε|² dx

<+∞, δ_∂vε

∂ν/^∂vε_∂ν(x)⊗ε^½_Bε|^∂v_∂ν^ε|²dx μ^∗

(2.8)

when ε → 0. We say that the sequence (v_ε)_ε>0 generates the gradient Young-concentration measure (v, μ) or, in short, that the sequence (v_ε)_ε>0 generates the Young-concentration measure μ. Recall that the weak convergence^∗ above is defined by

Bε

εθ(x)ϕ ∂v_ε

∂ν

dx→

Ω¯ S^m−1θ(x)ϕ(ζ) dμ

for all θ ∈ C( ¯Ω) and allϕ∈ C(S^m−1). The set YC_abs is the subset of the elements (v, μ) ofYC such that the projection measureπof μon ¯Ω is absolutely continuous with respect to the measureH^N−1S.

Remark 2.5. Let (v_ε)_ε>0be a sequence ofW₀^1,2(Ω,R^m) which satisfies (2.3). Clearlyδ_∂vε

∂ν/^∂vε

∂ν_(x)⊗ε^½_Bε|^∂v_∂ν^ε|²dx is bounded inM⁺( ¯Ω×S^m−1). On the other hand, from Lemma 2.2, one can extract a subsequence of (v_ε)_ε>0 which converges in L²(Ω,R^m) to some v ∈ W_∂Ω^1,2(Ω\Σ,R^m). Then one can extract a subsequence such that (v_ε)_ε>0 generates some (v, μ) ∈ YC, that is (v_ε, δ_∂vε

∂ν /^∂vε

∂ν _(x)⊗ε^½_Bε|^∂v_∂ν^ε|² dx) (v, μ) ∈ YC. Note that, according to Lemma 2.1, the sequence (¯v_ε)_ε>0 of solutions of Dirichlet problems ¯v_ε ∈ argmin_v∈W1,2

0 (Ω,R^m)G_ε satisfies (2.3), then generates a gradient Young-concentration measure (v, μ).

3. Characterization of the set of gradient Young-concentration measures

This section is devoted to the proof of the following characterization ofYC:

Theorem 3.1 (characterization). A pair (v, μ =μ_x⊗π) belongs to YC if and only if v ∈ W_∂^1,2_Ω(Ω\Σ,R^m), π is concentrated onS¯ and, for every ϕ∈ C(S^m−1)such thatϕ^∗∗>−∞,

dπ dH^N−1S(x)

S^m−1ϕ(ζ) dμ_x≥ϕ^∗∗([v](x)) for H^N−1 a.e. x∈S

S^m−1

ϕ(ζ) dμ_x≥0 forπ_s a.e. x∈S¯

(3.1)

whereπ=_dHN−1^dπ _SH^N−1S+π_sis the Radon-Nikodym decomposition ofπwith respect to the measureH^N−1S. By analogy with the gradient Young measures (see [4]), the Sobolev function v will be refered to as the underlying deformation of the measure μ. Note that for each v ∈ W_∂Ω^1,2(Ω\Σ,R^m), the pair (v, δ[v]

|[v]|(x)⊗

|[v](x)|²H^N−1S) belongs to YC (more precisely toYCabs) and v is its underlying deformation. We call these pairselementary limit gradient Young-concentration measures.

Gradient Young-concentration measures which are not elementary limit gradient Young-concentration mea- sures and which are generated by a sequence of solutions of Dirichlet problems of the type (1.4) exist in abundance, as seen in the next example.

Example 3.2. Fix an arbitrary open intervalI inR, a functionφ:I→R^min W₀^1,2(I,R^m) andv∈W_∂Ω^1,2(Ω\

Σ,R^m). For H^N−1 almost every x ∈ Σ, define now the probability measure μ_x on S^m−1 by μ_x = ˜μ_x/μ˜_x, where for allϕ∈ C(S^m−1)

˜μ_x, ϕ:=−

Iϕ

[v](x) + dφ dy(y)

dy.

Consider the measureμ=μ_x⊗ ˜μ_xH^N−1S. According to Jensen’s inequality, clearly the pair (v, μ) fulfills all the conditions of Theorem 3.1, but it is a non elementary limit concentration measure when φ ≡ 0 (the elementary limit concentration measures correspond to the case whenφ≡0). In the scalar casem= 1 where S^m−1={−1,1}andμ_x=a(x)δ₁+b(x)δ₋₁, we can expressaandbcompletely thanks to Theorem3.1. Indeed ϕ∈ C(S^m−1) fulfills the conditionϕ^∗∗>−∞, if and only if its extension is of the form

ϕ(ζ) =

cζ², ifζ≥0 dζ², ifζ≤0

where c anddare two non-negative real numbers (note that in this caseϕ^∗∗ =ϕ). With these considerations taking successivelyc >0,d= 0 and c= 0,d >0, we easily deduce

a(x) =

[[v](x)>−^dφ_dy][v](x) +^dφ_dy(y)² dy

I[v](x) +^dφ_dy(y)² dy , b(x) =

[[v](x)<−^dφ_dy][v](x) +^dφ_dy(y)² dy

I[v](x) +^dφ_dy(y)² dy ·

Let us now exhibit a sequence (v_ε)_ε>0 generating (v, μ). To this end we define R_ε : W^1,2(Ω\Σ,R^m) → W^1,2(Ω,R^m) by

R_εv(ˆx, x_N) = xN

ε [v(ˆx, ε/2)−v(ˆx,−ε/2)] +¹₂[v(ˆx, ε/2) +v(ˆx,−ε/2)] ifx∈Σ_ε

v ifx∈Ω\Σ_ε, (3.2)

where v_ε(ˆx, ε/2) and v_ε(ˆx,−ε/2) should be taken within the meaning of traces on Σ + ₂^εe_N and Σ− ^ε₂e_N respectively. TakeI= (−¹₂,¹₂) and setv_ε(x) :=R_εv(x)+φ(^x_ε^N),B=S×I. For allθ∈ C( ¯Ω) and allϕ∈ C(S^m−1)

we have ε

Bε

θ(x)ϕ ∂v_ε

∂ν

dx = ε

Bε

θ(x)ϕ 1

ε

v(ˆx, ε/2)−v(ˆx,−ε/2) +dφ dν

x_N ε

dx

= 1

ε _Bεθ(x)ϕ

v(ˆx, ε/2)−v(ˆx,−ε/2) +dφ dν

x_N ε

dx

=

Bθ(ˆx, εx_N)ϕ

v(ˆx, ε/2)−v(ˆx,−ε/2) +dφ dν(x_N)

dx which tends to

Sθ(ˆx,0)

Iϕ

[v](ˆx) +dφ dy(y)

dy

dˆx= ˜μ_x⊗ H^N−1S, θ⊗ϕ.

Since moreover clearlyv_ε→v in L²(Ω,R^m), the sequence (v_ε)_ε>0generates (v, μ).

We claim thatu_ε=√

εv_ε, wherev_ε is the function constructed above, is the solution to a Dirichlet problem with large source. In order to shorten the calculation we prove the claim in the scalar case (m = 1) and we begin by the one dimensional case (N = 1).

Take φ∈ C₀²(I), v ∈ C²((−1,0)∪(0,1)) with v(−1) =v(1) = 0 and denote by [v]_ε the approximate jump v(ε/2)−v(−ε/2). Then by an elementary computation, it is readily seen thatu_εis solution to

(σ_εu_ε) =^√1_εFε in (−1,1) u_ε(−1) =u_ε(1) = 0, whereF_ε is the measure

v

−1,−ε 2

∪ε 2,1

+1

εφ t

ε

−ε 2,ε

2

+

[v]_ε−v −ε

2

δ₋^ε

2 +

v −ε

2

−[v]_ε

δ^ε

2.

Note that, choosing v such thatv = 0, i.e. v of the form v(t) =a⁻(t+ 1) on (−1,0) and v(t) =a⁺(t−1) on (0,1), the measureFεis concentrated on (−^ε₂,^ε₂).

We treat now the case N = 2 (the calculation in the general case is similar). We set Ω = (−1,1)² and Σ = (−1,1)× {0}. Takeφ∈ C₀²(I) andv ∈ C²(Ω\Σ) satisfyingv = 0 on∂Ω. One can easily show thatu_ε is the solution of the Dirichlet problem

div(σ_ε∇uε) = √¹εFεin Ω

u_ε= 0 on∂Ω,

whereσ_ε:= ¹_ε^½_Ω\Σε+^½_Σε, and Fε is given by Fε = Δu_ε(Ω\Σ_ε)

+

[v]_ε(x₁)− ∂v

∂x₂

x₁,−ε 2

H¹ S−ε

2e₂

+ ∂v

∂x₂

x₁,ε 2

−[v]_ε(x₁)

H¹ S+ε

2e₂

+

x₂ ∂²v

∂²x₁

x₁,ε 2

− ∂²v

∂²x₁

x₁,−ε 2

+ε

2 ∂²v

∂²x₁

x₁,ε 2

+ ∂²v

∂²x₁

x₁,−ε 2

Σε+1 ε

d²φ d²x₂

x₂ ε

Σε.

The approximated jump [v]_ε is given now by [v]_ε(x₁) :=v(x₁,₂^ε)−v(x₁,−^ε₂). Note that, takingv solution of the Dirichlet-Neumann problem

⎧⎨

⎩

Δv= 0 in Ω⁺ v= 0 on∂Ω⁺\Σ

∂x∂v2 prescribed on Σ,

⎧⎨

⎩

Δv= 0 in Ω⁻ v= 0 on∂Ω⁻\Σ

∂x∂v2 prescribed on Σ, the sourceFε is concentrated on Σ_ε.

The proof of Theorem 3.1 is divided into two propositions, the necessary condition (Prop. 3.3) and the sufficient condition (Prop. 3.5).

Proposition 3.3(necessary condition). Assume that(v, μ=μ_x⊗π)belongs toYC. Thenv∈W_∂Ω^1,2(Ω\Σ,R^m), π is concentrated onS¯ and, for every ϕ∈ C(S^m−1)such thatϕ^∗∗>−∞,(3.1)holds.

The proof of Proposition3.3is based on next Lemma 3.4.

Lemma 3.4. Let (v_ε)_ε>0 be a sequence in W₀^1,2(Ω,R^m) satisfying (2.3) and converging to v in L²(Ω,R^m).

Then v∈W_∂Ω^1,2(Ω\Σ,R^m); moreover for all continuous functions ϕon S^m−1 and all non-negative functions θ in C( ¯S), we have

lim inf

ε→0 ε

Bε

θ(ˆx)ϕ ∂v_ε

∂ν

dx≥

Sθ(ˆx)ϕ^∗∗([v]) dˆx. (3.3) Proof. According to Lemma2.2, one hasv∈W_∂Ω^1,2(Ω\Σ,R^m). Consider the Moreau-Yosida proximal approxi- mationϕ_p of the continuous functionϕ:S^m−1→Rdefined by

ϕ_p(ζ) = inf

ξ∈S^m−1

ϕ(ξ) +p|ξ−ζ|² .

It is well known that ϕ_p is a Lipschitz function and that the sequence (ϕ_p)_p∈N converges increasing to ϕ(for a proof consult [1], Thm. 2.6.4). Consequently, according to Dini’s theorem (ϕ_p)_p∈N converges uniformly toϕ onS^m−1. Letη >0 andp(η)∈Nbe such that

sup

ξ∈S^m−1|ϕ(ξ)−ϕ_p(η)(ξ)|< η. (3.4) We establish the existence of a non-negative constantCindependent on εandη such that

lim inf

ε→0 ε

Bε

θ(ˆx)ϕ_p(η) ∂v_ε

∂ν

dx≥

S

θ(ˆx)ϕ^∗∗([v]) dˆx−Cη. (3.5)

Consider the functionw_ε:=v_ε−R_εv, whereR_ε:W^1,2(Ω\Σ,R^m)→W^1,2(Ω,R^m) is the map defined in (3.2).

It is readily seen thatw_εconverges to 0 inL²(Ω,R^m), sincev_ε→vinL²(Ω,R^m). By using the locally Lipschitz property (2.7) satisfied by the 2-homogeneous extension ofϕ_p(η)(recall thatϕ_p(η) is Lipschitz), we have

lim inf

ε→0 ε

Bε

θ(ˆx)ϕ_p(η) ∂v_ε

∂ν

dx = lim inf

ε→0

1

ε _Bεθ(ˆx)ϕ_p(η)

ε∂v_ε

∂ν

dx

= lim inf

ε→0

1

ε _Bεθ(ˆx)ϕ_p(η)

ε∂w_ε

∂ν +v(ˆx, ε/2)−v(ˆx,−ε/2)

dx

= lim inf

ε→0

1

ε _Bεθ(ˆx)ϕ_p(η)

[v] +ε∂w_ε

∂ν

dx. (3.6)