Dimension reduction for functionals on solenoidal vector fields

DOI:10.1051/cocv/2010051 www.esaim-cocv.org

DIMENSION REDUCTION FOR FUNCTIONALS ON SOLENOIDAL VECTOR FIELDS

Stefan Kr¨ omer

Abstract. We study integral functionals constrained to divergence-free vector fields inL^p on a thin domain, under standard p-growth and coercivity assumptions, 1 < p < ∞. We prove that as the thickness of the domain goes to zero, the Gamma-limit with respect to weak convergence inL^pis always given by the associated functional with convexified energy density wherever it is finite. Remarkably, this happens despite the fact that relaxation of nonconvex functionals subject to the limiting constraint can give rise to a nonlocal functional as illustrated in an example.

Mathematics Subject Classification.49J45, 35E99.

Received April 21, 2010.

Published online December 2, 2010.

1. Introduction

This article is devoted to the study of the “effective” value per unit volume of functionals constrained to solenoidal (i.e., divergence-free) vector fields defined on a thin domainω×(0, ε), in the limit as the thicknessε goes to zero. We assume that on a domain with finite thickness, our functional (which we call the “energy”, although its meaning might be different from a physical point of view) is given by a integral of the form

G_ε(v) :=

⎧⎪

⎨

⎪⎩ 1 ε

ω×(0,ε)

g

y, v(y)

dy ifv∈ Vε,

+∞ ifv∈L^p(ω×(0, ε);R^N)\ V_ε

where N ≥2, ω is a bounded domain in R^N−1,y = (y, y_N)∈ ω×(0, ε),g : ω×R^N →Ris a given energy density, andG_εis finite only in the class of solenoidal vector fields onω×(0, ε) inL^p for some 1< p <∞,i.e.,

Vε:= v∈L^p(ω×(0, ε);R^N)|divv= 0 .

Here and throughout the rest of this article, differential constraints as forv above are understood in the sense of distributions, in particular, divv= 0 for av∈L^p(ω×(0, ε);R^N) means that

ω×(0,ε)v· ∇ϕdy= 0 for all test functionsϕ∈C_c^∞(ω×(0, ε)) (smooth functions with compact support, scalar-valued). Using rescaled variables

Keywords and phrases. Divergence-free fields, Gamma-convergence, dimension reduction.

1 Universit¨at zu K¨oln, K¨oln, Germany. skroemer@math.uni-koeln.de

Article published by EDP Sciences c EDP Sciences, SMAI 2010

given by x= (x, x_N) = (y, ε⁻¹y_N) andu(x) = v(x, εx_N), G_ε is transformed into a functional defined on a fixed domain:

F_ε(u) := ^Ωf x, u(x)

dx, ifu∈ Uε, with Ω :=ω×(0,1), +∞ ifu∈L^p(Ω;R^N)\ Uε,

wheref(x,·) =g(x,·) forx= (x, x_N)∈R^N−1×R,

Uε:= u∈L^p(Ω;R^N)div_εu= 0 and

div_εu:= divu+¹_ε∂_Nu^N :=_N−1

α=1∂_αu^α

+¹_ε∂_Nu^N

for u = (u, u^N) = (u¹, . . . , u^N). As this does not further complicate our approach, we allow f to explicitly depend onx_N as well below. We assume that

f : Ω×R^N →Ris a Carath´eodory function² (f:0) satisfying the following structural conditions:

(growth) |f(x, μ)| ≤C|μ|^p+C, (f:1)

(coercivity) f(x, μ) ≥ _C¹ |μ|^p−C, (f:2)

with constantsC >0 and 1< p <∞, for everyμ∈R^N and a.e.x∈Ω.

Using the notion of Γ-convergence introduced by De Giorgi [9,10], the effective energy in the limit ε→0⁺ is expressed by the Γ-limit ofF_εwith respect to weak convergence inL^p. For an introduction to the theory of Γ-convergence, the reader is referred to [4,7]. We use the notation

Γ(L^p_weak)−lim infF_ε(u) := inf lim inf

ε→0⁺ F_ε(u_ε)u_ε uweakly inL^p , Γ(L^p_weak)−lim supF_ε(u) := inf lim sup

ε→0⁺

F_ε(u_ε)u_ε uweakly inL^p .

Below, we omit the topology indicated in brackets as throughout this paper, this is always the weak topology in L^p. We say that Γ−limF_ε exists if Γ−lim infF_ε and Γ−lim supF_ε coincide, in which case this quantity is denoted by Γ−limF_ε. In particular, the use of the weak topology inL^p causes a process of relaxation in the limit, roughly speaking because energetically favorable microstructures of a characteristic size converging to zero asε→0 are allowed along the sequences generating the effective (macroscopic) limiting energy.

The corresponding problem of dimension reduction for functionals depending on gradients instead of divergence-free fields was investigated by Le Dret and Raoult [18–20] and stimulated a great deal of further research, including the study of different scalings, partially with energy densities that are realistic from the point of view of hyperelasticity (see [15] and the references therein), as well as extensions to non-flat limiting surfaces [22,23].

Recently, dimension reduction problems for Ginzburg-Landau-type functionals, involving a magnetic potential which is divergence-free as a choice of gauge, were studied in [1,6]. In both cases, the relevant parts of the energy density (apart from compact perturbations) are convex and thus no relaxation occurs during the limit process, avoiding the main difficulty of our problem. Relaxation and homogenization of functionals constrained to solenoidal matrix fields were treated in [2,29] (for related results and some physical background also see [16,28]), as well as in [5,12,14] for a more general constraint of the formAu= 0. In this context,Ais a linear differential operator assumed to satisfy Murat’s condition of constant rank [26], and apart from the examples in [21,24,31],

2I.e.measurable in its first and continuous in its second variable.

very little is known if this condition is violated. In our framework, div_ε satisfies the condition of constant rank for eachε, but the associated limiting operator div₀(div₀u:=∂_Nu^N foru: Ω→R^N) does not. From the point of view of the theory forA-free fields developed in [5,14], this means that important bounds for the projection operator onto div_ε-free fields and its complementary projection are not uniform inε and projecting tends to create large errors as ε→0⁺ (cf. Rem. 2.8). Hence, we can (and do) use the projection only along sequences that are asymptotically div_ε-free in a very strong sense (cf.Lem.2.9).

As we shall see, the divergence-free dimension reduction problem with nonconvex energy density exhibits some intriguing features that do not occur in the gradient case. In particular, it turns out that dimension reduction and direct relaxation in the limit setting do not yield the same result in general. While the former simply leads to convexification by our main theorem stated below, the latter may give rise to a nonlocal functional as illustrated by the example discussed in Proposition3.3.

Unless indicated otherwise, we assume throughout that

N ≥2, ω⊂R^N−1 is open and bounded, Ω :=ω×(0,1) and 1< p <∞.

Theorem 1.1. Suppose that (f:0)–(f:2)are satisfied. Then Γ−lim_ε→0+F_ε (with respect to weak convergence in L^p) exists, and it has the representation

Γ−limF_ε(u) =

F^∗∗(u) :=

Ωf^∗∗(x, u) dx if u∈ U0,

+∞ if u∈L^p(Ω;R^N)\ U0, where for each x,f^∗∗(x,·) denotes the convex envelope of f(x,·)and

U₀:= u∈L^p(Ω;R^N) ∂_Nu^N = 0in Ω .

It is fairly easy to see that both Γ−lim supF_ε(u) and Γ−lim infF_ε(u) are finite if and only if u ∈ U0

(Lems. 2.2 and 2.3), and the lower bound for Γ−lim infF_ε(u) is of course a simple consequence of the weak lower semicontinuity of convex functionals (Prop.2.6). However, the upper bound, Γ−lim supF_ε(u)≤F^∗∗(u) for u ∈ U0, is far more difficult than in the gradient case. The main issue here is that a priori, we do not know whether or not Γ−lim supF_εis a local integral functional. The usual trick for a proof of this property, based on “localizing” a sequence u_ε that weakly converges to zero by multiplying it with suitable smooth cut- off functions with the desired support. In the gradient case, the fields uwith finite Γ-limit are independent of x^N, whence it suffices to localize in the first N −1 variables. By contrast, in our setting, only the last componentu^N is independent ofx_N, and we have to somehow localize inx_N as well. But using a cut-off inX_N as described above does not work, because the distance of the modified sequence to the set of div_ε-free fields inL^pmay be of an order approaching 1/εwhich is an error too large to handle. Indeed, our proof of the upper bound in Section 4 (culminating in Prop. 4.9) does not use this kind of truncation in direction x_N, instead relying on a rather explicit construction of suitable sequences with small support in direction ofx_N which are asymptotically div_ε-free in the sense that their distance to Uε with respect to the norm of L^p goes to zero as ε→0⁺ (by Lem.2.9). A prototype of this construction for a simple example is presented in Proposition3.5.

2. Preliminary observations

We first observe that both Γ−lim supF_ε(u) and Γ−lim infF_ε(u) are finite if and only if u ∈ U0. The following simple density result turns out to be useful.

Lemma 2.1. With respect to the strong topology in L^p(Ω;R^N),U₀∩C^∞(Ω;R^N) is dense inU₀.

Proof. Letu∈ U0, and extendu= (u¹, . . . , u^N) to a function inL^p_loc(R^N;R^N) such thatu^j = 0 onR^N\Ω for j = 1, . . . , N−1,u^N = 0 onR^N \(ω×R) andu^N(x, x_N) is still constant inx_N for a.e.x ∈ω. Mollifying in the usual way yields a sequence (u_k)_k∈Nin C^∞(R^N;R^N)∩ U₀ withu_k→ustrongly inL^p(Ω;R^N).

Lemma 2.2. Letv_n be a bounded sequence inL^p(Ω)withv_n v_∞weakly inL^p(Ω), and suppose that∂_Nv_n→0 in the sense of distributions. Thenv_∞ is constant in x_N. In particular, if(u_ε)⊂ Uεandu_ε uweakly inL^p, thenu∈ U0.

Proof. For everyϕ∈C₀^∞(ω) and every η∈C₀^∞((0,1)), we have 0 = lim

n→∞

ω

(0,1)

v_n(x, x_N)ϕ(x) ˙η(x_N) dx_Ndx=

ω

(0,1)

v_∞(x, x_N)ϕ(x) ˙η(x_N) dx_Ndx. In particular, sinceϕwas arbitrary, we get

(0,1)

v_∞(x, x_N) ˙η(x_N) dx_N = 0 for a.e.x ∈ω and everyη ∈C₀^∞((0,1)),

which in turn implies thatv_∞(x, x_N) is constant inx_N.

Lemma 2.3. For everyu∈ U₀, there exists a sequence (u_ε)⊂ U_εsuch that u_ε−u→0 inL^p(Ω;R^N).

Remark 2.4. Using Lebesgue’s theorem, (f:0) and (f:1), we get that limF_ε(u_ε) =

Ωf(x, u) dx <∞, and thus Γ−lim supF_ε(u)<∞for everyu∈ U₀.

Proof of Lemma 2.3.

Step 1: Assume in addition thatu∈C¹(Ω;R^N).

Forj= 1, . . . , N−1 defineu^j_ε:=u^j, and let

u^N_ε(x, x_N) :=u^N(x, x_N)−ε _xN

0

divu(x, t) dt,

where divu= ∂₁u¹+. . .+∂_N−1u^N−1. We thus have that div_εu_ε = 0 and u_ε →u strongly in L^p, whence v_ε:=u_ε has the asserted properties.

Step 2: The general case.

By Lemma2.1, there exists a sequence (u_k)⊂C¹(Ω;R^N)∩U₀withu_k →ustrongly inL^p(Ω;R^N) ask→ ∞.

For eachkand eachε, we defineu_k,ε∈ U_εas in the first step, usingu_k instead ofu. Now choose (k(ε))_ε>0with k(ε)→ ∞ slow enough such thatεu_k(ε)

C¹(Ω;R^N) →0 as ε→0. As a consequence, u_ε :=u_k(ε),ε converges

toustrongly inL^p, and it satisfies div_εu_ε= 0 by construction.

To prove the lower bound Γ−lim infF_ε(u)≥F^∗∗(u) foru∈ U0, we first recall the well known characterization of weak lower semicontinuity of convex functionals:

Theorem 2.5(see [17] or [13],e.g.). Suppose thatf satisfies (f:0). Then the functionalJ :L^p(Ω,R^N)→[0,∞], J(u) :=

Ωf(x, u) dx, is lower semicontinuous with respect to weak convergence in L^p if and only if f(x,·)is convex for a.e.x∈Ω.

As an immediate consequence, we have:

Proposition 2.6 (lower bound). Suppose that the assumptions of Theorem1.1 hold. Then for every u∈ U₀, Γ−lim infF_ε(u)≥F^∗∗(u).

For the upper bound, we have to construct a suitable sequence (u_ε) ⊂ U_ε such that u_ε u in L^p and F_ε(u_ε) → F^∗∗(u), starting from a given u ∈ U₀. The main problem here is the constraint div_εu_ε = 0. In particular, we rely on a projection onto div_ε-free fields, which is based on the following special case of the projection used in [14].

Lemma 2.7. Let 1< p <∞ and letQ⊂R^N be an open cube. For every ε >0, there exists a linear operator Pε:L^p(Q;R^N)→L^p(Q;R^N) with the following properties:

(i) div_εP_εu= 0 onR^N for every u∈L^p(Q;R^N), whereP_εuis extendedQ-periodically.

(ii) P_εw=wfor everyw∈L^p(Q;R^N)such thatdiv_εw= 0onR^N, wherewis identified with itsQ-periodic extension toR^N.

(iii) Pεu_Lp(Q;R^N)≤C_εu_Lp(Q;R^N) for everyu∈L^p(Q;R^N), with a constantC_ε>0 independent ofu.

(iv) (I− P_ε)u_Lp(Q;R^N)≤C_εdiv_εu_W^−1,p_(Q) for every u∈L^p(Q;R^N), with a constant C_ε>0 indepen- dent of u.

Here, on a given domain W^−1,p denotes the dual space ofW₀^1,p with p =p/(p−1).

Proof. Forξ= (ξ, ξ^N)∈R^N \ {0}, (ξ,¹_εξ)∈R^1×N has full rank independent ofξ= 0, which means that for fixedε, div_εsatisfies Murat’s condition of constant rank [26]. Hence, Lemma 2.14 in [14] applies withA:= div_ε

andT=Pε.

Remark 2.8. Ifp= 2 (avoiding the use of general Fourier multiplier theorems), it is easy to see from the proof of Lemma 2.14 in [14] that (iii) and (iv) actually hold with constants independent ofε. However, we do not exploit this fact, and in any case, the factor ¹_ε hidden in the div_εon the right hand side of (iv) is still a major obstacle even if the constant in (iv) does not blow up asε→0⁺.

For technical reasons, it is important for us to be able to work with sequences which are not div_ε-free but can be projected to div_ε-free sequences with an error that is negligible in the limit ε → 0⁺. The following application of Lemma 2.7gives a useful sufficient criterion for sequences with this property.

Lemma 2.9. Let Ω⊂R^N be open and bounded, let 1< p <∞and letε_n→0⁺. Then there exists a sequence σ_n→0⁺ such that the following holds: For every sequence(u_n)⊂L^p(Ω;R^N)withu_n0 inL^p and

div_εnu_n

W^−1,p(Ω)+ u_n,_ε¹

nu^N_n

W^−1,p(Ω;R^N)≤σ_n, (2.1)

whereu_n:= (u¹_n, . . . , u^N_n⁻¹), there exists a sequence(v_n)⊂L^p(Ω;R^N)such thatdiv_εnv_n= 0inΩandu_n−v_n→ 0 inL^p(Ω;R^N).

Proof. For every k ∈ N choose a function ϕ_k ∈ C_c^∞(Ω; [0,1]) such that ϕ_k(x) = 1 for every x ∈ Ω with dist (x;∂Ω)≥ ¹_k. Moreover, choose a cubeQcontaining Ω and a sequence ˜σ_n→0⁺ such thatC_εnσ˜_n→0 with the constants of Lemma2.7(iv) (which also depend onQ). We define

σ_n:=ϕ_j(n)⁻¹

W^2,∞(Ω)σ˜_n and ˜u_n:=ϕ_j(n)u_n

with a sequence of integersj(n)→ ∞(fast enough) such thatu_n−u˜_n→0 inL^p(Ω;R^N). Since div_εn(ϕ_ku_n) =ϕ_kdiv_εnu_n+∇ϕ_k·

u_n,_ε¹

nu^N_n , we have that

divεn(ϕ_ku_n)_W−1,p ≤ ϕ_kW2,∞(Ω)

divεnu_nW−1,p+ u_n,_ε¹

nu^N_n

W^−1,p

.

Hence, (2.1) implies that

C_εndiv_εnu˜_nW−1,p(Ω)=C_εndiv_εnu˜_nW−1,p(Q)≤C_εnσ˜_n →0

as n → ∞. The sequence v_n := P_nu˜_n ∈ L^p(Q;R^N), restricted to Ω, now has the desired properties by

Lemma2.7.

Applying Lemma 2.9 is not easy because σ_n might converge to zero extremely fast. Nevertheless, it turns out to be possible for certain sequences constructed below, first in Proposition 3.5 for a simple example and then in Proposition4.3as the first step in proof of the upper bound.

3. An example and a related relaxation problem

When studying the dimension reduction problem for functionals depending on gradients (instead of divergence- free functions), one usually relies on a characterization of the associated relaxed functional in the limit setting, both as a lower semicontinuity result for the lower bound and as a first step in the construction of a sequence for the upper bound. In our framework, the associated relaxed functional in the limit setting corresponds to the functional ˜F₀introduced below. Although ˜F₀does not play a role in the proof our main result, we briefly discuss it here to point out the somewhat surprising fact that ˜F₀ does not always give the right limiting model for the divergence-free dimension reduction problem and may even be nonlocal, in sharp contrast to the gradient case.

In addition, the crucial idea for the proof of the upper bound in our main result is developed in Proposition3.5 for a simple model problem.

In the following, we consider the functional

F(u) :=˜ ^Ωf(x, u) dx ifu∈ U₀, +∞ ifu /∈ U₀.

By definition, the relaxed functional associated to ˜F is given by the lower semicontinuous hull of ˜F with respect to weak convergence inL^p. Foru∈L^p(Ω;R^N), it can be expressed by

F˜₀(u) := Γ−lim ˜F(u) = inf

lim inf ˜F(u_n)u_n uweakly inL^p

. (3.1)

Here, note that since ˜F does not depend on n, Γ−lim inf ˜F = Γ−lim sup ˜F. Moreover, U₀ is weakly closed in L^p, whence ˜F₀(u) is finite if and only ifu∈ U₀.

Proposition 3.1 (partial representation of ˜F₀). Let f : ω×R^N → [0,∞) (identified with f : Ω×R^N →R constant inx_N)satisfy (f:0)–(f:2). Then for everyu∈L^p(ω;R^N) (identified withu= (u¹, . . . , u^N)∈L^p(Ω;R^N) with u^j independent ofx_N for everyj= 1, . . . , N), we haveF˜₀(u) =F^∗∗(u), the convexified functional.

Proof. Since f ≥f^∗∗ and F^∗∗ is weakly lower semicontinuous in L^p(Ω;R^N), it is clear that ˜F₀(u)≥F^∗∗(u).

On the other hand, for anyu∈L^p(Ω;R^N) which is constant inx_N, we have F˜₀(u) = inf

lim inf ˜F(u_n)u_n uweakly inL^p(Ω;R^N),∂_Nu^N_n = 0∈R

≤inf

lim inf ˜F(u_n)u_n uweakly inL^p(Ω;R^N),∂_Nu_n= 0∈R^N

= inf

lim inf

ω

f(x,u˜_n) dx

u˜_n uweakly inL^p(ω;R^N)

=

ω

f^∗∗(x, u) dx=

Ω

f^∗∗(x, u) dx, where we used that

ωf^∗∗(x, v) dx is the weakly lower semicontinuous hull ofv→

ωf(x, v) dx in L^p. Example 3.2. Letp= 6, letN= 2, letf :R²→Rbe the three-well potential given by

f(μ) :=|μ−ζ₁|²|μ−ζ₂|²|μ−ζ₃|²,

withζ₁:= (0,−1), ζ2:= (1,0), ζ₃:= (0,1),

and consider the functionu₀∈ U₀given by u₀(x₁, x₂) :=

(0,0) ifx₂∈(0,¹₂], (1,0) ifx₂∈(¹₂,1).

Proposition 3.3 (possible nonlocal character of ˜F₀). In the situation of Example 3.2, we have that F˜₀(u₀)>0 = ^|ω×(0,

12 )|

|Ω| F˜₀((0,0)) +^|ω×_|Ω|^{( 12,1)|}F˜₀((1,0)).

In particular, F˜₀(u) cannot be written in the form

ΩV(u) dx with some function V :R²→R, and F˜₀(u₀)>

F^∗∗(u₀).

Remark 3.4. As recently discovered in [8], the lower semicontinuous hull with respect to strong convergence in L² of certain integral functionals of the formu→

Ωf(u,∇u) dxcan also be nonlocal, if there is a lack of coercivity with respect to the gradient variable.

Proof of Proposition 3.3. Sincef^∗∗= 0 on the closed triangle formed byζ₁,ζ₂andζ₃, ˜F₀((0,0)) = ˜F₀((1,0)) = 0 by Proposition3.1. To prove that ˜F₀(u₀)>0, we proceed indirectly. Suppose that ˜F₀(u₀) = 0. By a standard diagonalization argument, we may choose a sequence u_n ∈ U₀ with u_n u₀ weakly in L⁶(Ω,R²) such that F˜₀(u₀) = lim ˜F(u_n). By passing to a subsequence (not relabeled), we may assume thatu_n generates a Young measure ν_x, which for a.e.x∈Ω is a probability measure onR², and by the fundamental theorem for Young measures (see [3,13,25],e.g.), also exploiting thatf ≥0, we get that

0 = ˜F₀(u₀) = lim

Ω

f(u_n)dx≥

Ω

R²

f(ξ)dν_x(ξ)dx.

Sincef vanishes only on{ζ₁, ζ₂, ζ₃}, this implies that ν_x is supported in{ζ₁, ζ₂, ζ₃}for a.e.x,i.e., ν_x=₃

j=1σ_j(x)δ_ζj, (3.2)

where δ_z denotes the Dirac mass concentrated at the pointz in R². Moreover, since

R²ξdν_x(ξ) =u₀(x) and ν_xis a probability measure for a.e.x, the coefficientsσ_j(x)∈[0,1] are determined by the linear system

₃

j=1σ_j(x)ζ_j=u₀(x) and ₃

j=1σ_j(x) = 1.

One easily checks that the unique solution of this system is given by

σ₁(x) = ¹₂, σ₂(x) = 0, σ₃(x) = ¹₂ ifx₂≤ ¹₂ (i.e.,u₀(x) = (0,0)),

σ₁(x) = 0, σ₂(x) = 1, σ₃(x) = 0 ifx₂> ¹₂ (i.e.,u₀(x) = (1,0)), (3.3) wherex= (x₁, x₂). In addition, the marginal ofν_xon the second coordinate axis,

ν_x²(A) :=ν_x(R×A) forA⊂RBorel-measurable,

is the Young measure generated byu²_n. Sinceu_n∈ U₀,u²_n(x) is independent ofx₂ for eachnand consequently, ν²_xonly depends onx₁. This contradicts (3.2), because the latter implies thatν_x²=σ₁(x)δ₋₁+σ₂(x)δ₀+σ₃(x)δ₁, and the coefficients given by (3.3) are not constant inx₂ (only piecewise constant).

The dimension reduction problem is different because the constraint div_εu_ε = 0 is actually genuinely less restrictive than∂_Nu^N_ε = 0:

Proposition 3.5. In the situation of Example 3.2, for every given pair of sequences ε_n →0⁺ andσ_n →0⁺, there exists a bounded sequence (u_n)⊂L^∞(Ω;R^N) such thatu_n0 inL^p,

Ω

f(u_n+u₀) dx→

Ω

f^∗∗(u₀) dx= 0, (3.4)

and

divεnu_nW−1,p(Ω)+ u_n,_ε¹

nu^N_n

W^−1,p(Ω;R^N)≤σ_n (3.5)

for every n. In particular, u_n can be projected onto U_εn with an error that goes to zero strongly in L^p by Lemma2.9, and since div_εnu₀= divu₀= 0, this entails thatΓ−lim infF_εn(u₀)≤0<F˜₀(u₀).

Proof. For eachnfix a functionϕ_n ∈C_c^∞((0,1); [0,1]) such that ϕ_n = 1 on [ε_n,1−ε_n], and fork∈Nlet

w_k(t) =

w¹_k(t), w_k²(t) :=

⎧⎨

⎩

ζ₃= (0,1) if 0< t≤ _2k¹, ζ₁= (0,−1) if _2k¹ < t≤ ¹_k, extended periodically to a functionw_k:R→R² with period _k¹. Note that

w_k ¹₂ζ₃+¹₂ζ₁= (0,0) weakly inL^p(T;R²) (3.6) for any bounded open setT ⊂R. We definev_k,n ∈L^p(Ω;R²) by

v_k,n(x₁, x₂) :=

⎧⎨

⎩

ϕ_n(2x₂)w_k(_ε¹

nx₁) if 0< x₂< ¹₂, 0 if ¹₂ ≤x₂<1.

Observe that althoughv_k,n is not continuous, its jumps do not contribute to div_εnv_k,n (as a distribution), and thus the latter is actually a function with

div_εnv_k,n(x₁, x₂) =_ε²

nϕ˙_n(2x₂)w²_k

ε1nx₁ .

In particular, as k → ∞ for fixed n, div_εnv_k,n 0 weakly in L^p(Ω) as a consequence of (3.6), and thus div_εnv_k,n → 0 strongly in W^−1,p(Ω), by compact embedding. Analogously, we get that (v_k,n¹ ,_ε¹

nv_k,n² ) → 0 in W^−1,p(Ω;R^N) as k → ∞. Hence, we may choose k = k(n) with k(n) → ∞ as n → ∞ fast enough such that (3.5) holds foru_n:=v_k(n),n. Again using (3.6), it is not difficult to check thatu_n0 weakly inL^p, and sincef(ζ₁) =f(ζ₂) = 0, u_n is bounded inL^∞ and{u_n=ζ₁} ∩ {u_n=ζ₂}→0, (3.4) holds as well.

Remark 3.6. The choice of the dimensionN = 2 is not crucial for Example3.2, it is just the simplest possible case. In fact, a completely analogous argument can be used for suitable potentialsf withN + 1 wells inR^N for anyN ≥2.

4. The upper bound

In this section, we provide the remaining part of the proof of Theorem1.1, namely the upper bound Γ−lim supF_ε(u)≤F^∗∗(u) foru∈ U₀,

by constructing a suitable recovery sequence. In particular, we need some results from convex analysis:

Lemma 4.1 (Carath´eodory’s theorem, see [30], e.g.). Let g : R^N → [0,∞) be continuous. Then for every ξ ∈ R^N and every δ > 0, there exists an m ∈ {0, . . . , N} and ξ_j ∈ R^N, θ_j ∈ (0,1], j = 0, . . . , m, such that

jθ_j= 1,ξ=

jθ_jξ_j,

g^∗∗(ξ) ≤ _m

j=0θ_jg(ξ_j) ≤ g^∗∗(ξ) +δ,

and the vectorsξ_j−ξ₀,j= 1, . . . , m, are linearly independent. Here,g^∗∗ denotes the convex envelope ofg.

Lemma 4.2. Suppose that the assumptions of Lemma4.1hold. If, in addition, there exist constantsp >1and C >0 such that

C1 |μ|^p−C≤g(μ)≤C|μ|^p+C for everyμ∈R^N, (4.1) then the assertion of Lemma 4.1stays true even forδ= 0, and in this case,

|ξ_j| ≤K(|ξ|+ 1) for j= 0, . . . , m, (4.2) whereK is a constant that only depends onpandC.

Proof. With some background in convex analysis, this is not hard to prove, and we just sketch some details: It is well known that the convex envelope of gcan be represented as

g^∗∗(ξ) = sup A(ξ)|A:R^N →Raffine andA≤g

, ξ∈R^N.

If g is (lower semi-)continuous and has superlinear growth, the supremum is attained at a suitable affine functionA(see [13],e.g.), andAalways touchesgfrom below at suitable pointsξ_j as in Lemma4.1withδ= 0.

In addition, as a consequence of (4.1), we have that

C1 |μ|^p−C≤A_ξ(μ)≤C|μ|^p+C for everyμ∈co{ξ_j}

(the convex hull of the points ξ_j, j = 0, . . . , m). Clearly, the existence of an affine function satisfying the latter implies that co{ξ_j} is bounded for fixedξ, and it is not difficult to obtain more precise estimates that

yield (4.2).

The following result is the crucial step towards the upper bound for Γ−lim supF_εin the general case.

Proposition 4.3. Let N ≥2, let 1 ≤ p <∞, let I ⊂(0,1) be an open interval and let ε_n → 0⁺. Then for every sequence τ_n →0⁺ and every pair of points ζ₁, ζ₂ ∈R^N and numbersγ₁, γ₂ ∈(0,1) such that ζ₁^N =ζ₂^N, γ₁ζ₁+γ₂ζ₂= 0 andγ₁+γ₂= 1, there exists a sequence (v_n)⊂L^∞(R^N;R^N) such that

v_nL∞≤max{|ζ₁|,|ζ₂|}, (4.3)

div_εnv_nW−1,p(Ω)+ v_n,_ε¹

nv^N_n

W^−1,p(Ω;R^N)≤τ_n (4.4)

for every n∈N,

v_n0 in L^p_loc(R^N;R^N)asn→ ∞, supp(v_n)⊂R^N−1×

z∈Z

ε_nz+ε_nI^[εn]

, (4.5)

whereI^[ε] :={t∈I|dist (t;∂I)≥ε}, and

|{v_n=ζ_j} ∩U| −→

n→∞γ_j|U| |I| for every measurable setU ⊂R^N (4.6) andj= 1,2.

Remark 4.4. The assumption ζ₁^N = ζ₂^N is actually obsolete. The case of equality is only excluded above because it is much simpler and will be treated separately in Proposition4.6below.

Proof of Proposition 4.3. For each n ∈ N fix a function ϕ_n ∈ C_c^∞(R; [0,1]) such that ϕ_n = 1 on I^[2εn] and ϕ_n= 0 onR\I^[εn], and define

ψ_n ∈C^∞(R; [0,1]), ψ_n :=

z∈Z

ϕ_n(·+z).

Furthermore, fork∈Nlet

w_k(t) :=

ζ₁ if 0< t≤γ_1k¹, ζ₂ if−γ_2k¹ < t≤0, extended periodically to a functionw_k:R→R^N with period ¹_k. Note that

w_k γ₁ζ₁+γ₂ζ₂= 0 weakly in L^p_loc(R;R^N). (4.7) With a fixed unit vectorζ₁₂^⊥ ∈R^N perpendicular toζ₁−ζ₂, we definev_k,n∈L^∞(R^N;R^N) by

v_k,n(x) :=ψ_n

ε1nx_N w_k

ε1²_nx,_ε¹

nx_N

·ζ₁₂^⊥ .

Observe that although x→ w_k (_ε¹₂

nx,_ε¹

nx_N)·ζ₁₂^⊥

is not continuous, it is div_εn-free (as a distribution), and thus div_εnv_k,n is actually a function with

div_εnv_k,n(x) = _ε¹₂

n

ψ˙_n

ε1nx_N w^N_k

ε1²nx,_ε¹

nx_N

·ζ₁₂^⊥ .

In particular, as k → ∞ for fixed n, div_εnv_k,n 0 weakly in L^p(Ω) due to (4.7), and thus div_εnv_k,n → 0 strongly in W^−1,p(Ω), by compact embedding. Analogously, we get that (v_k,n,_ε¹

nv^N_k,n)→ 0 inW^−1,p(Ω;R^N) as k → ∞. Hence, we may choosek=k(n) with k(n)→ ∞ as n→ ∞fast enough such that (4.4) holds for

v_n :=v_k(n),n, and (4.3), (4.5) and (4.6) hold by construction.

Carath´eodory’s theorem requires convex combination of up toN+ 1 points, but Proposition4.3only admits two points. The following elementary lemma allows us to handle general convex combinations by breaking them into suitable pairs of two. Essentially, it states that if ξ=

jθ_jξ_j is a convex combination withξ∈H, where H is an affine hyperplane, then ξ can be rewritten as a convex combination of points ¯ξ_ij ∈H, such that each ξ¯_ij is a convex combination of two of the original points,i.e., ¯ξ_ij =β_ijξ_j+β_jiξ_i:

Lemma 4.5. Let m ≤ N, let ξ_j ∈ R^N, θ_j ∈ (0,1] for j = 0, . . . , m such that _m

j=0θ_j = 1 and the vectors ξ_j−ξ₀,j = 1, . . . , m, are linearly independent, and define

β_ij :=

⎧⎪

⎨

⎪⎩

ξ^Ni −ξ^N

ξ^N_i −ξ_j^N if (ξ_i^N−ξ^N)(ξ_j^N−ξ^N)<0, 1 if i=j andξ^N_j =ξ^N, 0 else.