Introduction In the last few years, many mathematical efforts have been devoted to variational problems arising in the modelling of phase transitions in solids, see e.g

PHASE TRANSITIONS IN THIN FILMS

Bernardo Galv˜ao-Sousa Department of Mathematics and Statistics

McMaster University Hamilton, ON, Canada

beni@mcmaster.ca

Vincent Millot Universit´e Paris Diderot - Paris 7 CNRS, UMR 7598 Laboratoire J.L. Lions

F-75005 Paris, France millot@math.jussieu.fr

Abstract.Motivated by solid-solid phase transitions in elastic thin films, we perform a Γ-convergence analysis for a singularly perturbed energy describing second order phase transitions in a domain of vanishing thickness.

Under a two-wells assumption, we derive a sharp interface model with an interfacial energy depending on the asymptotic ratio between the characteristic length scale of the phase transition and the thickness of the film.

In each case, the interfacial energy is determined by an explicit optimal profile problem. This asymptotic problem entails a nontrivial dependance on the thickness direction when the phase transition is created at the same rate as the thin film, while it shows a separation of scales if the thin film is created at a faster rate than the phase transition. The last regime, when the phase transition is created at a faster rate than the thin film, is more involved. Depending on growth conditions of the potential and the compatibility of the two phases, we either obtain a sharp interface model with scale separation, or a trivial situation driven by rigidity effects.

Keywords: phase transitions, thin films, singular perturbations, double-well potential, Γ-convergence Mathematics Subject Classification 2000: 35G99, 49J40, 49J45, 49K20, 74K35, 74N99

1. Introduction

In the last few years, many mathematical efforts have been devoted to variational problems arising in the modelling of phase transitions in solids, see e.g. [6,16,17,18]. These problems often involve singularly perturbed functionals of the form

E_ε(u,B) = Z

B

1

εW(∇u) +ε|∇²u|²dx, (1.1)

whereu:B⊂R³→R³represents the displacement of an elastic bodyB,ε >0 is a small parameter, and W is a (nonnegative) free energy density with multiple minima corresponding to martensitic materials. Due to the multiple well structure, nucleation of phases in a given configuration may occur without increasingR

W(∇u), so that the free energy may admit many (eventually constrained) minimizers. In order to select preferred configurations, the Van der Waals-Cahn-Hilliard theory adds higher order terms leading to functionals of the form (1.1). In such functionals a competition occurs between the two terms: the free energy favors gradients close to a minimum value ofW, while|∇²u|² penalizes transitions from one minima to another.

The Γ-convergence method provides a suitable framework to study the asymptotic behavior of singularly perturbed energies likeE_ε (seee.g. [12,19] for a more detailed overview of this subject).

One of the first applications of Γ-convergence was actually obtained in [30,31,35] in the context of fluid-fluid phase transitions (seee.g.[25]). Here the authors deal with energy functionals of the form R ₁

εW(v) +ε|∇v|²dx where the potentialW has a double well structure, i.e., {W = 0}={α, β}. It

1

is shown that such family of energies Γ-converges (in a suitable topology) asε→ 0 to a functional which calculates the area of the interface between the two phasesαandβ, for limitingBV-functions v with values in {α, β}. Since then this result has been generalized in many different ways (see e.g.[1,5,7,10,24,32]), in particular in [23] for an intermediate situation where the singular perturbation

|∇v|²is replaced by the higher order term |∇²v|².

The first Γ-convergence result for functionals acting on gradient vector fields has been obtained in [16]. Assuming that{W = 0}={A, B}for some rank-one connected matricesAandB (and some additional constitutive conditions onW), the authors prove the Γ-convergence ofEε asε→0. Once again the effective functional returns the total area of the interfaces separating the phasesA andB, for limiting functions usatisfying ∇u ∈ {A, B} a.e. and ∇u∈ BV. Here the rank-one connection between the wells A and B turns out to be necessary for the existence of non-affine u’s satisfying

∇u ∈ {A, B}, and the interfaces must be planar and oriented according to the connection, see [6].

However the two-wells assumption onW is quite unrealistic for the modelling of martensitic materials.

Indeed, to take into account the frame-indifference, one should assume instead that the zero level set of W is of the form SO(3)A∪SO(3)B. Due in particular to compactness issues, this situation is much more involved, but [16] still represents an important step towards a better understanding of the SO(3)-invariant problem. In two dimensions the Γ-convergence of frame-indifferent functionals has been recently solved in [17] (see also [18] in the context of linear elasticity).

Another topic of increasing interest related to solid mechanics concerns thin elastic films, that is the derivation of limiting models starting from 3D nonlinear elasticity in domains of vanishing thickness. Also in this context, the Γ-convergence method provides a suitable framework as it has been shown in [28,13] for the membrane theory, and more recently in [21,22] for nonlinear plate models. In the regime of membranes, several studies have focused their attention on the impact of a higher order perturbation on the behavior of thin films. The first variational approach has been addressed in [9] where the authors add the perturbationε²R

|∇²u|²to the free energy R

W(∇u) for a domain of small thickness h. They obtain in the limit h→0 a 2D energy density which depends on the deformation gradient of the mid-surface, and the Cosserat vectorbwhich gives an asymptotic description of the out-of-plane deformation. An important consequence of the results of [9] is that for many interesting materials the low energy states in the thin film limit are different from the ones in three dimensional samples. However [9] does not treat possible correlations between the thicknessh and the parameterε. This issue was first conducted in [34], and more intensively in [20] to keep track of the Cosserat vector. It is shown in [20] that the limiting model is determined by the asymptotic ratioh/εash→0 andε→0, and it depends whetherh/ε∼0,h/ε∼ ∞, orh/ε∼1.

The general idea of this paper is to study a class of singularly perturbed functionals describing phase transitions in thin films. In this direction, some models have been recently analyzed, see [8], and also [15,26] for models without singular perturbation leading to sharp interfaces. Here we want to carry out an analysis in the spirit of [16] including correlations between the strength of an interfacial energy and the thickness of the film. As in [9,20,34] we consider a “membrane scaling”, and we introduce the normalized functionalF^h_ε defined for u∈H²(Ωh;R³) by

F^h_ε(u) := 1 h

Z

Ωh

W(∇u) +ε²|∇²u|²dx,

where Ω_h := ω×hI ⊂ R³, I := (−¹₂,¹₂), and the midsurface ω ⊂ R² is a bounded, connected, Lipschitz open set. Considering configurations uwith energy of orderε, that isF^h_ε(u)6O(ε), and renormalizing by 1/εwe are led to the energy _h¹E_ε in the thin domain Ω_h. We are interested in the variational convergence of{¹_hEε(·,Ωh)}ash→0 andε→0. To this aim, we introduce the standard rescaling

u(x) =u(x) with (x₁, x₂, x₃) = (x₁,x₂,x₃ h),

which yields functionals{F_ε^h}defined for u∈H²(Ω;R³) by F_ε^h(u) :=

Z

Ω

1

εW(∇hu) +ε|∇²_hu|²dx , where Ω := Ω₁, and∇_h:= ∂₁, ∂₂,_h¹∂₃

is the rescaled gradient operator.

The Γ-convergence as ε → 0 and h → 0 of the family {F_ε^h} is the problem we would like to address. As a first attempt in this direction, we will assume some simplifying conditions. First we assume that

(H₀) ω is a bounded andconvexopen set ofR².

The other conditions concern the potential W which will be of double-well type as in [16], but with a relaxed condition on the connection between the wells similar to [6] saying that the two wells are compatible in the plane but not necessarily in the bulk. More precisely, we assume that W :R^3×3→[0,+∞) is continuous and satisfies the following assumptions:

(H₁) {W = 0} ={A, B} where A = (A⁰, A₃) ∈ R^3×2×R³ and B = (B⁰, B₃) ∈ R^3×2×R³ are distinct matrices satisfyingA⁰−B⁰= 2a⊗ν¯for somea∈R³ and ¯ν ∈S¹;

(H2) _C¹

1|ξ|^p−C16W(ξ)6C1|ξ|^p+C1, forp>2 and some constantC1>0 .

Under this first set of assumptions, we shall derive compactness properties for sequences with uni- formly bounded energy. In our setting the limiting configurations space turns out to be

C :=

(u, b)∈W^1,∞(Ω;R³)×L^∞(Ω;R³) : (∇⁰u, b)∈BV Ω;{A, B}

, ∂3u=∂3b= 0 , (1.2) where we write∇⁰ := (∂1, ∂2). Throughout the paper we might identify pairs (u, b)∈C with functions defined on the mid-surfaceω, that isu(x) =u(x⁰),b(x) =b(x⁰) withx⁰ := (x1, x2). In particular, for any (u, b)∈C, we can write

(∇⁰u, b)(x⁰) = 1−χE(x⁰)

A+χE(x⁰)B forL²-a.e. x⁰ ∈ω , (1.3) whereE⊂ωis a set of finite perimeter inω, andχE denotes its characteristic function. ForA⁰ 6=B⁰ the (reduced) boundary ofE consists of countably many planar interfaces with normal ¯ν, whileE is an arbitrary set of finite perimeter inω ifA⁰=B⁰ (see Theorem 2.1, and [6]).

The general compactness result is formulated in Theorem 1.1 below. As a matter of fact this theorem does not provide optimal compactness in the case ε << h. Indeed, in this regime we may expect the film to behave like a three dimensional sample. For instance, if the wells are not compatible in the bulk, i.e., rank(A−B) > 1, it reasonable to believe that sequences with bounded energy converge to trivial limits. This question will be addressed in the last section with some partial results (see Theorems 6.3 and 6.4).

Theorem 1.1 (Compactness). Assume that ω is a bounded, connected, Lipschitz open set, and that (H1)−(H2)hold. Let hn →0⁺ andεn →0⁺ be arbitrary sequences, and let {un} ⊂H²(Ω;R³) be such thatsup_nF_ε^h_nⁿ(un)<+∞. Then there exist a subsequence (not relabeled) and(u, b)∈C such that u˜n:=un− −R

Ωundx→uinW^1,p(Ω;R³)and _h¹

n∂3un→b inL^p(Ω;R³).

To describe our Γ-convergence results we need some additional assumptions on the internal struc- ture of the potentialW (these assumptions will be used only in the construction of recovery sequences).

First of all we may assume without loss of generality thatA=−B and ¯ν =e⁰₁:= (1,0), so that A⁰=−B⁰=a⊗e⁰₁ and A3=−B3. (1.4) Indeed, the general case can be reduced to (1.4) by considering a modified bulk energy densityW_mod defined by Wmod(ξ) := W(ξR+C) where C = 1/2(A+B) and R = diag(R⁰,1) with R⁰ ∈ SO(2) satisfying R⁰ν¯ = e⁰₁. This new potential Wmod obviously satisfies hypotheses (H1) and (H2) with (1.4). According to (1.4) we assume that

(H3) there exist constants% >0 andC2>0 such that 1

C2

dist ξ,{A, B}p

6W(ξ)6C₂dist ξ,{A, B}p

if dist ξ,{A, B}

6%; (H₄) W(ξ₁,0, ξ₃)6W(ξ) for allξ= (ξ₁, ξ₂, ξ₃)∈R^3×3;

(H5) ifA⁰=B⁰= 0, then W(ξ⁰, ξ3) =V |ξ⁰|, ξ3

for someV : [0,+∞)×R³→[0,+∞).

Here | · | stands for the usual Euclidean norm, ξ⁰ := (ξ₁, ξ₂) ∈ R^3×2 and |ξ⁰|² = |ξ₁|²+|ξ₂|². We observe that in the case A⁰ = B⁰ (so that (1.4) yields A⁰ = B⁰ = 0), assumptions (H4) and (H5) require the function r 7→ V r, ξ₃

to be a nondecreasing function ofr for every ξ₃ ∈ R³. All these assumptions are satisfied by the prototypical potentialW given byW(ξ) = min |ξ−A|²,|ξ−B|²

. We now consider the family of functionalsF_ε^h: [L¹(Ω;R³)]²→[0,+∞] defined by

F_ε^h(u, b) :=

(F_ε^h(u) ifu∈H²(Ω;R³) andb= ¹_h∂3u , +∞ otherwise.

We will prove that the behavior ofF_ε^hdepends on the asymptotic ratio ^h_ε →γ∈[0,+∞] ashandε tend to 0, and that the sequence {F_ε^h} Γ-converges to a functional Fγ : [L¹(Ω;R³)]² → [0,+∞]

given by

Fγ(u, b) :=

(KγPerω(E) if (u, b)∈C,

+∞ otherwise, (1.5)

where (∇⁰u, b)(x) = (1−χ_E(x⁰))A+χ_E(x⁰)B as in (1.3), and Per_ω(E) :=H¹(∂^∗E∩ω) denotes the perimeter ofEinω. Here the constantKγ >0 is determined by an optimal profile problem depending on the value of γ. By assumption (H4), the description of Kγ involves in each case the 2D energy densityW:R^3×2→[0,+∞) defined by

W(ζ1, ζ2) :=W(ζ1,0, ζ2),

and two reference maps ¯u0∈W_loc^1,∞(R;R³) and ¯b0∈L^∞(R;R³) given by

¯

u₀(t) :=|t|a and ¯b₀(t) :=

(A3 ift>0,

B3 ift <0. (1.6)

Our first convergence result deals with the critical regime where the thickness of the film and the strength of the interfacial energy are of the same order, that isγ∈(0,+∞).

Theorem 1.2 (Critical Regime). Assume that (H₀)−(H₅) hold with (1.4). Let h_n → 0⁺ and εn →0⁺ be arbitrary sequences such that hn/εn → γ for someγ ∈(0,+∞). Then the functionals {F_ε^hn

n} Γ-converge for the strongL¹-topology to the functional Fγ given by (1.5)with Kγ:= inf

1 γ

Z

`I×γI

W(∇v) +|∇²v|²dy : ` >0, v∈C<sup>2</sup>(`I×γI;R³),

∇v(y) = ¯u⁰₀,¯b0

(y1) nearby

|y1|=`/2

. (1.7) We observe that the formula for Kγ (with γ ∈(0,+∞)) entails a nontrivial dependence on the vertical direction in the asymptotic problem, at least forA36=B3. Actually, even in the caseA3=B3, one can find potentialsW for which a nontrivial dependance onx₃still occurs, see [16, Section 8]. Note that in many second order phase transitions problems, optimal profiles usually have an oscillatory behavior along the limiting interface, see [16,27] and references therein (see also Theorem 1.4 below).

In contrast with the critical regime, one may expect the case γ = 0 (i.e., h << ε) to lead to a simpler behavior with respect to thex3-variable by separation of scales. Indeed, the energies formally

behave like two dimensional ones, and optimal transition layers should only depend on the distance to the interface by assumptions (H4)−(H5). We will illustrate this facts with more details in Section 5 (see Remark 5.4). Our results for this regime can be summarized in the following theorem.

Theorem 1.3 (Subcritical Regime). Assume that(H0)−(H5)hold with (1.4). Lethn →0⁺ and εn→0⁺ be arbitrary sequences such thathn/εn →0. Then the functionals{F_ε^h_nⁿ}Γ-converge for the strongL¹-topology to the functionalF0 given by (1.5)with

K0:= inf Z `

−`

W φ(t)

+|φ⁰₁(t)|²+ 2|φ⁰₂(t)|²dt : ` >0, φ= (φ<sub>1</sub>, φ<sub>2</sub>)∈C<sup>1</sup> [−`, `];R^3×2

, φ(`) = ¯u⁰₀,¯b₀

(`)andφ(−`) = ¯u⁰₀,¯b₀ (−`)

. (1.8) In the supercritical caseγ= +∞(i.e.,ε << h), one may again expect a separation of scales to hold.

In other words, we should be able to recover the limiting functional by taking first the limitε→0, and then operating the thin film limith→0. Hence, to obtain a nontrivial Γ-limit, it is natural to ask for AandB to be compatible in the bulk with a non vertical connection. As already mentioned, we have exhibited rigidity effects in the opposite case, at least for some particular potentials (see Theorems 6.3 and 6.4). For this reason we assume in the supercritical regime thatA−B is a rank-one matrix, and thatA⁰ 6=B⁰. Under the structure (1.4), this assumption is equivalent to the existence ofλ∈Rsuch thatA₃=−B3=λa. Then the wells AandB can be written as

A=−B=a⊗(e1+λe3) (a6= 0). (1.9) Unfortunately we have only obtained partial results for this regime through lower and upper bounds for the Γ-lim inf and Γ-lim sup. However, our estimates turn out to be nearly optimal in the sense that upper and lower bounds agree wheneverλ= 0, p= 2, andW is symmetric with respect toξ3. In this latter case, it follows that the separation of scales is indeed true by [16, Theorem 1.4].

Theorem 1.4 (Supercritical Regime). Assume that(H0)−(H4)and (1.9)hold for someλ∈R. Let hn →0⁺ andεn→0⁺ be arbitrary sequences such thathn/εn→+∞. Then,

cF∞6Γ L¹

−lim inf

n→+∞F_ε^hn

n and Γ L¹

−lim sup

n→+∞

F_ε^hn

n 6F∞, for a constantc >0, where the functional F∞ is given by (1.5)with

K_∞:= (1 +λ²)¹²inf Z

Q⁰_λ

`W(∇v) +1

`|∇<sup>2</sup>v|<sup>2</sup>dy : ` >0, v∈C²(R²;R³),

∇v(y) = ¯u⁰₀,¯b₀

(y₁)nearby

|y·ν_λ|= 1/2 , v is 1-periodic in the direction orthogonal toν_λ

, whereν_λ:= ^{√ 1}

1+λ²(1, λ)∈S¹, andQ⁰_λdenotes the unit cube ofR²centered at the origin with two faces orthogonal to νλ. Moreover, ifp= 2 in(H1),λ= 0in (1.9), and W satisfiesW(ξ⁰, ξ3) =W(ξ⁰,−ξ3) for allξ∈R^3×3, then the functionals{F_ε^h_nⁿ}Γ-converge for the strongL¹-topology to F∞.

The paper is organized as follows. We start in Section 2 with a structure result for the classC of limiting maps, and the compactness theorem is proved in Section 3. The proofs of Theorems 1.2, 1.3, and 1.4 are given in Sections 4, 5, and 6 respectively. We complete Section 6 with the aforementioned rigidity results in the supercritical case.

2. Preliminaries

Throughout the paper,Q andQ⁰ denote the standard open unit cubes centered at the origin of R³ andR² respectively, whileI:= (−¹₂,¹₂). For simplicity, the differential operators _∂x^∂

i and _∂x^∂2

i∂xj are

written ∂i and ∂²_ij respectively. The rescaled gradient ∇h and rescaled Hessian ∇²_h operators are given by

∇hu=

∂1u, ∂2u,1 h∂3u

and ∇²_hu=









∂₁₁² u ∂₁₂²u ¹_h∂₁₃² u

∂₁₂² u ∂₂₂²u ¹_h∂₂₃² u

1

h∂²₁₃u _h¹∂₂₃² u _h¹₂∂₃₃² u







 .

For a Borel setB ⊂R³ and an admissible mapuwe write F_ε^h(u, B) :=

Z

B

1

εW(∇hu) +ε|∇²_hu|²dx ,

Throughout the paper we will consider two reference maps,u₀ andb₀, defined forx∈R³ by u₀(x) := ¯u₀(x₁) and b₀(x) := ¯b₀(x₁), (2.1) where ¯u₀and ¯b₀ are given by (1.6).

We follow [4] for the standard results and notations on functions of bounded variation. We only recall that, given an open set O ⊂R^N, a Borel E ⊂ O is said to be of finite perimeter inO if its characteristic functionχE belongs toBV(O). In such a case, the perimeter ofE inO, that we write Per_O(E), is the total variation|DχE|(O), and it is equal toH^N−1(∂^∗E∩O) where∂^∗E denotes the reduced boundary ofE.

We now state a structure result for the class C of limiting configurations (see (1.2)). To this purpose, let us define

α_min:= inf

x₁∈R:x= (x₁, x₂)∈ω and α_max:= sup

x₁∈R:x= (x₁, x₂)∈ω . (2.2) We have the following theorem as a consequence of [6,16].

Theorem 2.1. Assume that (H₀)and (1.4) hold. Then for every pair(u, b)∈C,(∇⁰u, b)is of the form

∇⁰u(x), b(x)

= (1−χE(x⁰))A+χE(x⁰)B ,

whereE⊂ω is a set of finite perimeter inω. Moreover, if A⁰6=B⁰ thenuis of the form

u(x) =c0+x1a−2ψ(x1)a , (2.3) where c0 ∈ R³, c0·a = 0, ψ ∈ W^1,∞((αmin, αmax);R) and ψ⁰ ∈ BVloc((αmin, αmax);{0,1}). In particular, ifA⁰6=B⁰ then E is layered perpendicularly to e⁰₁, i.e.,

∂^∗E∩ω= [

i∈I

{α_i} ×J_i, (2.4)

where I ⊂ Z is made by successive integers, {αi} ⊂ (αmin, αmax) is locally finite in (αmin, αmax), α_i< α_i+1, and the setsJ_i:={t∈R: (α_i, t)∈ω} are open bounded intervals.

Proof. Step 1. We start with the case whereA⁰6=B⁰. Given (u, b)∈C, we can write

(∇⁰u, b) = (1−χF)A+χFB , (2.5) for some set F ⊂Ω of finite perimeter in Ω. Since∂3u= 0, we have∇u∈BV(Ω;{(A⁰,0),(B⁰,0)}).

Then we observe that (1.4) yields (A⁰,0)−(B⁰,0) =a⊗e1. Thanks to the convexity ofω, we can apply [16, Theorem 3.3] to deduce thatuis of the form (2.3). From (2.3), (2.5), and the convexity of ω, it readily follows thatχ_F =χ_E×I a.e. for some setE⊂ω of finite perimeter inω satisfying (2.4).

Step 2.We now consider the caseA⁰ =B⁰. Given (u, b)∈C, we have b∈BV(Ω;{A₃, B₃}), so that b=χFA3+ (1−χF)B3for some setF ⊂Ω of finite perimeter in Ω. By standard slicing results (see[4, Section 3.11]),b^x0 :=b(x⁰,·) belongs toBV(I;R³) forL²-a.e. x⁰ ∈ω, and L²bω⊗Db^x0 =∂3b. Since

∂3b= 0, we deduce thatDb^x0 = 0 forL²-a.e.x⁰∈ω. On the other hand, we can find a representative

b^∗ of bsuch that (b^∗)^x0 :=b^∗(x⁰,·) is a good representative ofb^x0 forL²-a.e. x⁰∈ω. Since Db^x0 = 0, we conclude that (b^∗)^x0 is constant for L²-a.e. x⁰ ∈ω, that is b^∗(x) = b^∗(x⁰). Then it follows that χF =χE×I a.e. for some setE⊂ω of finite perimeter inω.

3. Compactness

This section is devoted to the proof of Theorem 1.1, and we assume that (H1) and (H2) hold. We consider arbitrary sequences hn → 0⁺, εn →0⁺ as n→+∞, and{un}n∈N ⊂H²(Ω;R³) such that sup_nF_ε^h_nⁿ(un)<+∞. Throughout this section we writebn:= _h¹

n∂3un.

Proof of Theorem 1.1. Step 1. We claim that there exist a subsequence {εn} (not relabeled), a pair (u, b)∈W^1,∞(Ω;R³)×L^∞(Ω;R³) satisfying∂₃u= 0, andθ∈L^∞(Ω; [0,1]) such that

un− − Z

Ω

un * uweakly inW^1,p(Ω;R³), bn * bweakly inL^p(Ω;R³) asn→+∞, (3.1) and

(∇⁰u, b)(x) = 1−θ(x)

A+θ(x)B forL³-a.e.x∈Ω. (3.2) Indeed, we first deduce from the growth assumption (H₂) that

Z

Ω

|∇⁰un|^p+|bn|^p dx6C

Z

Ω

W(∇⁰un, bn)dx+ 1

6C εnF_ε^h_nⁿ(un) + 1 6C . Therefore {bn} is uniformly bounded in L^p(Ω;R³), and {un − −R

Ωun} is uniformly bounded in W^1,p(Ω;R³), thanks to the Poincar´e-Wirtinger Inequality. Hence we may extract a subsequence such that (3.1) holds for some pair (u, b)∈W^1,p(Ω;R³)×L^p(Ω;R³). Sincek∂3u_nkL^p(Ω)6Ch_n, we deduce that∂3u≡0.

Next we observe that the sequence {(∇⁰u_n, b_n)} also generates a Young measure{ν_x}_x∈Ω. From the fundamental theorem on Young measures (seee.g.[33, Theorem 6.11]), we derive

Z

Ω

Z

R^3×3

W(ξ)dνx(ξ)dx6 lim

n→+∞

Z

Ω

W ∇⁰un, bn

dx= 0, so that suppνx⊂ {A, B} forL³-a.e.x∈Ω. Hence there existsθ∈L¹ Ω; [0,1]

such that νx= 1−θ(x)

δξ=A+θ(x)δξ=B forL³-a.e. x∈Ω.

Multiplying this last equality byξand integrating with respect toξyields (3.2), which completes the proof of the claim.

Step 2.We claim that (u, b)∈C. We shall distinguish two distinct cases.

Case a).We first assume thatA⁰6=B⁰. ForM >0 andξ⁰∈R^3×2, we define ϕ(ξ⁰) := inf

Z 1 0

minq

W0 g(s) , M

|g⁰(s)|ds : g∈W^1,∞([0,1];R^3×2),

g(0) =A⁰ andg(1) =ξ⁰

, whereW₀(ξ⁰) := min{W(ξ⁰, z) :z∈R³}is a continuous function of ξ⁰. One may easily check thatϕ is Lipchitz continuous,ϕ(ξ⁰) = 0 if and only if ξ⁰=A⁰, and that

|∇ϕ(ξ⁰)|6minp

W0(ξ⁰), M forL^3×2-a.e.ξ⁰ ∈R^3×2. (3.3) We claim that

ϕ(∇⁰un) is uniformly bounded inW^1,1(Ω;R). Indeed, estimate first Z

Ω

∇ ϕ(∇⁰un) dx6

Z

Ω

q

W0 ∇⁰un(x) ∇(∇⁰un) dx61

2F_ε^h_nⁿ(un)6C ,

and by (3.3),

Z

Ω

ϕ ∇⁰un(x)

dx6M Z

Ω

|∇⁰un(x)|dx+ϕ(0)L³(Ω). Hence, up to a further subsequence (not relabeled),

ϕ(∇⁰un)→H in L¹(Ω) asn→+∞, (3.4)

for someH ∈BV(Ω). On the other hand, the Young measure{µ_x}_x∈Ω generated by

ϕ(∇⁰u_n) is given by

µ_x= 1−θ(x)

δ_t=ϕ(A0)+θ(x)δ_t=ϕ(B0). Then the strong convergence in (3.4) yieldsµx=δ_t=H(x), so that

δ_t=H(x)= 1−θ(x)

δ_t=ϕ(A0)+θ(x)δ_t=ϕ(B0). As a consequenceθ(x)∈ {0,1}forL³-a.e. x∈Ω, and

H(x) = 1−θ(x)

ϕ(A⁰) +θ(x)ϕ(B⁰) =θ(x)ϕ(B⁰) forL³-a.e. x∈Ω. Since ϕ(B⁰)6= 0, it yields θ ∈ BV Ω;{0,1}

, and we may now write θ =χ_F for some set F ⊂ Ω of finite perimeter in Ω. In view of (3.2), we obtain (∇⁰u, b) = (1−χF)A+χFB L³-a.e. in Ω, and thus (∇⁰u, b) belongs toBV Ω;{A, B}

. Since ∂3u= 0, we have ∇u∈BV Ω;{(A⁰,0),(B⁰,0)}

and it follows from [16, Theorem 3.3] thatF =E×I for some setE ⊂ω of finite perimeter inω, which in turn implies that∂3b= 0, and thus (u, b)∈C.

Case b).Let us now assume thatA⁰=B⁰ andA36=B3. ForM >0 andz∈R³, we define ψ(z) := inf

Z 1 0

minq

W1 g(s) , M

|g⁰(s)|ds : g∈W^1,∞([0,1];R³), g(0) =A3 andg(1) =z

, whereW1(z) := min{W(ξ⁰, z) :ξ⁰∈R^3×2}is a continuous function ofz. As previouslyψis Lipschitz continuous, and ψ(z) = 0 if and only if z =A3. Arguing as in Case a), we obtain that {ψ(bn)} is uniformly bounded inW^1,1(Ω;R), and

1 hn

Z

Ω

∂3 ψ(bn)

dx6 1 hn

Z

Ω

pW1(bn)|∂3bn|dx6 1

2F_ε^h_nⁿ(un)6C . Therefore, up to a subsequence,

ψ(bn)→G inL¹(Ω) asn→+∞, (3.5)

for someG∈BV(Ω) satisfying∂3G= 0. Arguing again as in Case a), we deduce from (3.5) that δ_t=G(x)= (1−θ(x))δ_t=ψ(A3)+θ(x)δ_t=ψ(B3),

which yieldsθ∈BV(Ω;{0,1}). Since∂3G= 0 we can argue as in the proof of Theorem 2.1, Step 2, to deduce thatθ(x) =χ_E(x⁰) for some setE⊂ω of finite perimeter inω. In view of (3.2), we conclude that (∇⁰u, b)∈BV Ω;{A, B}

and∂3b= 0, and thus (u, b)∈C.

Step 3.In view of the previous steps, we know that (∇un, bn)*(∇u, b) weakly inL^p(Ω), and that {(∇⁰un, bn)}_n∈Ngenerates the Young measure{νx}_x∈Ωgiven by

νx(ξ) =χE(x⁰)δξ=A+ 1−χE(x⁰)

δξ=B =δ_ξ=(∇⁰u,b).

By standard results on Young measures (seee.g.[33, Proposition 6.12]), it follows that (∇⁰un, bn)→ (∇⁰u, b) strongly inL^p(Ω;R^3×3), and the proof Theorem 1.1 is complete.

4. Γ-convergence in the critical regime

This section is devoted to the proof of Theorem 1.2. The Γ-liminf and Γ-limsup inequalities are derived in Theorem 4.1 and Theorem 4.2 respectively, while Corollary 4.2 shows that the lower and upper inequalities actually coincide. In proving lower inequalities, we adopt the approach of [16] once adapted to the dimension reduction setting. Throughout this section the parameter γ ∈ (0,+∞) is given.

4.1. TheΓ-lim inf inequality We introduce the constant

K_γ^?:= inf

lim inf

n→+∞F_ε^h_nⁿ(un, Q) : hn→0⁺ andεn→0⁺ withhn/εn→γ , {un} ⊂H²(Q;R³),(un, 1

hn

∂3un)→(u0, b0) in [L¹(Q;R³)]²

, (4.1) where the functionsu0andb0 are given by (2.1). The constantK_γ^? turns out to be finite, as one may check by considering an admissible sequence{un} made of suitable regularizations ofu0and b0 (see also the proof of Theorem 4.2). In this subsection we shall prove that under assumptions (H₀)−(H₂) and (H5), the lower Γ-limit evaluated at any (u, b)∈C is bounded from below byK_γ^?times the length of the jump set of (∇⁰u, b). We first prove this statement in the case of an elementary jump set.

Proposition 4.1. Assume that (H1), (H2)and (1.4) hold. Let hn →0⁺ andεn →0⁺ be arbitrary sequences such that h_n/ε_n → γ. Let ρ >0 and α∈ R, let J ⊂R be a bounded open interval, and consider the cylinder U := (α−ρ, α+ρ)×J×I. Let(u, b)∈W^1,∞(U;R³)×L^∞(U;R³) satisfying

∂3u=∂3b= 0, and

(∇⁰u, b) =χ_{x1<α}B+χ_{x1>α}A or (∇⁰u, b) =χ_{x16α}A+χ_{x1>α}B . (4.2) Then for any sequence{un} ⊂H²(U;R³)satisfying(un,_h¹

n∂3un)→(u, b) in[L¹(U;R³)]², we have lim inf

n→+∞ F_ε^h_nⁿ(un, U)>K_γ^?H¹(J).

The proof of Proposition 4.1 relies on scaling properties and the translation invariance of the energy functionalF_ε^h. To determine the corresponding properties in the limit we introduce the following set function. For an open bounded setJ ⊂Randρ >0, we writeJρ:=ρI×J×I, and we define

Eγ(J, ρ) := inf

lim inf

n→+∞F_ε^h_nⁿ(un, Jρ) : hn →0⁺ andεn→0⁺ withhn/εn→γ, {un} ⊂H²(Jρ;R³),(un, 1

hn

∂3un)→(u0, b0) in [L¹(Jρ;R³)]²

. Noticing thatK_γ^?=Eγ(I,1) we now state the following lemma.

Lemma 4.1. Assume that (H1),(H2)and (1.4)hold. Then (i) E_γ(t+J, ρ) =E_γ(J, ρ)for allt∈R;

(ii) ifJ1⊂J2 andρ16ρ2, then Eγ(J1, ρ1)6Eγ(J2, ρ2); (iii) ifJ1∩J2=∅, thenEγ(J1∪J2, ρ)>Eγ(J1, ρ) +Eγ(J2, ρ);

(iv) if α >0, thenEγ(αJ, αρ) =αEγ(J, ρ); (v) if0< α <1, thenEγ(αJ, ρ)>αEγ(J, ρ);

(vi) if J is an interval then Eγ(J, ρ) =H¹(J)Eγ(I, ρ); (vii) ifJ is an interval then Eγ(J, ρ) =Eγ(J, δ)for allδ >0.

Proof. We first observe that(i)follows from the translation invariance of the functional F_ε^h. Then (ii) is due to the fact that any admissible sequence forEγ(J2, ρ2) yields an admissible sequence for E_γ(J₁, ρ₁) whenever J₁⊂J₂ andρ₁6ρ₂. The proof of claim(iii)follows a similar argument.

Proof of (iv). Let hn → 0⁺ and εn → 0⁺ be such that hn/εn → γ, and let {un} be an ad- missible sequence for Eγ(αJ, αρ). We define for x ∈ Jρ, vn(x) := _α¹un(αx⁰, x3), ˜hn := hn/α and

˜

ε_n := ε_n/α. By homogeneity of u₀ and b₀, we infer that v_n(x) → _α¹u₀(αx⁰, x₃) = u₀(x) and

1

˜hn∂3vn→b0inL¹(Jρ;R³) asn→+∞. In particular,{vn}with{(˜hn,ε˜n)}is admissible forEγ(J, ρ).

Changing variables then yieldsF_ε˜^˜hn

n(v_n, J_ρ) = _α¹F_ε^hn

n u_n,(αJ)_αρ

. Lettingn→+∞ we deduce that lim infn F_ε^h_nⁿ un,(αJ)αρ

>αEγ(J, ρ). In view of the arbitrariness of{(hn, εn)}and{un}, we conclude thatEγ(αJ, αρ)>αEγ(J, ρ). The reverse inequality follows from the arbitrariness ofα >0.

Proof of (v). If 0< α <1 then (αJ)αρ ⊂(αJ)ρ, and we derive from(ii) and (iv)that Eγ(αJ, ρ)>

Eγ(αJ, αρ) =αEγ(J, ρ).

Proof of (vi). WriteJ =Z∪(SN

k=1J_k) for some finite set Z and some family {Jk}^N_k=1 of mutually disjoint open intervals of the formJk =ak+αkI with 0< αk<1. In particular,H¹(J) =PN

k=1αk. We infer from(i),(iii)and(v)that E_γ(J, ρ)>(P

kα_k)E_γ(I, ρ), leading toE_γ(J, ρ)>H¹(J)E_γ(I, ρ).

The reverse inequality can be obtained in the same way inverting the roles ofIandJ. Proof of (vii).Claim(vii)is a straightforward consequence of(iv)and (vi).

Remark 4.1. As a consequence of (vi) and (vii) in Lemma 4.1, we have Eγ(J, ρ) = K_γ^?H¹(J) for everyρ >0 and every bounded open intervalJ ⊂R.

An important consequence of Lemma 4.1 is that the energy of optimal sequences for E_γ(J, ρ) is concentrated near the limiting interface. We shall make use of Corollary 4.1 in the next subsection in order to compare the constantsK_γ^? andKγ.

Corollary 4.1. Assume that (H1),(H2)and (1.4)hold. Let 0< δ < ρand let J ⊂R be a bounded open interval. For any sequences h_n →0⁺,ε_n →0⁺ and {u_n} ⊂H²(J_ρ;R³)such that h_n/ε_n →γ, (un,_h¹

n∂3un)→(u0, b0)in [L¹(Jρ;R³)]², andlimnF_ε^h_nⁿ(un, Jρ) =K_γ^?H¹(J), we have

n→+∞lim F_ε^h_nⁿ(un, Jρ\Jδ) = 0. Proof. SinceJ_δ ⊂J_ρ, we deduce from Remark 4.1 that lim sup_n F_ε^hn

n(u_n, J_δ)6E_γ(J, ρ)<+∞. On the other hand the sequence{un}is admissible for Eγ(J, δ). In view of Lemma 4.1, we infer that

lim inf

n→+∞F_ε^h_nⁿ(un, Jδ)>Eγ(J, δ) =Eγ(J, ρ) = lim

n→+∞F_ε^h_nⁿ(un, Jρ).

Therefore limnF_ε^h_nⁿ(un, Jδ) = limnF_ε^h_nⁿ(un, Jρ) which clearly implies the announced result.

Proof of Proposition 4.1. By the translation invariance of F_ε^h_nⁿ, we have lim infnF_ε^h_nⁿ(un, U) = lim inf_nF_ε^hn

n(τ u_n, J_ρ), where τ u_n(x) := u_n(x₁+α, x₂, x₃). Obviously (τ u_n,_h¹

n∂₃τ u_n)→ (τ u, τ b) in [L¹(Jρ;R³)]² with τ u(x) :=u(x1+α, x2, x3) and τ b(x) :=b(x1+α, x2, x3). If the first case in (4.2) holds, then (τ u, τ b) = (u₀+c₀, b₀) for some constantc₀∈R³. Subtracting the constantc₀, we derive from the definition ofEγ and Remark 4.1 that

lim inf

n→+∞F_ε^h_nⁿ(un, U) = lim inf

n→+∞F_ε^h_nⁿ(τ un−c0, Jρ)>Eγ(J, ρ) =K_γ^?H¹(J).

If the alternative case in (4.2) holds, then (−τ u, τ b)(−x) = (u0+c0, b0)(x) for some constantc0∈R³. Observe thatF_ε^h_nⁿ(τ un, Jρ) =F_ε^h_nⁿ(vn, Jρ) withvn(x) =−τ un(−x)−c0. Then (vn,_h¹

n∂3vn)→(u0, b0) in [L¹(J_ρ;R³)]². Hence,

lim inf

n→+∞F_ε^hn

n(u_n, J_ρ) = lim inf

n→+∞F_ε^hn

n(v_n, J_ρ)>Eγ(J, ρ) =K_γ^?H¹(J), and the proof is complete.

Remark 4.2. SettingK_∞^? to be the constant defined by (4.1) withγ= +∞(see (6.1)), and assuming thatK_∞^? <+∞, one may check that Proposition 4.1 and Corollary 4.1 still hold in the caseγ= +∞

with the same proofs.

We are now ready to prove the main result of this subsection which extends Proposition 4.1 to the general case. The proof for A⁰ 6=B⁰ will be a direct consequence of Proposition 4.1, while the caseA⁰=B⁰ will require an additional analysis based on a blow-up argument.

Theorem 4.1. Assume that(H0)−(H2),(H5)and (1.4)hold. Lethn→0⁺andεn→0⁺be arbitrary sequences such thath_n/ε_n→γ. Then, for any(u, b)∈C and any sequences{un} ⊂H²(Ω;R³)such that (un,_h¹

n∂3un)→(u, b) in[L¹(Ω;R³)]², we have lim inf

n→+∞F_ε^hn

n(u_n)>K_γ^?Per_ω(E), where(∇⁰u, b)(x) = 1−χ_E(x⁰)

A+χ_E(x⁰)B.

Proof.Step 1. We first assume that A⁰ 6=B⁰. Assuming that E is non trivial, by Theorem 2.1 we can write ∂^∗E∩ω as in (2.4). Then we have H¹(∂^∗E∩ω) = P

i∈I H¹(Ji) < +∞. Consider an arbitrarily smallδ >0 and choosek⁻ =k⁻(δ)∈I and k⁺ =k⁺(δ)∈I such thatk⁻ 6k⁺, and H¹(∂^∗E∩ω)6Pk⁺

i=k⁻H¹(Ji) +δ. For eachi=k⁻, . . . , k⁺, letJ_i⁰ ⊂⊂Jibe an open interval satisfying H¹(J_i)6H¹(J_i⁰) +_k+−k^δ−+1. Since{α_i} ×J_i⁰⊂⊂ω andα_i< α_i+1, we may findρ >0 small in such a way that the sets (αi−ρ, αi+ρ)×J_i⁰ are still compactly contained in ω, andαi+ρ < αi+1−ρ fori=k⁻, . . . , k⁺. Then we set for eachi=k⁻, . . . , k⁺,Ui:= (αi−ρ, αi+ρ)×J_i⁰×I⊂Ω. Observe that theU_i’s are pairwise disjoint, and that (∇⁰u, b) is of the form (4.2) in eachU_i.

Let us now fix an arbitrary sequence {un} ⊂ H² Ω;R³

such that (un,_h¹

n∂3un) → (u, b) in [L¹(Ω;R³)]². Using Proposition 4.1, we estimate

lim inf

n→+∞F_ε^hn

n(u_n)>

k⁺

X

i=k⁻

lim inf

n→+∞F_ε^hn

n u_n, U_i

>

k⁺

X

i=k⁻

K_γ^?H¹(J_i⁰)>K_γ^?H¹(∂^∗E∩ω)−2δK_γ^?, and the conclusion follows lettingδ→0.

Step 2.We now consider the case A⁰ =B⁰ (= 0 by (1.4)). Thenu₀= 0 and uis constant. Without loss of generality we may assume that u≡ 0. In the remaining of this proof, we shall identify any b∈L¹(Q⁰;R³) with its extension toQgiven byb(x) =b(x⁰). With this convention, we introduce for b∈L¹(Q⁰;R³),

Gγ(b) := infn lim inf

n→+∞F_ε^h_nⁿ un, Q

: hn→0 andεn→0 withhn/εn→γ, {un} ⊂H² Q;R³

,(un, 1

h_n∂3un)→(0, b) in [L¹ Q;R³ ]²o

. Notice that K_γ^? =G_γ(b₀). We shall require the sequentialL¹-lower semicontinuity of the functional Gγ stated below. The proof of Lemma 4.2 only involves a standard diagonalization argument, and we shall omit it.

Lemma 4.2. G_γ is sequentially lower semicontinuous for the strongL¹-topology.

Let{un} ⊂H²(Ω;R³) be an arbitrary sequence satisfying (un,_h¹

n∂3un)→(0, b) in [L¹(Ω;R³)]². Without loss of generality, we may assume that lim infnF_ε^h_nⁿ(un) = limnF_ε^h_nⁿ(un) <+∞. By The- orem 2.1 we have b(x) = b(x⁰) = 1−χ_E(x⁰)

A₃+χ_E(x⁰)B₃ for a set E ⊂ ω of finite perimeter inω.

Using Fubini’s theorem, we define a finite nonnegative Radon measureµ_n onω by setting µn:=

Z ¹₂

−¹₂

1 εn

W(∇h_nun) +εn

∇²_hnun

2dx3

! L²bω ,