A Γ-convergence result for the two-gradient theory of phase transitions

Sergio Conti Max-Planck-Institut für Mathematik in den Naturwissenschaften 04103 Leipzig, Germany

Irene Fonseca Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, PA 15213, U.S.A.

Giovanni Leoni Dipartimento di Scienze e Tecnologie Avanzate, Università del Piemonte Orientale, Alessandria, 15100 Italy

July 13, 2001

Abstract

The generalization to gradient vector fields of the classical double-well, singularly perturbed functionals,

I_{ε} (u; Ω) := ∫_{Ω} \frac{1}{ε} W (∇u) + ε|∇^{2}u|^{2} dx,

where W (ξ) = 0 if and only if ξ = A or ξ = B, and A − B is a rank-one matrix, is considered. Under suitable constitutive and growth hypotheses on W it is shown that I_{ε} Γ-converge to

I (u; Ω) = \{ \begin{matrix}K^{∗}H^{N−1}(S (∇u) ∩ Ω) & if u ∈ W^{1,1}(Ω; ℝ^{d}), ∇u ∈ BV (Ω; \{A, B\}), \\ +∞ & otherwise, \end{matrix}

where K^{∗} is the (constant) interfacial energy per unit area.

Keywords: phase transitions, double-well potential, singular perturbations, Γ-convergence, vertical matching, lateral matching 2000 Mathematics Subject Classification: 35G99, 35M99, 49J40, 49J45, 49K20, 74B20, 74G65, 74N99

Contents

1 Introduction 2

2 Preliminaries 7

3 Compactness 10

4 Γ−liminf: the lower bound 13 5 Γ−limsup: the upper bound. Geodesic hypotheses. 17

5.1 Characterization of K^{∗} . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 5.2 Special domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 5.3 x′-connected domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 5.4 General domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

6 Γ−limsup: the upper bound. The symmetry hypotheses. 38 6.1 Characterization of K^{∗} . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 6.2 x′-connected domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 6.3 General domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

7 Condition (H_{4})′ 51

8 Example of a non one-dimensional interface 54

9 Appendix 57

1 Introduction

The theories of phase transitions and minimal surfaces have led to extensive study of singularly-perturbed, nonconvex functionals of the form

J_{ε} (v; Ω) := ∫_{Ω} \frac{1}{ε}W (v) + ε|∇v|^{2} dx,  (1.1)

where W is a nonnegative potential with multiple minima. This functional was first studied by Modica and Mortola [35], and subsequently it was applied by Modica [34] to the van der Waals–Cahn-Hilliard theory of fluid-fluid phase transitions to solve an “optimal design” problem proposed by Gurtin [28]:

Minimize ∫_{Ω} W (u) dx, subject to a density constraint \frac{1}{|Ω|} ∫_{Ω} u dx = θa + (1 − θ)b,

for some θ ∈ (0, 1), and where W is a nonnegative bulk energy density with {W = 0} = {a, b}, a, b ∈ ℝ, a < b. The striking nonuniqueness of solutions (minimizers) is due to the fact that nucleation of phases may occur without an increase in energy. In order to select physically preferred solutions, the van der Waals– Cahn-Hilliard theory adds a gradient term which upon rescaling leads to (1.1). Using De Giorgi’s notion of Γ-convergence ([22]; see also [21, 16, 1]), it was shown in [35, 34], that

Γ − lim_{ε→0^{+}} J_{ε} (u_{0}; Ω) = \{ \begin{matrix}K_{0}Per_{Ω}(E) & if u = χ_{E}a + (1 − χ_{E})b, |E| = θ |Ω| , u ∈ BV (Ω; \{a, b\}) , \\ +∞ & otherwise, \end{matrix} (1.2)

where K_{0} := ∫_{a}^{b} \sqrt{W (s)}ds. We conclude, therefore, that in the limit as ε → 0 partitions with minimal interfacial area and given volume fraction θ are selected.

Generalizations of (1.1)-(1.2) were obtained by Bouchitté [15] and by Owen and Sternberg [38] for the undecoupled problem, in which the integrand in J_{ε} has the form ε^{−1}f(x, v(x), ε∇v(x)). We refer also to the work of Kohn and Sternberg [32] where the study of local minimizers for (1.1) was undertaken.

The vector-valued setting, where u : Ω → ℝ^{d}, Ω ⊂ ℝ^{N}, d, N > 1, was considered in [26, 10], where K_{0} is replaced by

K_{1} := inf {∫_{−L}^{L} W (g (s)) + |g′ (s)|^{2} ds : L > 0, g piecewise C^{1}, g(−L) = a, g(L) = b} .  (1.3)

The case where W has more than two wells was addressed by Baldo [7] (see also Sternberg [40]), and later generalized by Ambrosio [2].

The corresponding problem for gradient vector fields, where in place of J_{ε} we introduce

I_{ε} (u; Ω) := \{ \begin{matrix}_{∫Ω} \frac{1}{ε}W (∇u) + ε|∇^{2}u|^{2} dx & if u ∈ W^{2,2} (Ω; ℝ^{d}) , \\ +∞ & otherwise, \end{matrix}  (1.4)

arises naturally in the study of elastic solid-to-solid phase transitions [9, 19, 31, 36], and it has defied a considerable mathematical effort during the past decade. Here u : Ω → ℝ^{d} stands for the deformation, and taking into account frame-indifference we assume that {W = 0} = SO(N)A ∪ SO(N)B, where SO(N) is the set of rotations in ℝ^{N}. In order to guarantee the existence of “classical” (as opposed to measure-valued) non affine solutions for the limiting problem, and in view of Hadamard’s compatibility condition for layered deformations (see also Ball and James [9]), the two wells must be rank-one connected. Without loss of generality, we then assume that A − B = a ⊗ ν for some a ∈ ℝ^{N} and ν ∈ S^{N−1} := ∂B(0, 1) ⊂ ℝ^{N}. We are now able to construct gradients taking values only on {A, B} and layered perpendicularly to ν.

As a first simplification of the problem, we remove the frame-indifference constraint and we assume simply that

{W = 0} = {A, B} , A − B = a ⊗ ν.

Here interfaces of minimizers must be planar with normal ν (see [9]), therefore at first glance the analysis may seem to be greatly simplified as compared with the initial problem (1.1) which requires handling minimal surfaces. However, it turns out that the PDE constraint curl = 0 imposed on the admissible fields presents numerous difficulties to the characterization of the Γ-limsup. Precisely, if, say, ∇u has a layered structure with two interfaces then it is possible to construct a “realizing” (effective) sequence nearby each interface, but the task of gluing together the two sequences on a suitable low-energy intermediate layer is very delicate. This is where specific constitutive hypotheses placed on W will come into play (see Sections 5 and 6 below).

An intermediate case between (1.1) and (1.4), where the nonconvex potential depends on u and the singular perturbation on ∇^{2}u, has been recently studied by Fonseca and Mantegazza [25] (for other generalizations see [24]). Also, in the two-dimensional case and when W vanishes on the unit circle (1.4) reduces to the so-called Eikonal functional which arises in the study of liquid crystals [5] as well as in blistering of delaminated thin films [37]. Recently, the Eikonal problem has received considerable mathematical attention, but in spite of substantial partial progress (see [3, 6, 30, 23]) its Γ-limit remains to be identified.

In this work, and under the standing hypothesis

(H_{1}) W is continuous, W (ξ) = 0 if and only if ξ ∈ {A, B} , where A − B = a ⊗ ν for some a ∈ ℝ^{d}\ {0} and ν ∈ S^{N−1};

and additional assumptions on W, we show that as ε → 0^{+} the functionals I_{ε} Γ-converge to

I (u; Ω) = \{ \begin{matrix}K^{∗}H^{N−1}(S (∇u) ∩ Ω) & if u ∈ W^{1,1}(Ω; ℝ^{d}), ∇u ∈ BV (Ω; \{A, B\}), \\ +∞ & otherwise, \end{matrix}

where S (∇u) is the singular set of ∇u, i.e. the collection of interfaces,

K^{∗} := Γ − lim inf_{ε→0^{+}} I_{ε} (u_{0}; Q_{ν})

= inf {lim inf_{n→∞} I_{ε_{n}}(u_{n}; Q_{ν}) : ε_{n} → 0^{+}, u_{n} ∈ W^{2,2} (Q_{ν}; ℝ^{d}) , u_{n} → u_{0} in L^{1} (Q; ℝ^{d})} ,

where Q_{ν} is a unit cube in ℝ^{N} centered at the origin and with two of its faces orthogonal to ν, and

∇u_{0} := \{ \begin{matrix}A & if x · ν > 0, \\ B & if x · ν < 0. \end{matrix}

The main results of this paper are:

Theorem 1.1 (Compactness) Assume that the double well potential W satisfies conditions (H_{1}) and (H_{2}) there exists C > 0 such that

W (ξ) ≥ C |ξ| − \frac{1}{C}

for all ξ ∈ ℝ^{d×N} .

Let ε_{n} → 0^{+}. If {u_{n}} ⊂ W^{2,2} (Ω; ℝ^{d}) is such that

sup_{n} I_{ε_{n}} (u_{n}; Ω) < ∞,

then there exist a subsequence {u_{n_{k}}} and u ∈ W^{1,1} (Ω; ℝ^{d}) , with ∇u ∈ BV (Ω; {A, B}) , such that

u_{n_{k}} − \frac{1}{|Ω|} ∫_{Ω} u_{n_{k}} dx → u in W^{1,1} (Ω; ℝ^{d}) .

Theorem 1.2 (Γ-liminf) Assume that W satisfies condition (H_{1}) . Let u ∈ W^{1,1} (Ω; ℝ^{d}) , with ∇u ∈ BV (Ω; {A, B}) . Then

Γ − lim inf_{ε→0^{+}} I_{ε} (u; Ω) ≥ K^{∗} Per_{Ω}(E),

where ∇u (x) = (1 − χ_{E} (x)) A + χ_{E} (x) B for L^{N} a.e. x ∈ Ω.

In order to characterize the Γ-limsup, we will consider two sets of additional constitutive hypotheses on W. Without loss of generality we may assume that

A = −B = a ⊗ e_{N}.

First consider

(H_{2})′ W (ξ) → ∞ as |ξ| → ∞;

(H_{3}) W (ξ) ≥ W (0, ξ_{N}) where ξ = (ξ′, ξ_{N}) ∈ ℝ^{d×(N−1)} × ℝ^{d}.

Note that (H_{3}) is satisfied by the prototype bulk energy density

W (ξ) := min {|ξ − A|^{2} , |ξ − B|^{2}} .

Theorem 1.3 (Γ-lim) Let Ω ⊂ ℝ^{N} be an open, bounded, simply connected domain with Lipschitz boundary. Assume that W satisfies the conditions (H_{1}) , (H_{2})′ and (H_{3}) . Suppose, in addition, that W is differentiable at A and B. Let u ∈ W^{1,1} (Ω; ℝ^{d}) , with ∇u ∈ BV (Ω; {A, B}) . Then

Γ − lim_{ε→0^{+}} I_{ε} (u; Ω) = K^{∗} Per_{Ω}(E),

where ∇u = (1 − χ_{E} (x)) A + χ_{E} (x) B for L^{N} a.e. x ∈ Ω.

The hypothesis (H_{3}) entails a one dimensional character to the asymptotic problem. Indeed in this case the characterization of the constant K^{∗} can be greatly simplified. It can be shown (see Proposition 5.3) that K^{∗} reduces to the analog of the constant K_{1} introduced in (1.3), precisely, K^{∗} = K where

K := inf {∫_{−L}^{L} W (0, g (s)) + |g′ (s)|^{2} ds : L > 0, g piecewise C^{1}, g(−L) = −a, g(L) = a} .

Theorem 1.3 is related to work of Kohn and Müller [31] who studied the minimization of the functional

∫(0,L)×(0,1) (\frac{∂u}{∂x_{1}} )^{2} + ε |\frac{∂^{2}u}{∂x_{2}^{2}} | dx_{1}dx_{2}

Figure 1: Example of a domain where (1.5) and (1.6) hold.

Figure 2: Example of a domain where (1.5) holds but (1.6) fails.

subject to the constraint |\frac{∂u}{∂x_{2}} | = 1 and boundary conditions.

The main effort of the present paper is devoted to the construction of a realizing or effective sequence for the Γ−limsup. It turns out that this construction is strongly hinged to the geometry of the domain. We first assume that (see Figure 1)

for each t ∈ ℝ the horizontal section Ω_{t} := {(x′, x_{N}) ∈ Ω : x_{N} = t} is connected in ℝ^{N},  (1.5)

and that

t ↦ H^{N−1} (Ω_{t}) is continuous in (α, β) ,  (1.6)

where

α := inf {x_{N} : x ∈ Ω} , β := sup {x_{N} : x ∈ Ω} .

It is easy to see that convex domains or cylinders of the form ω × (a, b) , where ω ⊂ ℝ^{N−1}, satisfy conditions (1.5) and (1.6). This case is particularly simple since realizing sequences are one-dimensional, and the assumption that W is differentiable at A and B is not used (see Theorem 5.5).

If we now remove the assumption (1.6) (see Figure 2), then one-dimensional sequences cease to be optimal as they would yield K^{∗} Per\overline{Ω}(E) rather than K^{∗} Per_{Ω}(E) as desired. In this case, realizing sequences are one-dimensional except near horizontally flat parts of ∂Ω where H^{N−1} (∂Ω ∩ {(x′, x_{N}) ∈ ℝ^{N} : x_{N} = t}) > 0 (see Theorem 5.6).

The situation becomes considerably more complicated when one drops condition (1.5) (see Figure 3) since the gradient may change abruptly when two connected components of Ω_{t} meet. To solve this problem we glue realizing sequences near the boundary to appropriate “mollifications” of u.

Figure 3: Example of a domain where (1.5) does not hold.

Figure 4: Example of a domain where translations of rotations of Ω cause discontinuous changes in the perimeter of the interface.

We remark that the above mentioned difficulties cannot be resolved by performing rotations and translations of Ω nearby the identity because the perimeter of the interface may change discontinuously under these transformations (see Figure 4).

As we already mentioned, the hypothesis (H_{3}) is quite strong as it entails a one dimensional character to the asymptotic problem. In the second part of the paper we replace it with the isotropy assumption:

(H_{5}) W is even in each variable ξ_{i}, i = 1, ⋯, N −1, that is W (ξ_{1}, ⋯, −ξ_{i}, ⋯, ξ_{N}) = W (ξ_{1}, ⋯, ξ_{i}, ⋯, ξ_{N}) for each i = 1, ⋯, N − 1,

where

ξ = (ξ_{1}, ⋯, ξ_{N}) ∈ \underbrace{ℝ^{d} × ⋯ × ℝ^{d}}_{N times}, ξ′ = (ξ_{1}, ⋯, ξ_{N} − 1) ∈ \underbrace{ℝ^{d} × ⋯ × ℝ^{d}}_{N−1 times},

so that ξ = (ξ′, ξ_{N}) ∈ ℝ^{d×(N−1)} × ℝ^{d}.

In this case we can prove the following result

Theorem 1.4 Let Ω ⊂ ℝ^{N} be an open, bounded, simply connected domain with Lipschitz boundary. Assume that W satisfies the conditions (H_{1}) , (H_{5}) , and that there exist an exponent p ≥ 2, constants c, C, ρ > 0 and a convex function g : [0, ∞) → [0, ∞), with g (s) = 0 if and only if s = 0, such that g is derivable in s = 0, g (2t) ≤ cg (t) for all 0 ≤ t ≤ ρ,

g (|ξ − A|) ≤ W (ξ) ≤ cg (|ξ − A|) if |ξ − A| ≤ ρ,
g (|ξ − B|) ≤ W (ξ) ≤ cg (|ξ − B|) if |ξ − B| ≤ ρ,

and

\frac{1}{C} |ξ|^{p} − C ≤ W (ξ) ≤ C (|ξ|^{p} + 1)

for all ξ ∈ ℝ^{d×N}. Let u ∈ W^{1,1} (Ω; ℝ^{d}) , with ∇u ∈ BV (Ω; {A, B}) . Then

Γ − lim_{ε→0^{+}} I_{ε} (u; Ω) = K^{∗} Per_{Ω}(E),

where ∇u(x) = (1 − χ_{E} (x)) A + χ_{E} (x) B for L^{N} a.e. x ∈ Ω. Moreover K^{∗} = K_{per}, where

K_{per} := inf \{∫_{Q} L W (∇v) + \frac{1}{L} |∇^{2}v|^{2} dx : L > 0, v ∈ W^{2,∞} (Q; ℝ^{d}) ,
∇v = ±a ⊗ e_{N} nearby x_{N} = ±\frac{1}{2}, v periodic of period one in x′\} .

It would be interesting to know if Theorem 1.4 continues to hold without assuming the isotropy assumption (H_{5}) . We have not been able to prove this.

Most of the literature on singularly perturbed double-well potentials deals with asymptotic problems which are one-dimensional, i.e. with asymptotically optimal interface profiles which depend only on one coordinate (the distance to the interface). Such profiles have been shown to be optimal in the first-order gradient theory of phase transitions modeled by (1.1) (see also [25]), and the same has been conjectured in various other cases such as the Eikonal problem mentioned above. However, Jin and Kohn [30] have shown that a simple perturbation of the Eikonal functional has non one-dimensional minimizers. A one-dimensional ansatz is also often used in the physics literature on ferroelastic domain walls, even if its validity is still under debate (see e.g. [11, 39, 18, 20] and references therein).

In the problem of interest here, we defined K as the energy of the optimal one-dimensional interface profile, and K_{per} as the optimal energy of interface profiles which are periodic along the interface. As discussed above, under hypothesis (H_{3}) we show that K_{per} = K, hence that one-dimensional interface profiles are energetically preferred. In the final section of this paper, building upon the example by Jin and Kohn [30], we show that without hypothesis (H_{3}), in general, we may have

K_{per} < K

thereby proving that optimal interface profiles are, at least in some cases, not one-dimensional. This happens because generating finite in–plane gradients (i.e. having a dependence on the coordinates parallel to the interface) reduces the energy in the regions far away from the two potential wells. The zero-curl constraint leads then to an oscillatory pattern. In elasticity, this multidimensional behavior has been predicted in [18]. Similar mechanisms are at play in the theory of micromagnetism, where indeed various non-one-dimensional wall structures are known, such as cross-tie domain walls and charged zigzag walls in ferromagnetic thin films (see e.g. [29] and references therein). It would be interesting to know if K_{per} is smaller or equal to K for realistic ferroelastic potentials obtained from the Landau theory of phase transitions.

2 Preliminaries

We start with some notation. Here Ω ⊂ ℝ^{N} is an open, bounded Lipschitz domain, L^{N} and H^{N−1} are, respectively, the N dimensional Lebesgue measure and the N − 1 dimensional Hausdorff measure in ℝ^{N}. We shall label the first N −1 coordinates of a point x ∈ ℝ^{N} by x′, and the N-th one by x_{N}, so that x = (x′, x_{N}). We define A (Ω) as the class of all open subsets of Ω and S^{N−1} := \{x ∈ ℝ^{N} : |x| = 1\} . We let Q := (−\frac{1}{2}, \frac{1}{2})^{N} be the unit cube centered at the origin, and we set Q(x_{0}, ε) := x_{0} + εQ. In the sequel C and c will stand for generic real positive constants which may vary from line to line and expression to expression within the same formula.

For ε > 0 consider the functional

I_{ε} : L^{1} (Ω; ℝ^{d}) × A (Ω) → [0, +∞]

defined by

I_{ε} (u; U) := \{ \begin{matrix}_{∫U} \frac{1}{ε}W (∇u) + ε|∇^{2}u|^{2} dx & if u ∈ W^{2,2} (Ω; ℝ^{d}) , \\ +∞ & otherwise, \end{matrix}

where the double well potential W : ℝ^{d×N} → [0, ∞) satisfies the following standing hypotheses:

(H_{1}) W is continuous, W (ξ) = 0 if and only if ξ ∈ {A, B} , where A − B = a ⊗ ν, for some a ∈ ℝ^{d}\ {0} and ν ∈ S^{N−1};

(H_{2}) there exists C > 0 such that

W (ξ) ≥ C |ξ| − \frac{1}{C}

for all ξ ∈ ℝ^{d×N} .

For simplicity of notation, we shall assume that

A = −B = a ⊗ e_{N}.  (2.1)

The general case may be reduced to this situation by considering in place of W a new bulk energy density

Ŵ (ξ) := W ((ξ + ξ_{0})R^{T})

for suitable ξ_{0} ∈ ℝ^{d×N} and a rotation R with R e_{N} = ν. We recall that Ball and James [9] have shown that there exists a non-affine Lipschitz function u such that its gradient takes only the matrix values A and B if and only if A and B are rank-one connected, i.e. rank (A−B) = 1, in which case the jump sets (or interfaces) of ∇u are planar and orthogonal to the direction e_{N}. Under (2.1) the prototype blown-up macroscopic field with one interface in the unit cell Q = (−1/2, 1/2)^{N} is

u (x) := |x_{N}| a.  (2.2)

We review briefly some facts about functions of bounded variation which will be useful in the sequel. A function u ∈ L^{1}(Ω; ℝ^{d}) is said to be of bounded variation if for all i = 1, ⋯ d, and j = 1, ⋯ N, there exists a Radon measure μ_{ij} such that

∫_{Ω} u_{i}(x) \frac{∂v}{∂x_{j}} (x) dx = − ∫_{Ω} v(x) dμ_{ij}

for every v ∈ C_{0}^{1}(Ω; ℝ). The distributional derivative Du is the matrix–valued measure with components μ_{ij}. Given u ∈ BV (Ω; ℝ^{d}) the approximate upper and lower limit of each component u_{i}, i = 1, ⋯ d, are given by

u_{i}^{+}(x) := inf \{t ∈ ℝ : lim_{ε→0^{+}} \frac{1}{ε^{N}} L^{N} (\{y ∈ Ω ∩ Q(x, ε) : u_{i}(y) > t\}) = 0\}

and

u_{i}^{−}(x) := sup \{t ∈ ℝ : lim_{ε→0^{+}} \frac{1}{ε^{N}} L^{N} (\{y ∈ Ω ∩ Q(x, ε) : u_{i}(y) < t\}) = 0\} ,

while the jump set of u, or singular set, is defined by

S(u) := ⋃_{i=1}^{d}{x ∈ Ω : u_{i}^{−}(x) < u_{i}^{+}(x)}.

It is well known that S(u) is N − 1 rectifiable, i.e.

S(u) = ⋃_{n=1}^{∞} K_{n} ∪ E,

where H^{N−1}(E) = 0 and K_{n} is a compact subset of a C^{1} hypersurface. If x ∈ Ω\S(u) then u(x) is taken to be the common value of (u_{1}^{+}(x), ⋯, u_{d}^{+}(x)) and (u_{1}^{−}(x), ⋯, u_{d}^{−}(x)). It can be shown that u(x) ∈ ℝ^{d} for H^{N−1} a.e. x ∈ Ω\S(u). Furthermore, for H^{N−1} a.e. x ∈ S(u) there exist a unit vector ν_{u}(x) ∈ S^{N−1}, normal to S(u) at x, and two vectors u^{−}(x), u^{+}(x) ∈ ℝ^{d} (the traces of u on S(u) at the point x) such that

lim_{ε→0} \frac{1}{ε^{N}} ∫_{\{y∈Q(x_{0},ε): (y−x)·ν_{u}(x)>0\}} |u(y) − u^{+}(x)|^{N/(N−1)}dy = 0

and

lim_{ε→0} \frac{1}{ε^{N}} ∫_{\{y∈Q(x_{0},ε): (y−x)·ν_{u}(x)<0\}} |u(y) − u^{−}(x)|^{N/(N−1)}dy = 0.

Note that, in general, (u_{i})^{+} ≠ (u^{+})_{i} and (u_{i})^{−} ≠ (u^{−})_{i}. We denote the jump of u across S(u) by [u] := u^{+} − u^{−}. The distributional derivative Du may be decomposed as

Du = ∇u L^{N} + (u^{+} − u^{−}) ⊗ ν_{u} H^{N−1}⌊ S(u) + C(u),

where ∇u is the density of the absolutely continuous part of Du with respect to the N–dimensional Lebesgue measure L^{N} and C(u) is the Cantor part of Du. These three measures are mutually singular.

A set E ⊂ Ω is of finite perimeter if χ_{E} ∈ BV (Ω; ℝ) and we denote by Per_{Ω}(E) the perimeter of E in Ω. Let ε_{n} → 0^{+}. We say that a functional

I : L^{1} (Ω; ℝ^{d}) × A(Ω) → [0, +∞]

is the Γ−lim inf (resp. Γ−lim sup) of the sequence of functionals {I_{ε_{n}}} with respect to the strong convergence in L^{1} (Ω; ℝ^{d}) if for every u ∈ L^{1} (Ω; ℝ^{d})

I(u; Ω) = inf {lim inf_{n→∞} (resp. lim sup_{n→∞}) I_{ε_{n}}(u_{n}; Ω) : u_{n} ∈ L^{1} (Ω; ℝ^{d}) , u_{n} → u in L^{1} (Ω; ℝ^{d})} ,

and we write

I = Γ − lim inf_{n→∞} I_{ε_{n}} (resp. I = Γ − lim sup_{n→∞} I_{ε_{n}}) .

Since I_{ε} (v; U) = ∞ if v ∉ W^{2,2} (Ω; ℝ^{d}) , it is clear that we may write

I(u; Ω) = inf {lim inf_{n→∞} (resp. lim sup_{n→∞}) I_{ε_{n}}(u_{n}; Ω) : u_{n} ∈ W^{2,2} (Ω; ℝ^{d}) , u_{n} → u in L^{1} (Ω; ℝ^{d})} .

We say that the sequence {I_{ε_{n}}} Γ-converges to I if the Γ − lim inf and Γ − lim sup coincide, and we write

I = Γ − lim_{n→∞} I_{ε_{n}}.

The functional I is said to be the Γ−lim inf (resp. Γ−lim sup) of the family of functionals {I_{ε}} with respect to the strong convergence in L^{1} (Ω; ℝ^{d}) if for every sequence ε_{n} → 0^{+} we have that

I = Γ − lim inf_{n→∞} I_{ε_{n}} (resp. I = Γ − lim sup_{n→∞} I_{ε_{n}}) ,

and we write

I = Γ − lim inf_{ε→0} I_{ε} (resp. I = Γ − lim sup_{ε→0} I_{ε}) .

Finally, we say that I is the Γ-limit of the family of functionals {I_{ε}}, and we write

I = Γ − lim_{ε→0} I_{ε},

if Γ − lim inf and Γ − lim sup coincide.

3 Compactness

The following compactness result is a direct consequence of the one obtained in [26] for the functional (1.1), and the structure of the limit has been characterized in [9]. For completeness we give here a short self-contained proof.

Theorem 3.1 (Compactness) Assume that the double well potential W satisfies conditions (H_{1}) and (H_{2}) . Let ε_{n} → 0^{+}. If {u_{n}} ⊂ W^{2,2} (Ω; ℝ^{d}) is such that

sup_{n} I_{ε_{n}} (u_{n}; Ω) < ∞,

then there exist a subsequence {u_{n_{k}}} and u ∈ W^{1,1} (Ω; ℝ^{d}) , with ∇u ∈ BV (Ω; {A, B}) , such that

u_{n_{k}} − \frac{1}{|Ω|} ∫_{Ω} u_{n_{k}} dx → u in W^{1,1} (Ω; ℝ^{d}) .

Proof. We claim that the sequence \{u_{n} − \frac{1}{|Ω|} ∫_{Ω} u_{n} dx\} is weakly compact in W^{1,1} (Ω; ℝ^{d}) . Indeed, by (H_{2}) , and with c > 0 such that

sup_{n} I_{ε_{n}} (u_{n}; Ω) =: c < ∞,  (3.1)

we have

ε_{n}c ≥ ∫_{Ω} W (∇u_{n}) dx ≥ C ∫_{Ω} |∇u_{n}| dx − \frac{1}{C} |Ω| ,

and so {∇u_{n}} is uniformly bounded in L^{1} (Ω; ℝ^{d×N}) . By Poincaré-Friedrichs’ inequality we conclude that

the sequence \{u_{n} − \frac{1}{|Ω|} ∫_{Ω} u_{n} dx\} is uniformly bounded in W^{1,1} (Ω; ℝ^{d}) . Thus, to prove the claim it remains to show that the sequence {∇u_{n}} is equi-integrable. Fix ϵ > 0. By (H_{2}) we have

W (ξ) ≥ \frac{1}{2}C |ξ|

for all ξ ∈ ℝ^{d×N} with |ξ| ≥ L := \frac{2}{C^{2}} , and by (3.1) we have

0 ≤ \frac{1}{2}C ∫_{\{|∇u_{n}|>L\}} |∇u_{n}| dx ≤ ∫_{Ω} W (∇u_{n}) dx ≤ ε_{n}c → 0  (3.2)

as n → ∞. Hence there exists n_{ϵ} such that

∫_{{|∇u_{n}|>L}} |∇u_{n}| dx ≤ ϵ for all n > n_{ϵ}.

Since ∇u_{n} ∈ L^{1} (Ω; ℝ^{d×N}) for all n = 1, ⋯, n_{ϵ}, by taking L larger, if necessary, we may assume that the previous inequality holds for all n. This completes the proof of the claim.

Thus we may extract a subsequence (not relabelled) such that

u_{n} − \frac{1}{|Ω|} ∫_{Ω} u_{n} dx ⇀ u in W^{1,1} (Ω; ℝ^{d})  (3.3)

and {∇u_{n}} generates a gradient Young measure {ν_{x}}_{x∈Ω} . We claim that

ν_{x} = (1 − θ (x)) δ_{ξ=A} + θ (x) δ_{ξ=B} L^{N}a.e. in Ω,

where θ (x) ∈ [0, 1] . Indeed, since W is nonnegative and continuous, the Fundamental Theorem on Young Measures (see e.g. [8, 12, 41]) yields

0 = lim_{n→∞} ∫_{Ω} W (∇u_{n}) dx ≥ ∫_{Ω} ∫_{ℝ^{d×N}} W (ξ) dν_{x} (ξ) dx;

hence, for L^{N} a.e x ∈ Ω

∫_{ℝ^{d×N}} W (ξ) dν_{x} (ξ) = 0,

and thus by (H_{1}) the claim follows. In turn

∇u(x) = ∫_{ℝ^{d×N}} ξ dν_{x} (ξ) = (1 − θ (x)) A + θ (x) B L^{N}a.e. in Ω.  (3.4)

Let M > 0 and set

φ (ξ) := inf \{∫_{0}^{1} min \{\sqrt{W (h (s))}, M\} |h′ (s)| ds : h : [0, 1] → ℝ^{d×N} piecewise C^{1}, h (0) = ξ, h (1) = A\} .

Then φ is Lipschitz, φ (ξ) = 0 if and only if ξ = A, and

{φ (∇u_{n})} is uniformly bounded in W^{1,1} (Ω; ℝ) .  (3.5)

Indeed,

∫_{Ω} |∇ (φ ∘ ∇u_{n})| dx≤ ∫_{Ω} \sqrt{W (∇u_{n})} |∇^{2}u_{n}| dx ≤ \frac{1}{2}I_{ε_{n}} (u_{n}; Ω) ≤ \frac{1}{2}c

and

∫_{Ω} |φ ∘ ∇u_{n}| dx ≤ ∫_{Ω} M |∇u_{n}| dx + φ(0) |Ω| ,

where we have used the fact that φ(ξ) ≤ M |ξ − 0| + φ(0). Hence (3.5) holds, and up to a subsequence (not relabelled)

φ (∇u_{n}) → H in L^{1} (Ω; ℝ) ,  (3.6)

where H ∈ BV (Ω; ℝ) . On the other hand, the Young measure generated by {φ (∇u_{n})} is

μ_{x} = (1 − θ (x)) δ_{t=φ(A)} + θ (x) δ_{t=φ(B)} L^{N} a.e. in Ω,

and the strong convergence in (3.6) now yields θ (x) ∈ {0, 1} L^{N} a.e. in Ω, precisely

θ (x) = χ_{E} (x)

for some measurable set E ⊂ Ω. By (3.4)

∇u (x) = (1 − χ_{E} (x)) A + χ_{E} (x) B

and

H = (1 − χ_{E} (x)) φ (A) + χ_{E} (x) φ ( B)
= χ_{E} (x) φ ( B) ∈ BV (Ω; ℝ) ,

therefore the set E has finite perimeter and ∇u ∈ BV (Ω; {A, B}) . Moreover, since ν_{x} = δ_{ξ=∇u(x)} and by (3.2) we have that ∇u_{n} → ∇u in L^{1} (Ω; ℝ^{d×N}) .

Remark 3.2 (i) We remark that the conclusion of Theorem 3.1 still holds if we do not impose the condition (H_{2}) but, instead, we assume apriori that the sequence {u_{n}} converges weakly in W^{1,1}(Ω; ℝ^{d}). Indeed, the argument follows exactly that of the latter proof once (3.3) has been established.

(ii) If we assume that

W (ξ) ≥ C_{1} |ξ|^{p} for all ξ ∈ ℝ^{d×N} with |ξ| ≥ L,

and for some 1 ≤ p < ∞, then

u_{n_{k}} − \frac{1}{|Ω|} ∫_{Ω} u_{n_{k}} dx → u in W^{1,p} (Ω; ℝ^{d}) .

Indeed

C_{1} ∫_{{|∇u_{n_{k}}|≥L}} |∇u_{n_{k}}|^{p} dx ≤ ∫_{Ω} W (∇u_{n_{k}}) dx → 0.

On the other hand,

∫_{{|∇u_{n_{k}}|≤L}} |∇u_{n_{k}} − ∇u|^{p} dx ≤ (L + |A| + |B|)^{p−1} ∫_{Ω} |∇u_{n_{k}} − ∇u| dx → 0.

Theorem 3.3 Let u ∈ W^{1,1} (Ω; ℝ^{d}) , with ∇u ∈ BV (Ω; {A, B}) . Then the function u has the form

u (x) = u (x′, x_{N}) = γ_{0} + ax_{N} − 2ψ (x) a,

where γ_{0} ∈ ℝ^{d}, γ_{0} · a = 0, ψ ∈ W^{1,∞} (Ω; ℝ) satisfies ∇ψ (x) = χ_{E} (x) e_{N} for some set E ⊂ Ω with Per_{Ω}(E) < ∞, and E is layered perpendicularly to e_{N} , that is

∂^{∗}E ∩ Ω = ⋃_{i=1}^{∞} ω_{i} × {α_{i} } ,

where the sets ω_{i} ⊂ ℝ^{N−1} are connected, bounded and open, α_{i} ∈ ℝ. Moreover, in any open subset Ω′ of Ω with the property that for each t ∈ ℝ the horizontal section

{(x′, x_{N}) ∈ Ω′ : x_{N} = t} is connected in ℝ^{N},

we may write

u (x) = u (x′, x_{N}) = γ_{0} + ax_{N} − 2h (x_{N}) a a.e. in Ω′,

where h ∈ W^{1,∞} (ℝ; ℝ) , h′ ∈ BV (ℝ; {0, 1}).

Proof. As in [9], in view of the fact that for L^{N} a.e. x ∈ Ω.

∇u(x) = (1 − χ_{E} (x)) A + χ_{E} (x) B = (1 − χ_{E} (x)) a ⊗ e_{N} − χ_{E} (x) a ⊗ e_{N}
= a ⊗ e_{N} − 2χ_{E} (x) a ⊗ e_{N},

we may conclude that E is layered perpendicularly to e_{N} , and that the function u has the form

u (x) = u (x′, x_{N}) = γ_{0} + ax_{N} − 2ψ (x) a,

where γ_{0} ∈ ℝ^{d}, γ_{0} · a = 0, ψ ∈ W^{1,∞} (Ω; ℝ) , satisfies ∇ψ (x) = χ_{E} (x) e_{N}. Moreover, since

∇_{x′}u (x′, x_{N}) = ∇_{x′} (γ_{0} + ax_{N} − 2ψ (x) a) ≡ 0

we conclude that in any open subset Ω′ of Ω with the property that for each t ∈ ℝ the horizontal section

{(x′, x_{N}) ∈ Ω′ : x_{N} = t} is connected in ℝ^{N},

we may represent u as

u (x) = ũ (x_{N}) = γ_{0} + ax_{N} − 2h (x_{N}) a a.e. in Ω′,

where h ∈ W^{1,∞} (ℝ; ℝ) , h′ ∈ BV (ℝ; {0, 1}).

4 Γ−liminf: the lower bound

In view of (2.2), we define

K^{∗} := Γ − lim inf_{ε→0^{+}} I_{ε} (|x_{N}| a; Q)  (4.1)

= inf {lim inf_{n→∞} I_{ε_{n}}(u_{n}; Q) : ε_{n} → 0^{+}, u_{n} ∈ W^{2,2} (Q; ℝ^{d}) , u_{n} → |x_{N}| a in L^{1} (Q; ℝ^{d})} ,

where, we recall, Q := (−\frac{1}{2}, \frac{1}{2})^{N} .

Theorem 4.1 Assume that W satisfies condition (H_{1}) . Let u ∈ W^{1,1} (Ω; ℝ^{d}) , with ∇u ∈ BV (Ω; {A, B}) . Then

Γ − lim inf_{ε→0^{+}} I_{ε} (u; Ω) ≥ K^{∗} Per_{Ω}(E),

where ∇u = (1 − χ_{E} (x)) A + χ_{E} (x) B.

The proof of Theorem 4.1 is hinged on the following lemma.

Lemma 4.2 Let ω ⊂ ℝ^{N−1} be a connected, bounded, open set, with H^{N−1} (∂ω) = 0, and consider the cylinder U := ω × (α − h, α + h) , where α ∈ ℝ and h > 0. If u_{0} ∈ W^{1,1} (U; ℝ^{d}) is such that

∇u_{0}(x′, x_{N}) = \{ \begin{matrix}A & if x_{N} > α, \\ B & if x_{N}< α, \end{matrix}

then

Γ − lim inf_{ε→0^{+}} I_{ε} (u_{0}; U) = K^{∗}H^{N−1} (ω) .

Assuming that Lemma 4.2 holds (its proof is left for the remaining of Section 4), we conclude the proof of Theorem 4.1.

Proof of Theorem 4.1. By Theorem 3.3 and since ∂Ω is Lipschitz, we may write

∂^{∗}E ∩ Ω = ⋃_{i=1}^{∞} ω_{i} × {α_{i} } , with ∑_{i=1}^{∞} H^{N−1} (ω_{i} × {α_{i} }) < ∞,

where the sets ω_{i} ⊂ ℝ^{N−1} are connected, bounded and open, with H^{N−1} (∂ω_{i}) = 0, α_{i} ∈ ℝ. Let δ > 0 and choose k > 1 such that

H^{N−1} (∂^{∗}E ∩ Ω) ≤ ∑_{i=1}^{k} H^{N−1} (ω_{i} × {α_{i} }) + δ.

Let ω′_{i} ⊂⊂ ω_{i} be connected, bounded and open, with H^{N−1} (∂ω′_{i}) = 0, and such that

H^{N−1} (ω_{i} × \{α_{i} \}) ≤ H^{N−1} (ω′_{i} × \{α_{i} \}) + \frac{δ}{k} .

Since each ω′_{i} × {α_{i} } ⊂⊂ Ω we may find h > 0 so small that

⋃_{i=1}^{k} ω′_{i} × (α_{i} − h, α_{i} + h ) ⊂⊂ Ω

and the sets ω′_{i} × (α_{i} − h, α_{i} + h) are mutually disjoint. Then for any sequences ε_{n} → 0^{+} and {u_{n}} ⊂ W^{2,2} (Ω; ℝ^{d}) such that u_{n} → u in L^{1} (Ω; ℝ^{d}) we have

lim inf_{n→∞} I_{ε_{n}}(u_{n}; Ω) ≥ lim inf_{n→∞} I_{ε_{n}} (u_{n}; ⋃_{i=1}^{k} ω′_{i} × (α_{i} − h, α_{i} + h))

≥ ∑_{i=1}^{k} lim inf_{n→∞} I_{ε_{n}} (u_{n}; ω′_{i} × (α_{i} − h, α_{i} + h))

≥ ∑_{i=1}^{k} K^{∗}H^{N−1} (ω′_{i} × {α_{i} })

≥ ∑_{i=1}^{k} K^{∗}H^{N−1} (ω_{i} × {α_{i} }) − δK^{∗}

≥ K^{∗}H^{N−1} (∂^{∗}E ∩ Ω) − 2δK^{∗},

where we have used Lemma 4.2 to assert that

lim inf_{n→∞} I_{ε_{n}} (u_{n}; ω′_{i} × (α_{i} − h, α_{i} + h )) ≥ K^{∗}H^{N−1} (ω′_{i} × {α_{i} }) .

Hence

Γ − lim inf_{ε→0^{+}} I_{ε} (u; Ω) ≥ H^{N−1} (∂^{∗}E ∩ Ω) − 2δK^{∗}

and it suffices to let δ → 0^{+}.

It remains to prove Lemma 4.2. Let ω ⊂ ℝ^{N−1} be a bounded, open set, with H^{N−1} (∂ω) = 0, and let h > 0. Define

F (ω; h) :=Γ − lim inf_{ε→0^{+}} I_{ε} (|x_{N}| a; ω × (−h, h))  (4.2)
= inf {lim inf_{n→∞} I_{ε_{n}}(u_{n}; ω × (−h, h)) : ε_{n} → 0^{+}, u_{n} ∈ W^{2,2} (ω × (−h, h) ; ℝ^{d}) ,
u_{n} → |x_{N}| a in L^{1} (ω × (−h, h) ; ℝ^{d})} .

Note that K^{∗} = F (Q′; \frac{1}{2}) , where Q′ := (−\frac{1}{2}, \frac{1}{2})^{N−1} .

Lemma 4.3 (i) F (x′ + ω; h) = F (ω; h) for every x′ ∈ ℝ^{N−1};

(ii) if ω_{1} ⊂ ω_{2} then F (ω_{1}; h) ≤ F (ω_{2}; h) ;

(iii) if ω_{1} ∩ ω_{2} = ∅ then F (ω_{1} ∪ ω_{2}; h) ≥ F (ω_{1}; h) + F (ω_{2}; h) ;

(iv) if α > 0 then F (αω; αh) = α^{N−1}F (ω; h) , while if 0 < α < 1 then F (αω; h) ≥ α^{N−1}F (ω; h) ;

(v) F (ω; h) = H^{N−1} (ω) F (Q′; h) ;

(vi) F (ω; h) = F (ω; δ) for each δ > 0.

We note that Lemma 4.2 is a direct consequence of Lemma 4.3 (v) and (vi).

Proof. (i) follows immediately from the translation invariance of the energies I_{ε_{n}}(u_{n}; ·), while (ii) and (iii) are consequences of the nonnegativeness of the energy densities of I_{ε_{n}}, together with the fact that admissible sequences for ω_{2} × (−h, h) are still admissible for ω_{1} × (−h, h) when ω_{1} ⊂ ω_{2}.

To prove (iv) let ε_{n} → 0^{+} and let { u_{n}} ⊂ W^{2,2} (ω × (−h, h) ; ℝ^{d}) be any sequence such that u_{n} → |x_{N}| a in L^{1} (ω × (−h, h) ; ℝ^{d}) . Set

v_{n} (x) := αu_{n}(x/α), x ∈ αω × (−αh, αh) .

Clearly |x_{N}| a = α |x_{N}/α| a and so v_{n} → |x_{N}| a in L^{1} (αω × (−αh, αh) ; ℝ^{d}) , and we have

F (αω; αh) ≤ lim inf_{n→∞} ∫_{αω×(−αh,αh)} \frac{1}{αε_{n}} W (∇v_{n}) + αε_{n}|∇^{2}v_{n}|^{2} dx

= lim inf_{n→∞} ∫_{αω×(−αh,αh)} \frac{1}{αε_{n}} W (∇u_{n} (\frac{x}{α})) + αε_{n} |\frac{1}{α}∇^{2}u_{n} (\frac{x}{α})|^{2} dx

= lim inf_{n→∞} α^{N} ∫_{ω×(−h,h)} \frac{1}{αε_{n}} W (∇u_{n} (y)) + αε_{n} |\frac{1}{α}∇^{2}u_{n} (y)|^{2} dy

= α^{N−1} lim inf_{n→∞} ∫_{ω×(−h,h)} \frac{1}{ε_{n}} W (∇u_{n} (y)) + ε_{n} |∇^{2}u_{n} (y)|^{2} dy.

Hence F (αω; αh) ≤ α^{N−1}F (ω; h) , and similarly

F (ω; h) = F (\frac{1}{α}αω; \frac{1}{α}αh) ≤ \frac{1}{α^{N−1}} F (αω; αh) .

This proves that F (αω; αh) = α^{N−1}F (ω; h) .

Next let 0 < α < 1; then F (αω; h) ≥ F (αω; αh) = α^{N−1}F (ω; h) .

To show (v) we use Vitali’s Covering Theorem to decompose

ω = ⋃_{i=1}^{∞}(a_{i} + η_{i}Q′) ∪ N_{0}

with H^{N−1} (N_{0}) = 0, a_{i} + η_{i}Q′ mutually disjoint, 0 < η_{i} < 1, Q′ := (−\frac{1}{2}, \frac{1}{2})^{N−1} , and

∑_{i=1}^{∞} η_{i}^{N−1} = H^{N−1} (ω) .

For all k ∈ ℕ, by (ii)-(iv)

F (ω; h) ≥ F (⋃_{i=1}^{k}(a_{i} + η_{i}Q′); h) ≥ ∑_{i=1}^{k} F (a_{i} + η_{i}Q′; h) ≥ ∑_{i=1}^{k} η_{i}^{N−1}F (Q′; h) .

By letting k → ∞ we conclude

F (ω; h) ≥ H^{N−1} (ω) F (Q′; h) .

Conversely, with

Q′ = ⋃_{i=1}^{∞}(b_{i} + δ_{i}ω) ∪ N_{1}

with H^{N−1} (N_{1}) = 0, b_{i} + δ_{i}ω mutually disjoint, and

∑_{i=1}^{∞} δ_{i}^{N−1} = \frac{1}{H^{N−1} (ω)},

we deduce that

F (Q′; h) ≥ \frac{1}{H^{N−1} (ω)}F (ω; h) ,

and we have concluded the proof of (v). This result will entail (vi) provided we show that

F (Q′; h) = F (Q′; δ)  (4.3)

for every δ > 0. We first claim that for all k ∈ ℕ

F (kQ′; δ) = k^{N−1}F (Q′; δ) .  (4.4)

Indeed, write

kQ′ = ⋃_{i=1}^{k^{N−1}}(a_{i} + Q′) ∪ N_{0}

with H^{N−1} (N_{0}) = 0, a_{i} + Q′ mutually disjoint. By (iv)

k^{N−1}F (Q′; δ) = F (kQ′; kδ) ≥ F (kQ′; δ) ≥ ∑_{i=1}^{k^{N−1}} F (a_{i} + Q′; δ) = k^{N−1}F (Q′; δ) .

Next we show that

F (Q′; δ) = F (Q′; \frac{δ}{k} ) .

Indeed by (iv) and (4.4)

F (Q′; δ) = \frac{1}{k^{N−1}} F (kQ′; kδ) = \frac{1}{k^{N−1}} k^{N−1}F (Q′; kδ) = F (Q′; kδ) ,

and thus

F (Q′; δ) = F (Q′; k \frac{δ}{k} ) = F (Q′; \frac{δ}{k} ) .

It follows that if p, k ∈ ℕ, k ≠ 0, then

F (Q′; \frac{p}{k} δ) = F (Q′; δ) ,  (4.5)

and to assert (4.3) it suffices now to establish the continuity of F (Q′; ·) . Let r_{n} → r and extract a subsequence r′_{n} → r. Without loss of generality assume that r′_{n} → r^{+} (similarly, if r′_{n} → r^{−}). Then

F (Q′; r) ≤ lim inf_{n→∞} F (Q′; r′_{n}) .

Let q_{n} ∈ ℚ such that q_{n}r > r′_{n} > r, q_{n} → 1. Then by (4.5)

F (Q′; r′_{n}) ≤ F (Q′; q_{n}r) = F (Q′; r) ,

and thus

lim sup_{n→∞} F (Q′; r′_{n}) ≤ F (Q′; r) .

This conclude the proof.

Remark 4.4 It follows immediately from Lemma 4.3(vi) that the effective energy concentrates near the interfaces. Precisely, if

F (ω; h) = lim_{n→∞} I_{ε_{n}}(u_{n}; ω × (−h, h))

then for each 0 < η < h

lim_{n→∞} I_{ε_{n}}(u_{n}; ω × [(−h, h) \ (−η, η)]) = 0.

Indeed, by Lemma 4.3(vi)

lim_{n→∞} I_{ε_{n}}(u_{n}; ω × (−h, h)) = F (ω; h) = F (ω; η) ≤ lim inf_{n→∞} I_{ε_{n}}(u_{n}; ω × (−η, η))

and thus

lim_{n→∞} I_{ε_{n}}(u_{n}; ω × [(−h, h) \ (−η, η)]) = 0.

5 Γ−limsup: the upper bound. Geodesic hypotheses.

In agreement with our adopted notation, in what follows the constants C and C_{δ} may change from line to line. Throughout this section we assume that W satisfies the following conditions:

(H_{1}) W is continuous, W (ξ) = 0 if and only if ξ ∈ {A, B} , where A = −B = a ⊗ e_{N} for some a ∈ ℝ^{d} \ {0};

(H_{2})′ W (ξ) → ∞ as |ξ| → ∞;

(H_{3}) W (ξ) ≥ W (0, ξ_{N}) where ξ = (ξ′, ξ_{N}) ∈ ℝ^{d×(N−1)} × ℝ^{d}.

Note that (H_{3}) is verified by the prototype bulk energy density

W (ξ) := min {|ξ − A|^{2} , |ξ − B|^{2}} .

5.1 Characterization of K^{∗}

In this subsection we prove that if (H_{1}), (H_{2})′, (H_{3}) hold then

K^{∗} = inf {∫_{−L}^{L} W (0, g (s)) + |g′ (s)|^{2} ds : L > 0, g piecewise C^{1}, g (L) = −g (−L) = a} .

Proposition 5.1 Assume that W satisfies condition (H_{2})′. Let \{ε_{n}\} ⊂ ℝ_{+} and \{u_{n}\} ⊂ W^{2,2} ((−\frac{1}{2}, \frac{1}{2}); ℝ^{d}) be two sequences such that ε_{n} → 0^{+} and

sup_{n∈ℕ} ∫− \frac{1}{2}^{\frac{1}{2}} \frac{W (0, u′_{n})}{ε_{n}} + ε_{n} |u′′_{n}|^{2} dt < +∞.

Then

sup_{n∈ℕ} sup_{t∈(− \frac{1}{2} , \frac{1}{2} )} |u′_{n} (t)| < +∞.

Proof. Since u′_{n} ∈ L^{∞}((−\frac{1}{2}, \frac{1}{2}); ℝ^{d}), without loss of generality we may suppose that 0 < ε_{n} < \frac{1}{4}. Let

c := sup_{n} ∫− \frac{1}{2}^{\frac{1}{2}} \frac{W (0, u′_{n})}{ε_{n}} + ε_{n} |u′′_{n}|^{2} dt < +∞,

and fix ε_{n} ∈ (0, \frac{1}{4}) and t ∈ (−\frac{1}{2}, \frac{1}{2}) . If t + ε_{n} < \frac{1}{2} then

\frac{1}{ε_{n}} ∫_{t}^{t+ε_{n}} W (0, u′_{n}) ds ≤ c,

and so there exists t_{n} ∈ (t, t + ε_{n}) such that

W (0, u′_{n} (t_{n})) ≤ c.

By (H_{2})′ we may find a constant C = C(c) such that sup_{n} |u′_{n} (t_{n})| ≤ C, and Hölder’s inequality now yields

|u′_{n} (t)| ≤ |u′_{n} (t_{n})| + (ε_{n} ∫_{t}^{t+ε_{n}} |u′′_{n}|^{2} ds)^{1/2} ≤ C + (ε_{n} ∫− \frac{1}{2}^{\frac{1}{2}} |u′′_{n}|^{2} ds)^{1/2} ≤ C + \sqrt{c}.  (5.1)

If t + ε_{n} ≥ \frac{1}{2} then t ≥ \frac{1}{2} − ε_{n} and so t − ε_{n} ≥ \frac{1}{2} − 2ε_{n} > −\frac{1}{2}. Therefore we may reason as above, using the interval (t − ε_{n}, t) in place of (t, t + ε_{n}) to obtain (5.1), and we conclude that sup_{t∈(− \frac{1}{2} , \frac{1}{2})} |u′_{n} (t)| ≤ C + \sqrt{c}.

Recall that (see (4.1) and (4.2))

K^{∗} = F (Q′; \frac{1}{2}) := inf\{lim inf_{n→∞} ∫_{Q} \frac{1}{ε_{n}} W (∇u_{n}) + ε_{n} |∇^{2}u_{n}|^{2} dx : ε_{n} → 0^{+}, \{u_{n}\} ⊂ W^{2,2} (Q; ℝ^{d}) ,
u_{n} → |x_{N}| a in L^{1} (Q; ℝ^{d})},

where Q := Q′ × (−\frac{1}{2}, \frac{1}{2}), Q′ := (−\frac{1}{2}, \frac{1}{2})^{N−1} , and introduce the “one-dimensional” version of K^{∗},

K_{∗} := inf\{lim inf_{n→∞} ∫− \frac{1}{2}^{\frac{1}{2}} \frac{W (0, u′_{n})}{ε_{n}} + ε_{n} |u′′_{n}|^{2} dt : ε_{n} → 0^{+}, u_{n} ∈ W^{2,2} ((−\frac{1}{2}, \frac{1}{2}); ℝ^{d}) ,

u_{n} → |t| a in L^{1} ((−\frac{1}{2}, \frac{1}{2}); ℝ^{d})\}.

An immediate consequence of the latter proposition and the compactness results of Section 3 is the following corollary.

Corollary 5.2 Assume that W satisfies conditions (H_{1}) and (H_{2})′. Given any 1 ≤ p < ∞ there holds

K_{∗} = inf\{lim inf_{n→∞} ∫− \frac{1}{2}^{\frac{1}{2}} \frac{W (0, u′_{n})}{ε_{n}} + ε_{n} |u′′_{n}|^{2} dt : ε_{n} → 0^{+}, u_{n} ∈ W^{2,2} ((−\frac{1}{2}, \frac{1}{2}); ℝ^{d}) ,

u_{n} → |t| a in W^{1,p} ((−\frac{1}{2}, \frac{1}{2}); ℝ^{d})\}.

Proof. In view of Proposition 5.1, energy bounded sequences admissible for K_{∗} are uniformly bounded in W^{1,∞} ((−\frac{1}{2}, \frac{1}{2}); ℝ^{d}), and, in particular, must converge weakly to |·| a in W^{1,1} ((−\frac{1}{2}, \frac{1}{2}); ℝ^{d}). The result now follows from Remark 3.2(i).

Proposition 5.3 Assume that W satisfies conditions (H_{1}) , (H_{2})′ and (H_{3}) . Then

K^{∗} = K_{∗} = K,

where

K = inf {∫_{−L}^{L} W (0, g (s)) + |g′ (s)|^{2} ds : L > 0, g piecewise C^{1}, g (L) = −g (−L) = a} .

Proof. We divide the proof in three steps.

Step 1: We prove first that K^{∗} = K_{∗}. Clearly

K_{∗} ≥ K^{∗} = F (Q′; \frac{1}{2}) .

Indeed, if \{ε_{n}\} ⊂ ℝ_{+} and \{u_{n}\} ⊂ W^{2,2} ((−\frac{1}{2}, \frac{1}{2}); ℝ^{d}) are two sequences such that ε_{n} → 0^{+} and u_{n} → |t| a in L^{1} ((−\frac{1}{2}, \frac{1}{2}); ℝ^{d}) , then the sequence v_{n} (x) := u_{n} (x_{N}) is admissible for F (Q′; \frac{1}{2}) . To prove the converse inequality, let {ε_{n}} ⊂ ℝ_{+} and {u_{n}} ⊂ W^{2,2} (Q; ℝ^{d}) be such that ε_{n} → 0^{+} and u_{n} → |x_{N}| a in L^{1} (Q; ℝ^{d}) . For H^{N−1} a.e. x′ ∈ Q′ the function u_{n}^{x′} (t) := u_{n} (x′, t) ∈ W^{2,2} ((−\frac{1}{2}, \frac{1}{2}); ℝ^{d}) and u_{n}^{x′} → |t| a in L^{1} ((−\frac{1}{2}, \frac{1}{2}); ℝ^{d}).

Using (H_{3}) and Fatou’s Lemma we have

lim inf_{n→∞} ∫_{Q} \frac{1}{ε_{n}} W (∇u_{n}) + ε_{n} |∇^{2}u_{n}|^{2} dx = lim inf_{n→∞} ∫_{Q′} (∫− \frac{1}{2}^{\frac{1}{2}} \frac{1}{ε_{n}} W (∇u_{n}) + ε_{n} |∇^{2}u_{n}|^{2} dt) dx′

≥ lim inf_{n→∞} ∫_{Q′} (∫− \frac{1}{2}^{\frac{1}{2}} \frac{1}{ε_{n}} W (0, \frac{du_{n}^{x′}}{dt} ) + ε_{n} |\frac{d^{2}u_{n}^{x′}}{dt^{2}} |^{2} dt) dx′

≥ ∫_{Q′} lim inf_{n→∞} (∫− \frac{1}{2}^{\frac{1}{2}} \frac{1}{ε_{n}} W (0, \frac{du_{n}^{x′}}{dt} ) + ε_{n} |\frac{d^{2}u_{n}^{x′}}{dt^{2}} |^{2} dt) dx′

≥ K_{∗} ∫_{Q′} dx′ = K_{∗}. Step 2: We now prove that K_{∗} = K_{1}, where

K_{1} := inf\{lim inf_{n→∞} ∫− \frac{1}{2}^{\frac{1}{2}} \frac{W (0, u′_{n})}{ε_{n}} + ε_{n} |u′′_{n}|^{2} dt : ε_{n} → 0^{+}, u_{n} ∈ W^{2,2} ((−\frac{1}{2}, \frac{1}{2}); ℝ^{d}) ,

u_{n} → |t| a in L^{1} ((−\frac{1}{2}, \frac{1}{2}); ℝ^{d}) , u′_{n} = ±a near t = ±\frac{1}{2} (resp.)\}.

Clearly K_{1} ≥ K_{∗}. To prove the converse inequality, let \{ε_{n}\} ⊂ ℝ_{+} and \{u_{n}\} ⊂ W^{2,2} ((−\frac{1}{2}, \frac{1}{2}); ℝ^{d}) be such that ε_{n} → 0^{+}, u_{n} → |t| a in L^{1} ((−\frac{1}{2}, \frac{1}{2}); ℝ^{d}) , and

K_{∗} = lim inf_{n→∞} ∫− \frac{1}{2}^{\frac{1}{2}} \frac{W (0, u′_{n})}{ε_{n}} + ε_{n} |u′′_{n}|^{2} dt.

Without loss of generality, and up to the extraction of a subsequence, we may assume that

lim inf_{n→∞} ∫− \frac{1}{2}^{\frac{1}{2}} \frac{W (0, u′_{n})}{ε_{n}} + ε_{n} |u′′_{n}|^{2} dt = lim_{n→∞} ∫− \frac{1}{2}^{\frac{1}{2}} \frac{W (0, u′_{n})}{ε_{n}} + ε_{n} |u′′_{n}|^{2} dt,

and, by Corollary 5.2, that u_{n} → |t| a in W^{1,1} ((−\frac{1}{2}, \frac{1}{2}); ℝ^{d}) . Since by Remark 4.4

lim_{n→∞} ∫ \frac{1}{2} \frac{1}{4} \frac{W (0, u′_{n})}{ε_{n}} + ε_{n} |u′′_{n}|^{2} + |u_{n} − ta| + |u′_{n} − a| dt = 0,  (5.2)

there exists t_{0} ∈ (\frac{1}{4}, \frac{1}{3}) such that (up to the extraction of a further subsequence, if necessary)

lim_{n→∞} [\frac{W (0, u′_{n} (t_{0}))}{ε_{n}} + ε_{n} |u′′_{n} (t_{0})|^{2} + |u_{n} (t_{0}) − t_{0}a| + |u′_{n} (t_{0}) − a|] = 0.  (5.3)

Define

w_{n} (t) := ψ_{n} (t) u_{n} (t) + (1 − ψ_{n} (t)) ((t − t_{0}) a + u_{n} (t_{0})) ,

where ψ_{n} is a smooth cut-off function such that ψ_{n} (t) = 0 for t ≥ t_{0} + ε_{n}, ψ_{n} (t) = 1 for t ≤ t_{0}, and

|ψ′_{n} (t)| ≤ C/ε_{n}, |ψ′′_{n} (t)| ≤ C/ε_{n}^{2}.

Then

w′_{n} (t) = ψ′_{n} (t) (u_{n} (t) − u_{n} (t_{0}) − (t − t_{0}) a) + ψ_{n} (t) u′_{n} (t) + (1 − ψ_{n} (t)) a,  (5.4)

and

w′′_{n} (t) = ψ′′_{n} (t) (u_{n} (t) − u_{n} (t_{0}) − (t − t_{0}) a) + 2ψ′_{n} (t) (u′_{n} (t) − a) + ψ_{n} (t) u′′_{n}.  (5.5)

By Hölder’s inequality, for t ∈ (t_{0}, t_{0} + ε_{n})

|u′_{n} (t) − a| ≤ |u′_{n} (t_{0}) − a| + (ε_{n} ∫_{t_{0}}^{t_{0}+ε_{n}} |u′′_{n}|^{2} ds)^{1/2} ,  (5.6)

and thus

|u_{n} (t) − u_{n} (t_{0}) − (t − t_{0}) a| ≤ ∫_{t_{0}}^{t_{0}+ε_{n}} |u′_{n} (s) − a| ds  (5.7)

≤ ε_{n} |u′_{n} (t_{0}) − a| + ε_{n} (ε_{n} ∫_{t_{0}}^{t_{0}+ε_{n}} |u′′_{n}|^{2} ds)^{1/2} .

Since W is continuous and W (0, a) = 0, we may find ρ > 0 and a modulus of continuity η = η (s) such that

W (0, ξ_{N}) ≤ η (|ξ_{N} − a|) for all |ξ_{N} − a| ≤ ρ.

By (5.4), (5.6) and (5.7), for t ∈ (t_{0}, t_{0} + ε_{n}) we have

|w′_{n} (t) − a| ≤ \frac{C}{ε_{n}} |u_{n} (t) − u_{n} (t_{0}) − (t − t_{0}) a| + |u′_{n} (t) − a|

≤ C |u′_{n} (t_{0}) − a| + C (ε_{n} ∫_{t_{0}}^{t_{0}+ε_{n}} |u′′_{n}|^{2} ds)^{1/2} ≤ ρ

for ε_{n} sufficiently small, where we have used (5.2) and (5.3). Hence for t ∈ (t_{0}, t_{0} + ε_{n}) and since η is increasing

W (0, w′_{n}) ≤ η (C |u′_{n} (t_{0}) − a| + C (ε_{n} ∫_{t_{0}}^{t_{0}+ε_{n}} |u′′_{n}|^{2} ds)^{1/2}) ,

and thus

∫_{t_{0}}^{t_{0}+ε_{n}} \frac{W (0, w′_{n})}{ε_{n}}dt ≤ η (C |u′_{n} (t_{0}) − a| + C (ε_{n} ∫_{t_{0}}^{t_{0}+ε_{n}} |u′′_{n}|^{2} ds)^{1/2}) → 0.

Similarly, by (5.5), (5.6) and (5.7) and for t ∈ (t_{0}, t_{0} + ε_{n}) ,

|w′′_{n} (t)| ≤ \frac{C}{ε_{n}^{2}} |u_{n} (t) − u_{n} (t_{0}) − (t − t_{0}) a| + \frac{C}{ε_{n}} |u′_{n} (t) − a| + |u′′_{n}|

≤ \frac{C}{ε_{n}} |u′_{n} (t_{0}) − a| + \frac{C}{ε_{n}} (ε_{n} ∫_{t_{0}}^{t_{0}+ε_{n}} |u′′_{n}|^{2} ds)^{1/2} + |u′′_{n}| ,

and we have

ε_{n} |w′′_{n} (t)|^{2} ≤ \frac{C}{ε_{n}} |u′_{n} (t_{0}) − a|^{2} + C ∫_{t_{0}}^{t_{0}+ε_{n}} |u′′_{n}|^{2} ds + Cε_{n} |u′′_{n}|^{2} .

Hence, in view of (5.2) and (5.3),

∫_{t_{0}}^{t_{0}+ε_{n}} ε_{n} |w′′_{n} (t)|^{2} dt ≤ C |u′_{n} (t_{0}) − a|^{2} + Cε_{n} ∫_{t_{0}}^{t_{0}+ε_{n}} |u′′_{n}|^{2} ds → 0.

Since w_{n} (t) = u_{n} (t) for t ≤ t_{0} and w′_{n} (t) = a for t ≥ t_{0} + ε_{n}, we conclude that

lim sup_{n→∞} ∫− \frac{1}{2}^{\frac{1}{2}} \frac{W (0, w′_{n})}{ε_{n}} + ε_{n} |w′′_{n}|^{2} dt ≤ lim_{n→∞} ∫− \frac{1}{2}^{\frac{1}{2}} \frac{W (0, u′_{n})}{ε_{n}} + ε_{n} |u′′_{n}|^{2} dt = K_{∗}.

By repeating the same construction nearby t = −\frac{1}{2} we ensure that the new sequence satisfies w′_{n} = ±a near t = ±\frac{1}{2} resp. Hence K_{1} ≤ K_{∗}, and the proof of Step 2 is complete.

Step 3: We finally prove that K_{∗} = K. Let \{ε_{n}\} ⊂ ℝ_{+} and \{u_{n}\} ⊂ W^{2,2} ((−\frac{1}{2}, \frac{1}{2}); ℝ^{d}) be such that ε_{n} → 0^{+}, u_{n} → |t| a in L^{1} ((−\frac{1}{2}, \frac{1}{2}); ℝ^{d}) , and

K_{∗} = lim inf_{n→∞} ∫− \frac{1}{2}^{\frac{1}{2}} \frac{W (0, u′_{n})}{ε_{n}} + ε_{n} |u′′_{n}|^{2} dt.

In light of Step 2, we may assume, without loss of generality, that u′_{n} = a near t = \frac{1}{2} and u′_{n} = −a near

t = −\frac{1}{2}. Define v_{n} (s) := \frac{1}{ε_{n}} u_{n} (ε_{n}s) for s ∈ [−\frac{1}{2ε_{n}} , \frac{1}{2ε_{n}} ] . Then v′_{n} (s) = u′_{n} (ε_{n}s), v′′_{n} (s) = ε_{n}u′′_{n} (ε_{n}s) , and so

∫− \frac{1}{2}^{\frac{1}{2}} \frac{W (0, u′_{n} (t))}{ε_{n}} + ε_{n} |u′′_{n} (t)|^{2} dt = ∫− \frac{1}{2εn}^{\frac{1}{2εn}} W (0, u′_{n} (ε_{n}s)) + ε_{n}^{2} |u′′_{n} (ε_{n}s)|^{2} ds

= ∫− \frac{1}{2εn}^{\frac{1}{2εn}} W (0, v′_{n} (s)) + |v′′_{n} (s)|^{2} ds ≥ K,

therefore K_{∗} ≥ K. Conversely, let g : [−L, L] → ℝ^{d} be a piecewise C^{1} curve, with g (L) = −g (−L) = a. Consider any sequence {ε_{n}} converging to 0^{+}, and define

u_{n} (t) := ∫_{0}^{t} v_{n} (s) ds, v_{n} (t) := \{ \begin{matrix}−a & if t < −ε_{n}L, \\ g (\frac{t}{ε_{n}} ) & if |t| ≤ ε_{n}L, \\ a & if t > ε_{n}L. \end{matrix}  (5.8)

As in [26], we have

v_{n} → v_{0} := \{ \begin{matrix}−a & if t ≤ 0, \\ a & if t > 0, \end{matrix} in L^{p} ((−\frac{1}{2}, \frac{1}{2}); ℝ^{d}) , for any 1 ≤ p < ∞,

and so u_{n} = ∫_{0}^{t} v_{n} (s) ds → |t| a in W^{1,p} ((−\frac{1}{2}, \frac{1}{2}); ℝ^{d}) . Moreover,

K_{∗} ≤ lim_{n→∞} ∫− \frac{1}{2}^{\frac{1}{2}} \frac{W (0, u′_{n})}{ε_{n}} + ε_{n} |u′′_{n}|^{2} dt = lim_{n→∞} ∫− \frac{1}{2}^{\frac{1}{2}} \frac{W (0, v_{n})}{ε_{n}} + ε_{n} |v′_{n}|^{2} dt

= ∫_{−L}^{L} W (0, g (s)) + |g′ (s)|^{2} ds,

and taking the infimum over all such functions g we get the desired inequality.

Remark 5.4 Note that the argument of Step 3 in the latter proof, together with Propositions 5.1 and 5.3, ensures that given any sequence {ε_{n}} converging to 0^{+} there exists a sequence {u_{n}} converging to |x_{N}|a in W^{1,p} for all p ∈ [1, +∞), and such that

K^{∗} = lim_{n→∞} ∫_{Q} \frac{1}{ε_{n}} W (∇u_{n}) + ε_{n} |∇^{2}u_{n}|^{2} dx .

Indeed, for each k ∈ ℕ construct via (5.8) a sequence {u_{n,k}} corresponding to a function g_{k} admissible for K and such that

∫_{−L}^{L} W (0, g_{k} (s)) + |g′_{k} (s)|^{2} ds ≤ K^{∗} + \frac{1}{k} .

Then, with u(x) := |x_{N}|a we obtain

lim_{k→∞} lim_{n→∞} ∥u_{n,k} − u∥_{W^{1,p}(Q;ℝ^{d})} = 0,

and

lim sup_{k→∞} lim_{n→∞} ∫_{Q} \frac{1}{ε_{n}} W (∇u_{n,k}) + ε_{n} |∇^{2}u_{n,k}|^{2} dx ≤ K^{∗}.

On the other hand, by Theorem 4.1 we always have the opposite inequality, and so

lim_{k→∞} lim_{n→∞} ∫_{Q} \frac{1}{ε_{n}} W (∇u_{n,k}) + ε_{n} |∇^{2}u_{n,k}|^{2} dx = K^{∗}.

It suffices now to extract a subsequence {k_{n}} of {k} such that the subsequence z_{n} := u_{n,k_{n}} satisfies (see Lemma 7.2 in [17])

lim_{n→∞} ∥z_{n} − u∥_{W^{1,p}(Q;ℝ^{d})} = 0,

and

lim inf_{n→∞} ∫_{Q} \frac{1}{ε_{n}} W (∇z_{n}) + ε_{n} |∇^{2}z_{n}|^{2} dx = K^{∗}.

5.2 Special domains

Let

α := inf {x_{N} : x ∈ Ω} , β := sup {x_{N} : x ∈ Ω} .  (5.9)

Throughout this subsection we assume that the domain Ω satisfies (see Figure 1):

for each t ∈ ℝ the horizontal section Ω_{t} := {(x′, x_{N}) ∈ Ω : x_{N} = t} is connected in ℝ^{N}  (5.10)

and

t ↦ H^{N−1} (Ω_{t}) is continuous in (α, β) .  (5.11)

Theorem 5.5 Assume that W satisfies the conditions (H_{1}) , (H_{2})′ and (H_{3}) . Let u ∈ W^{1,1} (Ω; ℝ^{d}) , with ∇u ∈ BV (Ω; {A, B}) . Then

Γ − lim_{ε→0^{+}} I_{ε} (u; Ω) = K^{∗} Per_{Ω}(E),  (5.12)

where ∇u(x) = (1 − χ_{E} (x)) A + χ_{E} (x) B for L^{N} a.e. x ∈ Ω.

Proof. In view of Theorem 4.1, to prove (5.12) it suffices to show that for any sequence {ε_{n}} ⊂ ℝ_{+} such that ε_{n} → 0^{+} we have

Γ − lim sup_{n→∞} I_{ε_{n}} (u; Ω) ≤ K^{∗} Per_{Ω}(E).

Thus we fix a sequence {ε_{n}} ⊂ ℝ_{+} converging to 0^{+}. For simplicity in the notation we drop the subscript n so that ε := ε_{n}. By Theorem 3.3 and (5.10) we may assume that

u (x) = ũ (x_{N}) = γ_{0} + ax_{N} − 2h (x_{N}) a a.e. in Ω,

where h ∈ W^{1,∞} (ℝ; ℝ) , h′ ∈ BV (ℝ; {0, 1}), with

S (∇u) ∩ Ω = ⋃_{i=1}^{∞} Ω_{l_{i}}

and Ω_{l_{i}} := {x = (x′, x_{N}) ∈ Ω : x_{N} = l_{i}} for some l_{i} ∈ ℝ. We divide the proof in two steps. Step 1: Assume that the number of interfaces is finite, that is

S (∇u) ∩ Ω = ⋃_{i=1}^{m} Ω_{l_{i}},

for some m ∈ ℕ, and some finite family l_{1} < ⋯ < l_{m}. Fix k ∈ ℕ, and in view of Proposition 5.3 consider a piecewise C^{1} curve g_{k} : [−L, L] → ℝ^{d}, with g_{k} (L) = −g_{k} (−L) = a, such that

∫_{−L}^{L} W (0, g_{k} (s)) + |g′_{k} (s)|^{2} ds ≤ K^{∗} + \frac{1}{k} .  (5.13)

Let ε > 0 be so small that l_{i} + ε L < l_{i+1} − ε L, for i = 1, ⋯, m − 1. Set

v_{ε,k} (s) := \{ \begin{matrix}g_{k} (− sgn (u′ (l_{i} − ε L) · a) \frac{s−l_{i}}{ε} ) & if l_{i} − ε L < s < l_{i} + ε L, i = 1, ⋯, m, \\ ũ′ (s) & otherwise in ℝ, \end{matrix}

where we have extended ũ′ constantly to (−∞, l_{1} − ε L) and (l_{m} + ε L, +∞) , and define

u_{ε,k} (x) = ũ_{ε,k} (x_{N}) := ũ (l_{1}) + ∫_{l_{1}}^{x_{N}} v_{ε,k} (s) ds, x ∈ ℝ^{N}.

Then

∫_{Ω} |∇u_{ε,k} − ∇u| dx ≤ C ∑_{i=1}^{m} ∫_{l_{i}−ε L}^{l_{i}+ε L} [|g_{k} (± \frac{s − l_{i}}{ε} )| + 1] ds ≤ 2Cmε L + C ∑_{i=1}^{m} ε ∫_{−L}^{L} |g_{k} (t)| dt,

and since u_{ε,k} (x) = u (x) when x_{N} = l_{1}, by Poincaré’s inequality we have that u_{ε,k} → u in W^{1,1} (Ω; ℝ^{d}) as ε → 0^{+}. Further,

lim_{ε→0^{+}} I_{ε} (u_{ε,k}; Ω) = lim_{ε→0^{+}} ∑_{i=1}^{m} ∫_{Ω∩\{l_{i}+ε L<x_{N} <l_{i+1}+ε L\}} \frac{W (0, g_{k} (±\frac{x_{N} −l_{i}}{ε} ))}{ε} + \frac{1}{ε} |g′_{k} (±\frac{x_{N} − l_{i}}{ε} )|^{2} dx

= lim_{ε→0^{+}} ∑_{i=1}^{m} ∫_{l_{i}−ε L}^{l_{i}+ε L} (\frac{W (0, g_{k} (±\frac{s−l_{i}}{ε} ))}{ε} + \frac{1}{ε} |g′_{k} (±\frac{s − l_{i}}{ε} )|^{2}) H^{N−1} (\{x ∈ Ω : x_{N} = s\}) ds

= lim_{ε→0^{+}} ∑_{i=1}^{m} ∫_{−L}^{L} (W (0, g_{k} (t)) + |g′_{k} (t)|^{2}) H^{N−1} ({x ∈ Ω : x_{N} = εt + l_{i}}) dt

= (∫_{−L}^{L} W (0, g_{k} (t)) + |g′_{k} (t)|^{2} dt) ∑_{i=1}^{m} H^{N−1} (Ω_{l_{i}}) ≤ (K^{∗} + \frac{1}{k} ) ∑_{i=1}^{m} H^{N−1} (Ω_{l_{i}}) ,

where we have used (5.11) and (5.13). Hence

lim sup_{k→∞} lim_{ε→0^{+}} I_{ε} (u_{ε,k}; Ω) ≤ K^{∗} ∑_{i=1}^{m} H^{N−1} (Ω_{l_{i}}) = K^{∗} Per_{Ω} (E) ,

and in view of Theorem 4.1 this inequality is actually an identity. As in Remark 5.4, it is possible to extract a subsequence {k_{ε}} of {k} such that the subsequence u_{ε} := u_{ε,k_{ε}} satisfies

lim_{ε→0^{+}} ∥u_{ε} − u∥_{W^{1,1}(Ω;ℝ^{d})} = 0,

and

lim_{ε→0^{+}} I_{ε} (u_{ε}; Ω) = K^{∗} Per_{Ω} (E) .

Step 2: Suppose now that the number of interfaces is infinite, that is

S (∇u) ∩ Ω = ⋃_{i=1}^{∞} Ω_{l_{i}},

where Ω_{l_{i}} := {x = (x′, x_{N}) ∈ Ω : x_{N} = l_{i}} , l_{i} ∈ ℝ. We claim that interfaces can only accumulate at the top or bottom of Ω, i.e. if l is an accumulation point of {l_{i}} then l ∈ {α, β}, where α and β are defined in (5.9). Indeed, if Ω_{l} ≠ ∅ then we may find x = (x′, l) ∈ Ω and, in turn, an open cylinder of the form B′ (x′, r) × (l − h, l + h) ⊂ Ω. For i_{k} large enough B′ (x′; r) × {l_{i_{k}}} ⊂ Ω, where {l_{i_{k}}} is a subsequence of {l_{i}} converging to l, and so

∞ > ∑_{i=1}^{∞} H^{N−1} (Ω_{l_{i}}) ≥ ∑_{i_{k}} H^{N−1} (Ω_{l_{i_{k}}} ) ≥ ∑_{i_{k}} H^{N−1} (B′ (x′; r)) = ∞,

and we have reached a contradiction.

Fix an integer m ∈ ℕ and consider

U_{m} := {x ∈ Ω : α + δ_{m} < x_{N} < β − δ_{m}} ,

where δ_{m} → 0^{+} and {l_{i}} ∩ {α + δ_{m}, β − δ_{m}} = ∅. Then L^{N} (Ω\U_{m}) → 0 as m → ∞, and ∇u has a finite number of interfaces in U_{m}. By Step 1 we may construct a sequence {u_{ε}^{m}} such that u_{ε}^{m} → u in W^{1,1} (U_{m}; ℝ^{d}) as ε → 0^{+}, u_{ε}^{m} = u in Ω\U_{m}, and

lim_{ε→0} I_{ε} (u_{ε}^{m}; U_{m}) = K^{∗} Per_{U_{m}} (E) .

We have

lim_{m→∞} lim_{ε→0} I_{ε} (u_{ε}^{m}; Ω) = lim_{m→∞} lim_{ε→0} I_{ε} (u_{ε}^{m}; U_{m}) = K^{∗} lim_{m→∞} Per_{U_{m}} (E) = K^{∗} Per_{Ω} (E) .

We can conclude as in Step 1.

5.3 x′-connected domains

Throughout this subsection we assume that for each t ∈ ℝ the horizontal section

Ω_{t} := {(x′, x_{N}) ∈ Ω : x_{N} = t} is connected in ℝ^{N}.  (5.14)

Here we allow for the possibility that t ↦ H^{N−1} (Ω_{t}) is not continuous (see Figure 2). In this case a more careful analysis is required near the boundary under the additional hypothesis that W is smooth at the wells.

Theorem 5.6 Assume (5.14) and let W satisfy the conditions (H_{1}) , (H_{2})′ and (H_{3}) . Suppose, in addition, that W is differentiable at A and B. Let u ∈ W^{1,1} (Ω; ℝ^{d}) , with ∇u ∈ BV (Ω; {A, B}) . Then

Γ − lim_{ε→0^{+}} I_{ε} (u; Ω) = K^{∗} Per_{Ω}(E),

where ∇u(x) = (1 − χ_{E} (x)) A + χ_{E} (x) B for L^{N} a.e. x ∈ Ω.

Proof. By virtue of Theorem 4.1 it suffices to prove that given an arbitrary sequence {ε_{n}} converging to 0^{+} we have

Γ − lim sup_{n→+∞} I_{ε_{n}}(u; Ω) ≤ K^{∗} Per_{Ω}(E).  (5.15)

Fix one such sequence {ε_{n}}, and for simplicity of notation abbreviate ε := ε_{n}. We divide the proof of (5.15) into three steps.

Figure 5: Construction for one interface in Step 1 of Theorem 5.6. The shaded region represents {0 < ψ_{δ} < 1}, and corresponds to A_{2δ}^{+}\A_{δ}^{+} where A_{2δ}^{+} := ω_{2δ}^{+} ×(−\frac{h}{2}, \frac{h}{2}) and A_{δ}^{+} := ω_{δ}^{+}×(−\frac{h}{3}, \frac{h}{3}). The function u_{ε,δ} coincides with u outside A_{2δ}^{+} and with z_{ε} inside A_{δ}^{+}.

Step 1: One interface. We assume first that u has the form

u (x) = |x_{N}| a a.e. in Ω,

so that there is only one interface Ω_{0} := {x = (x′, x_{N}) ∈ Ω : x_{N} = 0} , and we may write Ω_{0} = ω × {0} , where ω is an open bounded connected subset of ℝ^{N−1}. Let 0 < h < \frac{1}{4} min \{β, −α\} , where α and β are defined in (5.9), and consider a sequence {δ_{m}} converging to 0^{+} as m → +∞. We write δ := δ_{m}. For every δ we construct a smooth cut-off function ψ_{δ} ∈ C_{c}^{∞} (ℝ^{N}; [0, 1]) such that ψ_{δ} = 1 in ω_{δ}^{+} × (−\frac{h}{3}, \frac{h}{3}) and ψ_{δ} = 0 outside ω_{2δ}^{+} × (−\frac{h}{2}, \frac{h}{2}) , where for s > 0 we denote ω_{s}^{+} := \{x′ ∈ ℝ^{N−1} : dist (x′, ω) < s\} (see Figure 5). In view of Proposition 5.3 consider a piecewise C^{1} curve g : [−L, L] → ℝ^{d}, with g (L) = −g (−L) = a. Extend g to all of ℝ by setting g (t) = −g (−t) := a for all t > L, and define

u_{ε,δ} (x) := ψ_{δ} (x) z_{ε} (x) + (1 − ψ_{δ} (x)) u,  (5.16)

where z_{ε} (x) := ∫_{0}^{x_{N}} g (\frac{s}{ε}) ds (see Figure 5). Note that u ∈ W^{2,2} (Ω\ (ω_{δ}^{+} × (−\frac{h}{3}, \frac{h}{3})); ℝ^{d}) . Then

∫_{Ω} \frac{1}{ε}W (∇u_{ε,δ}) + ε |∇^{2}u_{ε,δ}|^{2} dx = ∫_{\{ψ_{δ}=1\}} \frac{1}{ε}W (∇z_{ε}) + ε |∇^{2}z_{ε}|^{2} dx  (5.17)

+ ∫_{\{0<ψ_{δ}<1\}} \frac{1}{ε}W (∇u_{ε,δ}) + ε |∇^{2}u_{ε,δ}|^{2} dx.

By the fact that g (t) = −g (−t) := a for all t > L we have

∫_{\{ψ_{δ}=1\}} \frac{1}{ε}W (∇z_{ε}) + ε |∇^{2}z_{ε}|^{2} dx ≤ H^{N−1} (ω_{δ}^{+}) ∫_{−L}^{L} W (0, g (s)) + |g′ (s)|^{2} ds.  (5.18)

Also, as W is continuous in ℝ^{d×N}, W (±a ⊗ e_{N}) = 0, and W is differentiable at ±a ⊗ e_{N}, we may find a modulus of continuity η with

lim_{s→0^{+}} \frac{η (s)}{s} = 0  (5.19)

and such that

W (ξ) ≤ min {η (|ξ − a ⊗ e_{N}|) , η (|ξ + a ⊗ e_{N}|)} for all ξ ∈ ℝ^{d×N}.  (5.20)

Hence, by (5.20),

∫_{\{0<ψ_{δ}<1\}} \frac{1}{ε}W (∇u_{ε,δ}) + ε |∇^{2}u_{ε,δ}|^{2} dx ≤ ∫_{\{0<ψ_{δ}<1\}} \frac{1}{ε}η (|∇u_{ε,δ} − ∇u|) + ε |∇^{2}u_{ε,δ}|^{2} dx.  (5.21)

To estimate the right hand side of (5.21) note that

|∇u_{ε,δ} − ∇u| ≤ |∇ψ_{δ}| |z_{ε} − u| + ψ_{δ} |∇z_{ε} − ∇u| ,  (5.22)
|∇^{2}u_{ε,δ}| ≤ |∇^{2}ψ_{δ}| |z_{ε} − u| + 2 |∇ψ_{δ}| |∇z_{ε} − ∇u| + ψ_{δ} |∇^{2}z_{ε}| .

Fix L_{1} ≥ L. For |x_{N}| ≥ εL_{1} we have

z_{ε} − |x_{N}| a = \{ \begin{matrix}ε ∫_{0}^{L} (g (s) − a) ds & for x_{N} ≥ εL_{1,} \\ −ε ∫_{−L}^{0} (g (s) + a) ds & for x_{N} ≤ −εL_{1,} \end{matrix}  (5.23)

and so from (5.22)

|∇u_{ε,δ} − ∇u| ≤ C_{δ} ε, |∇^{2}u_{ε,δ}| ≤ C_{δ} ε  (5.24)

for |x_{N}| ≥ εL_{1}, while for |x_{N}| ≤ εL_{1}

|∇ (z_{ε} − u)| ≤ |g (\frac{x_{N}}{ε} )| + ∥∇u∥_{∞} ≤ C, |∇^{2}z_{ε}| ≤ \frac{1}{ε} |g′ (\frac{x_{N}}{ε} )| ≤ \frac{C}{ε} ,  (5.25)

and

|z_{ε} (x) − u (x)| ≤ |z_{ε} (x′, εL_{1}) − u (x′, εL_{1})| + ∫_{−εL_{1}}^{εL_{1}} |∇ (z_{ε} − u)| dx_{N} ≤ εC.

Hence, from (5.22) we have for |x_{N}| ≤ εL_{1}

|∇u_{ε,δ} − ∇u| ≤ C_{δ} ε + C, |∇^{2}u_{ε,δ}| ≤ C_{δ} + \frac{C}{ε} .  (5.26)

It now follows from (5.24) and (5.26) that, for ε < h/3L_{1},

∫_{\{0<ψ_{δ}<1\}} \frac{1}{ε}η (|∇u_{ε,δ} − ∇u|) + ε |∇^{2}u_{ε,δ}|^{2} dx ≤ (\frac{1}{ε}η (εC_{δ}) + C_{δ}ε^{3}) H^{N−1} (ω_{2δ}^{+}) h  (5.27)

+ H^{N−1} (ω_{2δ}^{+}\ω_{δ}^{+}) 2L_{1}ε (\frac{1}{ε}η (εC_{δ} + C) + ε (C_{δ} + \frac{C}{ε^{2}} )) .

In view of (5.17), (5.18), (5.21) and (5.27)

∫_{Ω} \frac{1}{ε}W (∇u_{ε,δ}) + ε |∇^{2}u_{ε,δ}|^{2} dx ≤ (∫_{−L}^{L} W (0, g (s)) + |g′ (s)|^{2} ds) H^{N−1} (ω_{δ}^{+})  (5.28)

+ C (\frac{1}{ε}η (εC_{δ}) + C_{δ}ε^{3}) + C (η (εC_{δ} + C) + ε^{2}C_{δ} + C) H^{N−1} (ω_{2δ}^{+}\ω_{δ}^{+}) ,

and letting ε → 0^{+} and then δ → 0^{+} yields, by (5.19),

lim sup_{δ→0^{+}} lim sup_{ε→0^{+}} ∫_{Ω} \frac{1}{ε}W (∇u_{ε,δ}) + ε |∇^{2}u_{ε,δ}|^{2} dx ≤ (∫_{−L}^{L} W (0, g (s)) + |g′ (s)|^{2} ds) H^{N−1} (ω) . (5.29)

Next we claim that

lim_{δ→0^{+}} lim_{ε→0^{+}} ∥u_{ε,δ} − u∥_{W^{1,p}(Ω;ℝ^{d})} = 0  (5.30)

for any 1 ≤ p < ∞. Indeed, by (5.24) and (5.26) we have

∫_{Ω} |∇u_{ε,δ} − ∇u|^{p} dx ≤ ε^{p}C_{δ}H^{N−1} (ω_{2δ}^{+}) 2h + H^{N−1} (ω_{2δ}^{+}\ω_{δ}^{+}) 2L_{1}ε |εC_{δ} + C|^{p} → 0 as ε → 0.

Since u_{ε,δ} (x) = u (x) for |x_{N}| ≥ 2h, by Poincaré’s inequality we have that u_{ε,δ} → u in W^{1,p} (Ω; ℝ^{d}) .

To conclude the proof of this step, in view of Proposition 5.3 for every k ∈ ℕ consider a piecewise C^{1} curve g_{k} : [−L, L] → ℝ^{d}, with g_{k} (L) = −g_{k} (−L) = a, such that

∫_{−L}^{L} W (0, g_{k} (s)) + |g′_{k} (s)|^{2} ds ≤ K^{∗} + \frac{1}{k} .  (5.31)

If we denote by u_{ε,δ,k} the function defined in (5.16) and corresponding to g_{k}, by (5.29)–(5.31) we have

lim sup_{k→∞} lim sup_{δ→0^{+}} lim sup_{ε→0^{+}} ∫_{Ω} \frac{1}{ε}W (∇u_{ε,δ,k}) + ε |∇^{2}u_{ε,δ,k}|^{2} dx ≤ K^{∗}H^{N−1} (ω) ,  (5.32)

and

lim_{k→∞} lim_{δ→0^{+}} lim_{ε→0^{+}} ∥u_{ε,δ,k} − u∥_{W^{1,p}(Ω;ℝ^{d})} = 0.  (5.33)

On the other hand, Theorem 4.1 entails

lim inf_{k→∞} lim inf_{δ→0^{+}} lim inf_{ε→0^{+}} ∫_{Ω} \frac{1}{ε}W (∇u_{ε,δ,k}) + ε |∇^{2}u_{ε,δ,k}|^{2} dx ≥ K^{∗}H^{N−1} (ω) .

This, together with (5.32) and (5.33), allow us to diagonalize the triple-indexed sequence u_{ε,δ,k} to obtain v_{ε} := u_{ε,δ(ε),k(δ(ε))} satisfying

lim_{ε→0^{+}} ∥v_{ε} − u∥_{W^{1,p}(Ω;ℝ^{d})} = 0

and

lim_{ε→0^{+}} ∫_{Ω} \frac{1}{ε}W (∇v_{ε}) + ε |∇^{2}v_{ε}|^{2} dx = K^{∗}H^{N−1} (ω) .

The case where the function u has the form

u (x) = − |x_{N}| a a.e. in Ω,

may be treated similarly. We omit the details. Step 2: Finitely many interfaces. Assume that the number of interfaces is finite, that is

S (∇u) ∩ Ω = ⋃_{i=1}^{m} Ω_{l_{i}},

where Ω_{l_{i}} := {x = (x′, x_{N}) ∈ Ω : x_{N} = l_{i}} = ω_{i} × {l_{i}} , for some finite family l_{1} < ⋯ < l_{m}.

Fix 0 < h < \frac{1}{4} min \{l_{i+1} − l_{i} : i = 1, ⋯, m − 1\} and consider a piecewise C^{1} curve g : [−L, L] → ℝ^{d} as in Step 1. We may now apply the construction of Step 1 to each interface, precisely we define

u_{ε,δ} (x) := \{ \begin{matrix}ψ_{δ,n} (x) z_{ε,i} (x) + (1 − ψ_{δ,i} (x)) u (x) & for |x_{N} − l_{i}| < 2h, i = 1, ⋯, m − 1, \\ u (x) & otherwise, \end{matrix}

where now ψ_{δ,i} is a smooth cut-off function such that ψ_{δ,i} = 1 in ω_{δ,i}^{+} × (l_{i} − \frac{h}{3}, l_{i} + \frac{h}{3}) and ψ_{δ,i} = 0 outside ω_{2δ,i}^{+} × (l_{i} − \frac{h}{2}, l_{i} + \frac{h}{2}) , where for s > 0 we denote ω_{s,i}^{+} := \{x′ ∈ ℝ^{N−1} : dist (x′, ω_{i}) < s\} , and

z_{ε,i} (x) := u (x′, l_{i}) + \{ \begin{matrix}∫_{0}^{x_{N} −l_{i}} g (\frac{s}{ε}) ds & if ∇u (x) = a ⊗ e_{N}, \\ − ∫_{0}^{−x_{N} +l_{i}} g (\frac{s}{ε}) ds & if ∇u (x) = −a ⊗ e_{N}. \end{matrix}

As in Step 1, the argument leading to (5.28) now yields

∫_{Ω} \frac{1}{ε}W (∇u_{ε,δ}) + ε |∇^{2}u_{ε,δ}|^{2} dx ≤ (∫_{−L}^{L} W (0, g (s)) + |g′ (s)|^{2} ds) ∑_{i=1}^{m} H^{N−1} (ω_{δ,i}^{+})

+ C (\frac{1}{ε}η (εC_{δ}) + C_{δ}ε^{3}) + C (η (εC_{δ} + C) + ε^{2}C_{δ} + C) ∑_{i=1}^{m} H^{N−1} (ω_{2δ,i}^{+}\ω_{δ,i}^{+})

and the proof is concluded as before upon letting first ε → 0^{+}, then δ → 0^{+}, and finally k → +∞ where we consider in place of g a realizing sequence {g_{k}} for K^{∗}. Step 3: Countably many interfaces. If the number of interfaces is infinite, we may proceed exactly as in Step 2 of Theorem 5.5. We omit the details.

5.4 General domains

Let τ : ℝ → ℝ be an odd, C^{∞} function, with τ (0) = τ′ (0) = 0, and such that τ (t) = t if |t| ≥ 1. For ε > 0 define

u_{ε}^{AA} (x) := ετ (\frac{x_{N}}{ε} ) a, u_{ε}^{BB} (x) := −u_{ε}^{AA} (x) , u_{ε}^{AB} (x) := ετ (\frac{|x_{N}|}{ε}) a, u_{ε}^{BA} (x) := −u_{ε}^{AB} (x) .

Proposition 5.7 (Lateral matching) Assume that Ω = ω × (−h, h) , where ω ⊂ ℝ^{N−1} is a bounded, open connected set with H^{N−1} (∂ω) = 0, and let W satisfy (H_{1}) , (H_{2})′ and (H_{3}) . Suppose, in addition, that W is differentiable at A and B. Let u ∈ W^{1,1} (Ω; ℝ^{d}) , with ∇u ∈ BV (Ω; {A, B}) . Let {ε_{n}} ⊂ ℝ_{+} be a sequence converging to zero. Then there exists {u_{n}} ⊂ W^{2,2} (Ω; ℝ^{d}) such that u_{n} → u in W^{1,p} (Ω; ℝ^{d}) for all p ∈ [1, +∞), u_{n} = u nearby x_{N} = ±h,

u_{n} (x) = u_{ε_{n}}^{F G} (x) + u (x′, 0) nearby ∂ω × (−h, h) where ∇u = \{ \begin{matrix}F & if x_{N} > 0, \\ G & if x_{N} < 0, \end{matrix} F, G ∈ \{A, B\} ,

and

lim_{n→+∞} ∫_{Ω} \frac{1}{ε_{n}} W (∇u_{n}) + ε_{n} |∇^{2}u_{n}|^{2} dx = K^{∗}H^{N−1} (S (∇u) ∩ Ω) .

Proof. In light of the argument used in Remark 5.4, we claim that given an arbitrary sequence {ε_{n}} converging to 0^{+}, it suffices to construct a double indexed sequence {u_{n,k}} satisfying the prescribed boundary conditions, such that

lim_{k→+∞} lim_{n→+∞} ∥u_{k,n} − u∥_{L^{1}(Ω;ℝ^{d})} = 0,

and

lim sup_{k→+∞} lim sup_{n→+∞} ∫_{Ω} \frac{1}{ε_{n}} W (∇u_{k,n}) + ε |∇^{2}u_{k,n}|^{2} dx ≤ K^{∗}H^{N−1} (ω) .

We fix one such sequence {ε_{n}}, abbreviated ε := ε_{n}, and we divide the proof leading to the construction of the double indexed sequence u_{k,n} into two steps corresponding to the cases where ∇u has either no interfaces or one interface in the cylinder Ω. Step 1: Assume that ∇u ≡ sgn (x_{N}) a ⊗ e_{N} in Ω. Without loss of generality we may take u (x) = |x_{N}| a. We proceed as in Step 1 of the proof of Theorem 5.6 until (5.16) which should now be replaced by

u_{ε,δ} (x) := ψ_{δ} (x) z_{ε} (x) + (1 − ψ_{δ} (x)) u_{ε}^{AB},  (5.34)

where δ stands for the elements δ_{m} of a sequence converging to 0^{+} as m → +∞, z_{ε} (x) := ∫_{0}^{x_{N}} g (\frac{s}{ε}) ds as before, ψ_{δ} ∈ C_{c}^{∞} (ℝ^{N}; [0, 1]) is a a smooth cut-off function such that ψ_{δ} = 1 in ω_{2δ}^{−} × (−\frac{h}{3}, \frac{h}{3}) and ψ_{δ} = 0 Figure 6: Construction for one interface in Step 1 of Proposition 5.7. The shaded region represents {0 < ψ_{δ} < 1}, and corresponds to A_{δ}^{−}\A_{2δ}^{−} where A_{δ}^{−} := ω_{δ}^{−} × (−h/2, h/2) and A_{2δ}^{−} := ω_{2δ}^{−} × (−h/3, h/3). The function u_{ε,δ} coincides with u_{ε}^{AB} outside A_{δ}^{−} and with z_{ε} inside A_{2δ}^{−}. A similar construction is used in Step 2, with u_{ε}^{AB} replaced by u_{ε}^{AA}, and z_{ε} replaced by u.

outside ω_{δ}^{−} × (−\frac{h}{2}, \frac{h}{2}) , where for s > 0 we denote ω_{s}^{−} := \{x′ ∈ ω : dist (x′, ∂ω) > s\} (see Figure 6). In turn (5.17) becomes

∫_{Ω} \frac{1}{ε}W (∇u_{ε,δ}) + ε |∇^{2}u_{ε,δ}|^{2} dx = ∫_{\{ψ_{δ}=1\}} \frac{1}{ε}W (∇z_{ε}) + ε |∇^{2}z_{ε}|^{2} dx  (5.35)

+ ∫_{\{ψ_{δ}=0\}} \frac{1}{ε}W (∇u_{ε}^{AB}) + ε |∇^{2}u_{ε}^{AB}|^{2} dx

+ ∫_{\{0<ψ_{δ}<1\}} \frac{1}{ε}W (∇u_{ε,δ}) + ε |∇^{2}u_{ε,δ}|^{2} dx.

Then (5.18) should be replaced by

∫_{\{ψ_{δ}=1\}} \frac{1}{ε}W (∇z_{ε}) + ε |∇^{2}z_{ε}|^{2} dx ≤ H^{N−1} (ω_{2δ}^{−}) ∫_{−L}^{L} W (0, g (s)) + |g′ (s)|^{2} ds,  (5.36)

(5.21) continues to hold, while

∫_{\{ψ_{δ}=0\}} \frac{1}{ε}W (∇u_{ε}^{AB}) + ε |∇^{2}u_{ε}^{AB}|^{2} dx ≤ H^{N−1} (ω\ω_{δ}^{−}) ∫_{−1}^{1} W (0, τ′ (|t|) sgn (t) a) + |τ′′ (|t|)|^{2} dt (5.37)
≤ CH^{N−1} (ω\ω_{δ}^{−}) .

To estimate the right hand side of (5.21) we replace (5.22) with

|∇u_{ε,δ} − ∇u| ≤ |∇ψ_{δ}| |z_{ε} − u_{ε}^{AB}| + ψ_{δ} |∇z_{ε} − ∇u| + (1 − ψ_{δ}) |∇u_{ε}^{AB} − ∇u| ,  (5.38)
|∇^{2}u_{ε,δ}| ≤ |∇^{2}ψ_{δ}| |z_{ε} − u_{ε}^{AB}| + 2 |∇ψ_{δ}| |∇z_{ε} − ∇u_{ε}^{AB}| + ψ_{δ} |∇^{2}z_{ε}| + (1 − ψ_{δ}) |∇^{2}u_{ε}^{AB}| .

Let L_{1} := max {1, L} . Then for |x_{N}| ≥ εL_{1} we have u_{ε}^{AB} = |x_{N}| a and so from (5.23) and (5.38) the bound (5.24) continues to hold for |x_{N}| ≥ εL_{1}. Moreover

|∇ (u_{ε}^{AB} − u)| ≤ |τ′ (±\frac{x_{N}}{ε} ) a| + ∥∇u∥_{∞} ≤ C, |∇^{2}u_{ε}^{AB}| ≤ \frac{1}{ε} |τ′′ (±\frac{x_{N}}{ε} )| ≤ \frac{C}{ε}  (5.39)

for 0 < |x_{N}| ≤ εL_{1}, and thus

|z_{ε} (x) − u_{ε}^{AB} (x)| ≤ |z_{ε} (x′, εL_{1}) − u_{ε}^{AB} (x, εL_{1})| + ∫_{−εL_{1}}^{εL_{1}} |∇ (z_{ε} − u_{ε}^{AB})| dx_{N} ≤ εC.  (5.40)

Hence, from (5.25), (5.38), (5.39) and (5.40) the estimate (5.26) is still valid for |x_{N}| ≤ εL_{1}, while (5.27) becomes

∫_{\{0<ψ_{δ}<1\}} \frac{1}{ε}η (|∇u_{ε,δ} − ∇u|) + ε |∇^{2}u_{ε,δ}|^{2} dx ≤ (\frac{1}{ε}η (εC_{δ}) + C_{δ}ε^{3}) H^{N−1} (ω) 2h  (5.41)

+ H^{N−1} (ω_{δ}^{−}\ω_{2δ}^{−}) 2L_{1}ε (\frac{1}{ε}η (εC_{δ} + C) + ε (C_{δ} + \frac{C}{ε^{2}} )) .

In view of (5.35), (5.37), (5.36), (5.21) and (5.41), we now have

∫_{Ω} \frac{1}{ε}W (∇u_{ε,δ}) + ε |∇^{2}u_{ε,δ}|^{2} dx ≤CH^{N−1} (ω\ω_{2δ}^{−}) + (∫_{−L}^{L} W (0, g (s)) + |g′ (s)|^{2} ds) H^{N−1} (ω)

+ C (\frac{1}{ε}η (εC_{δ}) + C_{δ}ε^{3})

+ C (η (εC_{δ} + C) + ε^{2}C_{δ} + C) H^{N−1} (ω_{δ}^{−}\ω_{2δ}^{−}) ,

and letting ε → 0^{+} and then δ → 0^{+} yields, by (5.19),

lim sup_{δ→0^{+}} lim sup_{ε→0^{+}} ∫_{Ω} \frac{1}{ε}W (∇u_{ε,δ}) + ε |∇^{2}u_{ε,δ}|^{2} dx ≤ H^{N−1} (ω) ∫_{−L}^{L} W (0, g (s)) + |g′ (s)|^{2} ds.

We can now continue the argument of Step 1 in the proof of Theorem 5.6 from (5.29) onwards. Step 2: Assume that ∇u ≡ A = a ⊗ e_{N} in Ω. Without loss of generality we may assume that u (x) = ax_{N}. Fix δ > 0 and let ψ_{δ} be defined as in Step 1. Define

u_{ε,δ} (x) := ψ_{δ} (x) u (x) + (1 − ψ_{δ} (x)) u_{ε}^{AA} (x) .

For |x_{N}| ≥ ε we have u_{ε}^{AA} (x) = u (x) and so u_{ε,δ} (x) = u (x) . Since ψ_{δ} = 1 in ω_{2δ}^{−} × (−\frac{h}{3}, \frac{h}{3}) and ψ_{δ} = 0 outside ω_{δ}^{−} × (−\frac{h}{2}, \frac{h}{2}) , for 0 < ε < h we have

∫_{Ω} \frac{1}{ε}W (∇u_{ε,δ}) + ε |∇^{2}u_{ε,δ}|^{2} dx ≤ ∫_{(ω\ω_{δ}^{−})×(−ε,ε)} \frac{1}{ε}W (∇u_{ε}^{AA}) + ε |∇^{2}u_{ε}^{AA}|^{2} dx  (5.42)

+ ∫_{(ω_{δ}^{−}\ω_{2δ}^{−})×(−ε,ε)} \frac{1}{ε}W (∇u_{ε,δ}) + ε |∇^{2}u_{ε,δ}|^{2} dx

≤ CH^{N−1} (ω\ω_{δ}^{−}) + ∫_{(ω_{δ}^{−}\ω_{2δ}^{−})×(−ε,ε)} \frac{1}{ε}η (|∇u_{ε,δ} − a ⊗ e_{N}|) + ε |∇^{2}u_{ε,δ}|^{2} dx,

where we used (5.37) (which continues to hold, provided we replace the derivatives of τ (|·|) with those of τ (·)) and (5.20).

To estimate the last integral on the right hand side of (5.42) note that

|∇u_{ε,δ} (x) − ∇u (x)| ≤ |∇ψ_{δ} (x)| |u_{ε}^{AA} (x) − u (x)| + ψ_{δ} (x) |∇u_{ε}^{AA} (x) − ∇u (x)| ,  (5.43)
|∇^{2}u_{ε,δ} (x)| ≤ |∇^{2}ψ_{δ} (x)| |u_{ε}^{AA} (x) − u (x)| + 2 |∇ψ_{δ} (x)| |∇u_{ε}^{AA} (x) − ∇u (x)| + ψ_{δ} (x) |∇^{2}u_{ε}^{AA} (x)| .

The bounds (5.39) and (5.40) are still valid for |x_{N}| ≤ ε, with L_{1} := 1 and u_{ε}^{AB} replaced by u_{ε}^{AA}. Hence from (5.43) we deduce that (5.26) holds for |x_{N}| ≤ ε, and thus by (5.42)

∫_{Ω} \frac{1}{ε}W (∇u_{ε,δ}) + ε |∇^{2}u_{ε,δ}|^{2} dx ≤ CH^{N−1} (ω\ω_{δ}^{−}) + H^{N−1} (ω_{δ}^{−}\ω_{2δ}^{−}) 2 (η (εC_{δ} + C) + ε^{2}C_{δ} + C) ,

and by (5.19), letting ε → 0^{+} and then δ → 0^{+} yields

lim sup_{δ→0^{+}} lim sup_{ε→0^{+}} ∫_{Ω} \frac{1}{ε}W (∇u_{ε,δ}) + ε |∇^{2}u_{ε,δ}|^{2} dx = 0.

The argument of Remark 5.4 brings the proof of this case to a closure. The remaining two cases where ∇u ≡ −a ⊗ e_{N} in Ω and ∇u ≡ − sgn (x_{N}) a ⊗ e_{N} in Ω are treated in a way similar to Steps 1 and 2. We omit the details.

In preparation for the main result of this section, Theorem 5.10, we establish the following inequality for level sets.

Lemma 5.8 For each t ∈ ℝ let Ω_{t} := {(x′, x_{N}) ∈ Ω : x_{N} = t} denote a horizontal interface of Ω. Then

lim sup_{ε→0^{+}} \frac{1}{2ε} ∫_{−ε}^{ε} H^{N−1} (Ω_{s+t}) ds ≤ H^{N−1} (\overline{Ω}_{t}) .

Proof. Write \overline{Ω}_{t} = \overline{ω} × \{t\}, fix δ > 0, and consider an open set Σ ⊃⊃ \overline{ω} such that

H^{N−1} (Σ × \{t\}) ≤ H^{N−1} (\overline{Ω}_{t}) + δ.

We claim that if ε is sufficiently small then

{x ∈ Ω : |x_{N} − t| < ε} ⊂ Σ × (t − ε, t + ε) .

Indeed, if this was not the case then there would exist a sequence {x_{n}} ⊂ Ω, with x_{n} = (x′_{n}, (x_{n})_{N}) , such that

(x_{n})_{N} → t and x′_{n} ∉ Σ.  (5.44)

By extracting a subsequence, if necessary, we may assume that x_{n} → (x′, t) . But then (x′, t) ∈ \overline{Ω}_{t} ⊂ Σ×\{t\} , which is in contradiction with (5.44) since Σ is open. Hence the claim holds, and in turn by Fubini’s Theorem

∫_{−ε}^{ε} H^{N−1} (Ω_{s+t}) ds = |{x ∈ Ω : |x_{N} − t| < ε}| ≤ |Σ × (t − ε, t + ε)| = 2εH^{N−1} (Σ × {t}) .

Hence

\frac{1}{2ε} ∫_{−ε}^{ε} H^{N−1} (Ω_{s+t}) ds ≤ H^{N−1} (Σ × \{t\}) ≤ H^{N−1} (\overline{Ω}_{t}) + δ,

and by letting first ε → 0^{+} and then δ → 0^{+} we conclude the proof.

The next result is a generalization of the Isoperimetric Inequality (see [4, 14]).

Theorem 5.9 Let Ω ⊂ ℝ^{N} be an open, bounded connected domain with Lipschitz boundary. Then there exists a constant C_{iso}, depending on N and on Ω, such that for every set E ⊂ Ω of finite perimeter there holds

min \{|E| , |Ω\E|\}^{1− \frac{1}{N}} ≤ C_{iso} Per_{Ω}(E).  (5.45)

Proof. By Poincaré’s inequality there exists a constant C (N, Ω) such that

∫_{Ω} |v − v_{Ω}|^{N/(N−1)} dx ≤ C |Dv| (Ω)

for all v ∈ BV (Ω; ℝ) , where

v_{Ω} := \frac{1}{|Ω|} ∫_{Ω} v dx

and |Dv| (Ω) is the total variation of Dv. If we take v := χ_{E} then v_{Ω} := \frac{|E|}{|Ω|} and so

C Per_{Ω}(E) = C |Dv| (Ω) ≥ ∫_{Ω} |v − v_{Ω}|^{N/(N−1)} dx = ∫_{Ω} |χ_{E} − \frac{|E|}{|Ω|}|^{N/(N−1)} dx
= |Ω|^{−N/(N−1)} (|Ω| − |E|)^{N/(N−1)} |E| + |Ω|^{−N/(N−1)} |E|^{N/(N−1)} |Ω\E|
≥ |Ω|^{−N/(N−1)} min \{|E| , |Ω\E|\}^{1− \frac{1}{N}} (|E| + |Ω\E|)
= |Ω|^{−1/(N−1)} min \{|E| , |Ω\E|\}^{1− \frac{1}{N}} .

Setting C_{iso} := C(N, Ω) |Ω|^{1/(N−1)} we conclude the proof.

Theorem 5.10 Let Ω ⊂ ℝ^{N} be an open, bounded, simply connected domain with Lipschitz boundary. Assume that W satisfies the conditions (H_{1}) , (H_{2})′ and (H_{3}) . Suppose, in addition, that W is differentiable at A and B. Let u ∈ W^{1,1} (Ω; ℝ^{d}) , with ∇u ∈ BV (Ω; {A, B}) . Then

Γ − lim_{ε→0^{+}} I_{ε} (u; Ω) = K^{∗} Per_{Ω}(E),

where ∇u(x) = (1 − χ_{E} (x)) A + χ_{E} (x) B for L^{N} a.e. x ∈ Ω.

Proof. Just as in the proofs of the previous Γ-limit results, in view of Theorem 4.1 and of Remark 5.4 it suffices to show that given an arbitrary sequence {ε_{n}} converging to 0^{+} we may construct a double indexed sequence {u_{n,k}} such that

lim_{k→+∞} lim_{n→+∞} ∥u_{k,n} − u∥_{L^{1}(Ω;ℝ^{d})} = 0,

and

lim sup_{k→+∞} lim sup_{n→+∞} ∫_{Ω} \frac{1}{ε_{n}} W (∇u_{k,n}) + ε |∇^{2}u_{k,n}|^{2} dx ≤ K^{∗}H^{N−1} (ω) .

We fix one such sequence {ε_{n}}, abbreviated ε := ε_{n}, and we divide the proof leading to the construction of the double indexed sequence u_{k,n} into four steps. Step 1: One hyperplane– case 1. We assume first that

S (∇u) ∩ Ω ⊂ Ω_{0} := {x = (x′, x_{N}) ∈ Ω : x_{N} = 0}

and that (see Figure 1)

t ↦ H^{N−1} (Ω_{t}) is continuous at t = 0.  (5.46)

Then we may write

Ω_{0} = ⋃_{i=1}^{∞} ω_{i} × {0} ,

where the open sets ω_{i} ⊂ ℝ^{N−1} are pairwise disjoint and connected. Since Ω is bounded we know that

∑_{i=1}^{∞} H^{N−1} (ω_{i} × {0}) < ∞.

For fixed δ > 0, standing for an arbitrary element of a sequence {δ_{k}} converging to 0^{+}, choose M > 1 so large that

∑_{i=M+1}^{∞} H^{N−1} (ω_{i} × {0}) < δ,  (5.47)

Figure 7: Construction for the case of one hyperplane, with continuously varying interface area, in Step 1 of Theorem 5.10. The functions u_{ε,δ}^{1} and u_{ε,δ}^{2} used in the two cylinders have been constructed in Proposition 5.7, and on the cylinder boundaries agree with u_{ε}^{AB} and u_{ε}^{BB}, respectively. The shaded region represents the set E_{ε,δ}.

and for each i = 1, ⋯, M, let ω_{i,δ} ⊂⊂ ω_{i} be such that

H^{N−1} ((ω_{i}\ω_{i,δ}) × \{0\}) < \frac{δ}{M} .  (5.48)

Since ω_{i,δ} × {0} ⊂⊂ Ω there exists h > 0 such that the cylinders ω_{i,δ} × (−h, h) ⊂⊂ Ω for each i = 1, ⋯, M (see Figure 7). In each cylinder ω_{i,δ} × (−h, h) we may apply Proposition 5.7 to obtain sequences {u_{ε,δ}^{i}} ⊂

W^{2,2} (ω_{i,δ} × (−h, h) ; ℝ^{d}) such that u_{ε,δ}^{i} → u in W^{1,2} (ω_{i,δ} × (−h, h) ; ℝ^{d}) , u_{ε,δ}^{i} = u nearby x_{N} = ±h,

u_{ε,δ}^{i} (x) = u_{ε}^{F G} (x) + u (x′, 0) nearby ∂ω_{i,δ} × (−h, h) , where ∇u = \{ \begin{matrix}F & in ω_{i,δ} × (0, h) , \\ G & in ω_{i,δ} × (−h, 0) , \end{matrix} F, G ∈ \{A, B\} ,

and

lim_{ε→0^{+}} ∫_{ω_{i,δ}×(−h,h)} \frac{1}{ε}W (∇u_{ε,δ}^{i}) + ε |∇^{2}u_{ε,δ}^{i}|^{2} dx = K^{∗}H^{N−1} (S (∇u) ∩ (ω_{i,δ} × \{0\})) .  (5.49)

Define

v_{ε,δ} (x) := \{ \begin{matrix}u_{ε,δ}^{i} (x) & if x ∈ ω_{i,δ} × (0, h) , i = 1, ⋯, M, \\ u_{ε}^{AA} (x) + u (x′, 0) & if x ∈ Ω\ (⋃_{i=1}^{M} ω_{i,δ} × (−h, h)) and ∇u (x) = A, \\ u_{ε}^{BB} (x) + u (x′, 0) & if x ∈ Ω\ (⋃_{i=1}^{M} ω_{i,δ} × (−h, h)) and ∇u (x) = B. \end{matrix}  (5.50)

We claim that {v_{ε,δ}} ⊂ W^{2,∞} (Ω; ℝ^{d}) , v_{ε,δ} → u in L^{1} (Ω; ℝ^{d}) and

lim sup_{δ→0^{+}} lim sup_{ε→0^{+}} ∫_{Ω} \frac{1}{ε}W (∇v_{ε,δ}) + ε |∇^{2}v_{ε,δ}|^{2} dx ≤ K^{∗}H^{N−1} (S (∇u) ∩ Ω) .

Define

E_{ε,δ} := {x ∈ Ω\ (⋃_{i=1}^{M} ω_{i,δ} × (−h, h)) : |x_{N}| < ε} .

Since u_{ε}^{F G} (x) + u (x′, 0) = u (x) for |x_{N}| ≥ ε we have

∫_{Ω} \frac{1}{ε}W (∇v_{ε,δ}) + ε |∇^{2}v_{ε,δ}|^{2} dx = ∑_{i=1}^{M} ∫_{ω_{i,δ}×(−h,h)} \frac{1}{ε}W (∇u_{ε,δ}^{i}) + ε |∇^{2}u_{ε,δ}^{i}|^{2} dx  (5.51)

+ ∫_{E_{ε,δ}} \frac{1}{ε}W (∇v_{ε,δ}) + ε |∇^{2}v_{ε,δ}|^{2} dx

= ∑_{i=1}^{M} ∫_{ω_{i,δ}×(−h,h)} \frac{1}{ε}W (∇u_{ε,δ}^{i}) + ε |∇^{2}u_{ε,δ}^{i}|^{2} dx

+ \frac{1}{ε} ∫_{E_{ε,δ}} W (0, ±τ′ (±\frac{x_{N}}{ε} ) a) + |τ′′ (±\frac{x_{N}}{ε} )|^{2} dx.

By Fubini’s Theorem we have

\frac{1}{ε} ∫_{E_{ε,δ}} W (0, ±τ′ (±\frac{x_{N}}{ε} ) a) + |τ′′ (±\frac{x_{N}}{ε} )|^{2} dx ≤ C \frac{1}{ε} ∫_{−ε}^{ε} (H^{N−1} (Ω_{s}) − ∑_{i=1}^{M} L^{N−1} (ω_{i,δ})) ds

= C (\frac{1}{ε} ∫_{−ε}^{ε} H^{N−1} (Ω_{s}) ds − 2 ∑_{i=1}^{M} L^{N−1} (ω_{i,δ}))

→ 2C (H^{N−1} (Ω_{0}) − ∑_{i=1}^{M} H^{N−1} (ω_{i,δ} × {0}))

as ε → 0^{+}, and where we have used (5.46). By (5.47), (5.48), (5.49) and (5.51) we obtain

lim sup_{ε→0^{+}} ∫_{Ω} \frac{1}{ε}W (∇v_{ε,δ}) + ε |∇^{2}v_{ε,δ}|^{2} dx ≤ K^{∗} ∑_{i=1}^{M} H^{N−1} (S (∇u) ∩ (ω_{i,δ} × \{0\}))

+ C ∑_{i=M+1}^{∞} H^{N−1} (ω_{i} × {0}) + Cδ

≤ K^{∗}H^{N−1} (S (∇u) ∩ Ω) + Cδ.

It now suffices to let δ → 0^{+}. Step 2: One hyperplane– case 2. Next we remove condition (5.46), and thus we only assume that (see Figure 2)

S (∇u) ∩ Ω ⊂ Ω_{0} := {x = (x′, x_{N}) ∈ Ω : x_{N} = 0} .

Write

∂Ω ∩ {x = (x′, x_{N}) ∈ ℝ^{N} : x_{N} = 0} =: (Ξ × {0}) ∪ E,

where Ξ is an open subset of ℝ^{N−1} and H^{N−1} (E) = 0. Fix δ > 0 and consider a smooth cut-off function ψ_{δ} such that ψ_{δ} = 1 in Ξ_{2δ}^{−} × (−\frac{h}{3}, \frac{h}{3}) and ψ_{δ} = 0 outside Ξ_{δ}^{−} × (−\frac{h}{2}, \frac{h}{2}) , where for s > 0 we denote Ξ_{s}^{−} := {x′ ∈ Ξ : dist (x′, ∂Ξ) > s} (see Figure 8). For x ∈ Ω define

w_{ε,δ} (x) := ψ_{δ} (x) u (x) + (1 − ψ_{δ} (x)) v_{ε,δ},

where \{v_{ε,δ}\} is the sequence defined in (5.50). Note that u ∈ W^{2,2} (Ω ∩ Ξ_{δ}^{−} × (−\frac{h}{2}, \frac{h}{2}); ℝ^{d}) . Set

F_{ε,δ} := {x ∈ Ω\ ((Ξ_{δ}^{−} × (−h, h)) ∪ ⋃_{i=1}^{M} (ω_{i,δ} × (−h, h))) : |x_{N}| < ε} .

Figure 8: Construction for one hyperplane in Step 2 of Theorem 5.10. The function v_{ε,δ}, which is used everywhere except for the set B_{δ}^{−} := Ξ_{δ}^{−} × (−h/2, h/2), is the one constructed in Step 1. Inside B_{2δ}^{−} := Ξ_{2δ}^{−} × (−h/3, h/3), the limit u is used directly. The set F_{ε,δ} is the shaded region. The two boxes ω_{1,δ}^{−} ×(−h/2, h/2) and ω_{2,δ}^{−} × (−h/2, h/2) are also shown.

Using (5.51) we have

∫_{Ω} \frac{1}{ε}W (∇w_{ε,δ}) + ε |∇^{2}w_{ε,δ}|^{2} dx = ∫_{\{ψ_{δ}=0\}} \frac{1}{ε}W (∇v_{ε,δ}) + ε |∇^{2}v_{ε,δ}|^{2} dx

+ ∫_{\{0<ψ_{δ}<1\}} \frac{1}{ε}W (∇w_{ε,δ}) + ε |∇^{2}w_{ε,δ}|^{2} dx

≤ ∑_{i=1}^{M} ∫_{ω_{i,δ}×(−h,h)} \frac{1}{ε}W (∇u_{ε,δ}^{i}) + ε |∇^{2}u_{ε,δ}^{i}|^{2} dx + \frac{1}{ε} ∫_{F_{ε,δ}} W (0, ±τ′ (±\frac{x_{N}}{ε} ) a) + |τ′′ (±\frac{x_{N}}{ε} )|^{2} dx (5.52)

+ ∫_{\{0<ψ_{δ}<1\}} \frac{1}{ε}W (∇w_{ε,δ}) + ε |∇^{2}w_{ε,δ}|^{2} dx.

The second integral on the right hand side of (5.52) may be estimated as before to obtain

lim sup_{ε→0^{+}} \frac{1}{ε} ∫_{F_{ε,δ}} W (0, ±τ′ (±\frac{x_{N}}{ε} ) a) + |τ′′ (±\frac{x_{N}}{ε} )|^{2} dx  (5.53)

≤ lim sup_{ε→0^{+}} C \frac{1}{ε} ∫_{−ε}^{ε} (H^{N−1} (Ω_{s}) − L^{N−1} (Ξ_{δ}^{−}) − ∑_{i=1}^{M} L^{N−1} (ω_{i,δ})) ds

= C lim sup_{ε→0^{+}} (\frac{1}{ε} ∫_{−ε}^{ε} H^{N−1} (Ω_{s}) ds − 2L^{N−1} (Ξ_{δ}^{−}) − 2 ∑_{i=1}^{M} L^{N−1} (ω_{i,δ}))

≤ 2C (H^{N−1} (\overline{Ω}_{0}) − L^{N−1} (Ξ_{δ}^{−}) − ∑_{i=1}^{M} H^{N−1} (ω_{i,δ} × \{0\}))

≤ 2C (H^{N−1} ((Ξ\Ξ_{δ}^{−}) × {0}) + ∑_{i=M+1}^{∞} H^{N−1} (ω_{i} × {0}) + δ) ,

where we have used Lemma 5.8.

The estimate of the third integral on the right hand side of (5.52) is very similar to the proof of Step 2 of Proposition 5.7. Indeed, for |x_{N}| ≥ ε we have u_{ε}^{F G} (x) + u (x′, 0) = u (x) and so w_{ε,δ} = u in the set {x ∈ Ω : 0 < ψ_{δ} (x) < 1, |x_{N}| ≥ ε} . Hence

∫_{\{0<ψ_{δ}<1\}} \frac{1}{ε}W (∇w_{ε,δ}) + ε |∇^{2}w_{ε,δ}|^{2} dx ≤ ∫_{(Ξ_{δ}^{−}\Ξ_{2δ}^{−})×(−ε,ε)} \frac{1}{ε}η (|∇w_{ε,δ} − ∇u|) + ε |∇_{ε,δ}^{2}w|^{2} dx, (5.54)

where we used (5.20) and we have extended u to all of ℝ^{N} as an affine function. The estimates (5.38), (5.39) and (5.40) continue to hold for x ∈ (Ξ_{δ}^{−}\Ξ_{2δ}^{−}) × (−ε, ε), with u_{ε}^{AB} replaced by u_{ε}^{F G}. Hence from (5.38) we deduce that (5.26) is still valid for x ∈ (Ξ_{δ}^{−}\Ξ_{2δ}^{−}) × (−ε, ε) , and thus by (5.52)-(5.54) we obtain

lim sup_{ε→0^{+}} ∫_{Ω} \frac{1}{ε}W (∇w_{ε,δ}) + ε |∇^{2}w_{ε,δ}|^{2} dx ≤ K^{∗}H^{N−1} (S (∇u) ∩ Ω) + Cδ + H^{N−1} (Ξ_{δ}^{−}\Ξ_{2δ}^{−}) 2 (η (C) + C) .

Letting δ → 0^{+} concludes the proof. Step 3: Finitely many hyperplanes. Assume that

S (∇u) ∩ Ω ⊂ ⋃_{i=1}^{M} Ω_{l_{i}},

where Ω_{l_{i}} := {x = (x′, x_{N}) ∈ Ω : x_{N} = l_{i}} = ω_{i} × {l_{i}} for some finite family l_{1} < ⋯ < l_{M}.

Fix 0 < h < \frac{1}{2} min \{l_{i+1} − l_{i} : i = 1, ⋯, M − 1\} and let \{ε\} ⊂ ℝ_{+} be a sequence converging to zero. Define

Ω^{i} := {x ∈ Ω : |x_{N} − l_{i}| < 2h} .

By Step 2 applied to Ω^{i} we may find sequences {u_{ε}^{i}} ⊂ W^{2,2} (Ω^{i}; ℝ^{d}) such that u_{ε}^{i} → u in W^{1,2} (Ω^{i}; ℝ^{d}) , u_{ε}^{i} = u nearby x_{N} = l_{i} ± h, and

lim_{ε→0^{+}} ∫_{Ω^{i}} \frac{1}{ε}W (∇u_{ε}^{i}) + ε |∇^{2}u_{ε}^{i}|^{2} dx = K^{∗}H^{N−1} (S (∇u) ∩ Ω^{i}) .

It suffices to define

u_{ε} (x) := \{ \begin{matrix}u_{ε}^{i} (x) & if x ∈ Ω^{i}, i = 1, ⋯, M, \\ u (x) & otherwise. \end{matrix}

Then {u_{ε}} ⊂ W^{2,2} (Ω; ℝ^{d}) , u_{ε} → u in W^{1,2} (Ω; ℝ^{d}) , and

lim_{ε→0^{+}} ∫_{Ω} \frac{1}{ε}W (∇u_{ε}) + ε |∇^{2}u_{ε}|^{2} dx = K^{∗}H^{N−1} (S (∇u) ∩ Ω) .

Step 4: Countably many hyperplanes. Assume that

S (∇u) ∩ Ω = ⋃_{i=1}^{∞} ω_{i} × {l_{i}} ,

where ω_{i} ⊂ ℝ^{N−1} are connected open sets with ∂ (ω_{i} × {l_{i}}) ⊂ ∂Ω and {l_{i}} is a sequence of real numbers (not necessarily distinct). Fix 0 < δ < min \{1, \frac{1}{2} |Ω|\} . Since

H^{N−1} (S (∇u) ∩ Ω) = ∑_{i=1}^{∞} H^{N−1} (ω_{i} × {l_{i}}) < ∞,  (5.55)

there exists an integer M = M (δ) such that

∑_{i=M+1}^{∞} H^{N−1} (ω_{i} × \{l_{i}\}) < \frac{1}{c_{0}} δ < 1,

where c_{0} := max \{1, Ciso^{\frac{N}{N−1}} \} and C_{iso} is the isoperimetric constant introduced in (5.45). As Ω is simply

connected, for each i = M + 1, ⋯, the set Ω\ (ω_{i} × {l_{i}}) may be written as the union of two open connected disjoint sets E_{i} and Ω\E_{i}, where |E_{i}| ≥ \frac{1}{2} |Ω| . By Theorem 5.9

min \{|E_{i}| , |Ω\E_{i}|\} ≤ c_{0}H^{N−1} (∂E_{i} ∩ Ω)^{\frac{N}{N−1}} = c_{0}H^{N−1} (ω_{i} × \{l_{i}\})^{\frac{N}{N−1}} < δ < \frac{1}{2} |Ω| ,

and so |Ω\E_{i}| ≤ c_{0}H^{N−1} (ω_{i} × \{l_{i}\})^{\frac{N}{N−1}} . Set u^{M} := u, and for each i = M + 1, ⋯, define u^{i} as u^{i−1} in the set E_{i}, while we extend u^{i} as an affine function outside E_{i}. Thus ∇u^{i} is continuous across ω_{i} × {l_{i}} and

S (∇u^{i}) ∩ Ω ⊂ ( ⋃_{j=1}^{M} ω_{j} × {l_{j}}) ∪ ( ⋃_{j=i+1}^{∞} ω_{j} × {l_{j}}) .  (5.56)

Clearly ∇u^{i} ∈ BV (Ω; {A, B}) and by (5.55) and (5.56)

sup_{i} H^{N−1} (S (∇u^{i}) ∩ Ω) < ∞.

In addition, u^{i} = u on Ω\ (⋃_{j=M+1}^{∞} (Ω\E_{j})) = ⋂_{j=M+1}^{∞} E_{j}, with

| ⋂_{j=M+1}^{∞} E_{j}| ≥ |Ω| − c_{0} ∑_{j=M+1}^{∞} H^{N−1} (ω_{j} × \{l_{j}\})^{\frac{N}{N−1}} ≥ |Ω| − c_{0} ∑_{j=M+1}^{∞} H^{N−1} (ω_{j} × \{l_{j}\}) ≥ \frac{|Ω|}{2}. (5.57)

By Poincaré’s inequality we can extract a subsequence (not relabelled) converging in W^{1,1} (Ω; ℝ^{d}) to a function v^{M}, with ∇v^{M} ∈ BV (Ω; {A, B}) ,

v^{M} = u on ⋂_{i=M+1}^{∞} E_{i},  (5.58)

and

S (∇v^{M}) ∩ Ω ⊂ ⋃_{j=1}^{M} ω_{j} × {l_{j}} .

In order to assert the latter inclusion, consider a point x_{0} ∉ ⋃_{j=1}^{M} ω_{j} × {l_{j}} , and find r > 0 such that

Q (x_{0}, r) ⊂ Ω\ ( ⋃_{j=1}^{M} ω_{j} × {l_{j}}) .

If i is large enough so that H^{N−1} (ω_{i} × {l_{i}}) < r^{N−1} then, clearly, and in light of (5.56),

Q (x_{0}, r) ∩ S (∇u^{i}) = ∅.

We deduce, therefore, that ∇u^{i} ∈ W^{1,∞} (Q (x_{0}, r) ; ℝ^{d×N}) and thus ∇v^{M} ∈ W^{1,∞} (Q (x_{0}, r) ; ℝ^{d×N}) . In particular, x_{0} ∉ S (∇v^{M}) .

Let {ε} ⊂ ℝ_{+} be a sequence converging to zero. By Step 3 we may find sequences {v_{ε}^{M}} ⊂ W^{2,2} (Ω; ℝ^{d}) such that v_{ε}^{M} → v^{M} in W^{1,2} (Ω; ℝ^{d}) as ε → 0^{+}, and

lim_{ε→0^{+}} ∫_{Ω} \frac{1}{ε}W (∇v_{ε}^{M}) + ε |∇^{2}v_{ε}^{M} |^{2} dx = K^{∗}H^{N−1} (S (∇v^{M}) ∩ Ω) ≤ K^{∗} ∑_{i=1}^{M} H^{N−1} (ω_{i} × \{l_{i}\}) .

In turn

lim sup_{M→∞} lim_{ε→0^{+}} ∫_{Ω} \frac{1}{ε}W (∇v_{ε}^{M}) + ε |∇^{2}v_{ε}^{M} |^{2} dx ≤ K^{∗} ∑_{i=1}^{∞} H^{N−1} (ω_{i} × \{l_{i}\})
= K^{∗}H^{N−1} (S (∇u) ∩ Ω) .

Since v_{ε}^{M} → v^{M} in W^{1,2} (Ω; ℝ^{d}) as ε → 0^{+}, by means of a standard diagonalization process it suffices to prove that v^{M} → u in L^{1} (Ω; ℝ^{d}) as M → ∞. By construction we have

∫_{Ω} |∇v^{M} − ∇u| dx ≤ ∫_{⋃_{i=M+1}^{∞} Ω\E_{i}} |∇v^{M} − ∇u| dx

≤ 2 |a ⊗ e_{N}| | ⋃_{i=M+1}^{∞} Ω\E_{i}| ≤ C ∑_{i=M+1}^{∞} |Ω\E_{i}|

≤ C ∑_{i=M+1}^{∞} |Ω\E_{i}|^{1− \frac{1}{N}} ≤ C ∑_{i=M+1}^{∞} H^{N−1} (ω_{i} × \{l_{i}\}) ≤ Cδ,

and by Poincaré’s inequality and (5.58) we obtain ∫_{Ω} |v^{M} − u| dx ≤ C̃ ∫_{Ω} |∇v^{M} − ∇u| dx ≤ C̃δ,

where the Poincaré constant C̃ may be taken independently of δ in view of (5.57). It now suffices to let δ → 0^{+}.

6 Γ−limsup: the upper bound. The symmetry hypotheses.

We introduce the notation

ξ = (ξ_{1}, ⋯, ξ_{N}) ∈ \underbrace{ℝ^{d} × ⋯ × ℝ^{d}}_{N times}, ξ′ = (ξ_{1}, ⋯, ξ_{N} − 1) ∈ \underbrace{ℝ^{d} × ⋯ × ℝ^{d}}_{N−1 times},

so that ξ = (ξ′, ξ_{N}) ∈ ℝ^{d×(N−1)} × ℝ^{d}. Throughout this section we assume that (H_{1}) W is continuous, W (ξ) = 0 if and only if ξ ∈ {A, B} , where A = −B = a ⊗ e_{N}, for some a ∈ ℝ^{d} \ {0}; (H_{2})′′ there exist an exponent p ≥ 2 and a constant C > 1 such that

\frac{1}{C} |ξ|^{p} − C ≤ W (ξ) ≤ C (|ξ|^{p} + 1)

for all ξ ∈ ℝ^{d×N}; (H_{4}) there exist constants ρ, γ > 0 such that

\frac{1}{γ} |ξ − A|^{p} ≤ W (ξ) ≤ γ |ξ − A|^{p} if |ξ − A| ≤ ρ,
\frac{1}{γ} |ξ − B|^{p} ≤ W (ξ) ≤ γ |ξ − B|^{p} if |ξ − B| ≤ ρ.

(H_{5}) W is even in each variable ξ_{i}, i = 1, ⋯, N −1, that is W (ξ_{1}, ⋯, −ξ_{i}, ⋯, ξ_{N}) = W (ξ_{1}, ⋯, ξ_{i}, ⋯, ξ_{N})

for each i = 1, ⋯, N − 1. Hypothesis (H_{4}) may be improved as (H_{4})′ there exist a constant ρ > 0 and a convex function g : [0, ∞) → [0, ∞), with g (s) = 0 if and only if

s = 0, such that g is derivable in s = 0,

g (2t) ≤ cg (t) ,  (6.1)

for all 0 ≤ t ≤ ρ,

g (|ξ − A|) ≤ W (ξ) ≤ cg (|ξ − A|) if |ξ − A| ≤ ρ,

and

g (|ξ − B|) ≤ W (ξ) ≤ cg (|ξ − B|) if |ξ − B| ≤ ρ,

for some constant c > 0.

Condition (6.1) is called the doubling condition – it prevents g to be too degenerate near t = 0, precisely, it is satisfied if g (t) ∼const. t^{p} as t → 0^{+}, for some p ≥ 1, while it does not hold if g grows exponentially near the origin, i.e., g (t) ∼const. e^{−1/t^{2}} as t → 0^{+}.

Remark 6.1 In what follows, and without loss of generality, we will consider the model case where A = −B = a ⊗ e_{N}. It is easy to check that (H_{2})′′ and (H_{4}) yield

W (ξ) ≥ C_{1}|ξ′|^{p}

for all ξ ∈ ℝ^{d×N} and for some constant C_{1} > 0. Moreover, we claim that

W (ξ) ≤ C_{2} (W (η) + |ξ − η|^{p}) ,  (6.2)

for all ξ, η ∈ ℝ^{d×N}, and for some constant C_{2} > 0. Indeed, assume by contradiction that (6.2) does not hold. Then there exist two sequences {ξ_{n}} , {η_{n}} ⊂ ℝ^{d×N} such that

W (ξ_{n}) > n (W (η_{n}) + |ξ_{n} − η_{n}|^{p}) .  (6.3)

We have by (H_{2})′′

C (|ξ_{n}|^{p} + 1) ≥ W (ξ_{n}) > n (W (η_{n}) + |ξ_{n} − η_{n}|^{p}) ≥ n (\frac{1}{C} |η_{n}|^{p} − C + \frac{1}{C} |ξ_{n} − η_{n}|^{p})

≥ n (\frac{1}{2^{p−1}C} |ξ_{n}|^{p} − C) ,

where we have used the inequality |a|^{p} + |b|^{p} ≥ \frac{1}{2^{p−1}} |a − b|^{p} . This clearly implies that the sequence \{ξ_{n}\} is bounded, and by (6.3) it follows that W (η_{n}) + |ξ_{n} − η_{n}|^{p} → 0. In view of (H_{1}) we may assume, without loss of generality, that η_{n} → A. For n sufficiently large the quantities |ξ_{n} − A| , |η_{n} − A| are so small that (H_{2}) may be applied, and (6.3) yields

γ |ξ_{n} − A|^{p} ≥ W (ξ_{n}) > n (W (η_{n}) + |ξ_{n} − η_{n}|^{p})

≥ n (\frac{1}{γ} |η_{n} − A|^{p} + |ξ_{n} − η_{n}|^{p})

≥ \frac{1}{2^{p−1}} n min \{1, \frac{1}{γ} \} |ξ_{n} − A|^{p} ,

which is clearly a contradiction for n large.

For simplicity we will present the proof of the analog of the results of Section 5 first under hypothesis (H_{4}), and then on Section 7 we move on to the general case where (H_{4})′ holds.

6.1 Characterization of K^{∗}

In this subsection we prove that under conditions (H_{1}) , (H_{2})′′, (H_{4}) and (H_{5})

K^{∗} = inf \{∫_{Q} L W (∇v) + \frac{1}{L} |∇^{2}v|^{2} dx : L > 0, v ∈ W^{2,∞} (Q; ℝ^{d}) ,
∇v = ±a ⊗ e_{N} nearby x_{N} = ±\frac{1}{2}, v periodic of period one in x′\} .

The next two propositions will establish that, for cubes, realizing sequences may be taken periodic in the transversal directions, and that there is a matching of vertical boundary conditions, precisely:

Proposition 6.2 (Vertical matching) Assume that W satisfies (H_{1}) , (H_{2})′′ and (H_{4}) . Then there exist sequences {ε_{n}} ⊂ ℝ_{+}, {c_{n}} ⊂ ℝ and {z_{n}} ⊂ W^{2,2} (Q; ℝ^{d}) , such that ε_{n} → 0^{+}, c_{n} → 0, z_{n} → |x_{N}| a in W^{1,p} (Q; ℝ^{d}) ,

z_{n} (x) = −x_{N} a nearby x_{N} = −\frac{1}{2}, z_{n} (x) = x_{N} a + c_{n} nearby x_{N} = \frac{1}{2},  (6.4)

and

lim_{n→+∞} ∫_{Q} \frac{1}{ε_{n}} W (∇z_{n}) + ε_{n} |∇^{2}z_{n}|^{2} dx = K^{∗}.

Proof. By definition of K^{∗} there exist sequences {ε_{n}} ⊂ ℝ_{+}, {u_{n}} ⊂ W^{2,2} (Q; ℝ^{d}) , such that ε_{n} → 0^{+}, u_{n} → u_{0} := |x_{N}| a in L^{1} (Q; ℝ^{d}) and

lim_{n→+∞} ∫_{Q} \frac{1}{ε_{n}} W (∇u_{n}) + ε_{n} |∇^{2}u_{n}|^{2} dx = K^{∗}.

We abbreviate ε := ε_{n} and u_{ε} := u_{n}. Due to Theorem 3.1 and Remark 3.2 (ii), we may assume, up to extraction of a subsequence, that u_{ε} → u in W^{1,p} (Q; ℝ^{d}) . Partition Q′ × (\frac{1}{6}, \frac{1}{3}) into [\frac{1}{ε}] horizontal layers

of height [\frac{1}{ε}]^{−1} \frac{1}{6}. In view of Remark 4.4, choose one such layer, L_{ε} = Q′ × (θ_{ε} − [\frac{1}{ε}]^{−1} \frac{1}{6}, θ_{ε}) , such that

[\frac{1}{ε}] ∫_{L_{ε}} \frac{1}{ε}W (∇u_{ε}) + ε |∇^{2}u_{ε}|^{2} + |∇u_{ε} − a ⊗ e_{N}|^{p} + |u_{ε} − u_{0}|^{p} dx  (6.5)

≤ ∫_{Q′×(\frac{1}{6} , \frac{1}{3})} \frac{1}{ε}W (∇u_{ε}) + ε |∇^{2}u_{ε}|^{2} + |∇u_{ε} − a ⊗ e_{N}|^{p} + |u_{ε} − u_{0}|^{p} dx =: α_{ε} → 0.

In L_{ε} select a height z_{ε} ∈ (θ_{ε} − [\frac{1}{ε}]^{−1} \frac{1}{6}, θ_{ε}) such that

∫_{Q′} \frac{1}{ε}W (∇u_{ε} (x′, z_{ε})) + ε |∇^{2}u_{ε} (x′, z_{ε})|^{2} + |∇u_{ε} (x′, z_{ε}) − a ⊗ e_{N}|^{p}  (6.6)
+ |u_{ε} (x′, z_{ε}) − u_{0} (x′, z_{ε})|^{p} dx′ ≤ 6α_{ε}.

First matching: Set

v_{ε} (x) := u_{0} (x) + ū_{ε} (x) + φ_{ε} (x_{N}) (u_{ε} (x) − u_{0} (x) − ū_{ε} (x)) ,

where ū_{ε} (x) := u_{ε} (x′, z_{ε}) − ũ (z_{ε}), u_{0}(x) =: ũ(x_{N}), and let φ_{ε} be a smooth cut-off function such that \{0 < φ_{ε} < 1\} ⊂ L_{ε}, φ_{ε} = 1 if x_{N} < θ_{ε} − [\frac{1}{ε}]^{−1} \frac{1}{6}, φ_{ε} = 0 if x_{N} > θ_{ε}, and

∥φ′_{ε}∥_{∞} ≤ \frac{c}{ε}, ∥φ′′_{ε}∥_{∞} ≤ \frac{c}{ε^{2}} .

We claim that (i) ∫_{L_{ε}} |v_{ε} − u_{0}|^{p} dx → 0; (ii) \frac{1}{ε} ∫_{L_{ε}} |∇v_{ε} − a ⊗ e_{N}|^{p} dx → 0; (iii) ∫_{L_{ε}} \frac{1}{ε}W (∇v_{ε}) dx → 0; (iv) ∫_{L_{ε}} ε |∇^{2}v_{ε}|^{2} dx → 0.

It is easy to deduce (i) from (6.5) and (6.6). Now

\frac{1}{ε} ∫_{L_{ε}} |∇v_{ε} − a ⊗ e_{N}|^{p} dx ≤ C ∫_{L_{ε}} \frac{1}{ε} |∇_{x′}u_{ε} (x′, z_{ε})|^{p} + \frac{1}{ε} |∇u_{ε} − a ⊗ e_{N}|^{p} + \frac{1}{ε^{p+1}} |u_{ε} − u_{0} − ū_{ε}|^{p} dx (6.7)

≤ C \{∫_{Q′} |∇_{x′}u_{ε} (x′, z_{ε})|^{p} dx′ + \frac{1}{ε} ∫_{L_{ε}} |∇u_{ε} − a ⊗ e_{N}|^{p} dx\}

≤ C \{∫_{Q′} |∇u_{ε} (x′, z_{ε}) − a ⊗ e_{N}|^{p} dx′ + \frac{1}{ε} ∫_{L_{ε}} |∇u_{ε} − a ⊗ e_{N}|^{p} dx\} → 0

by (6.5) and (6.6), where we have invoked the Poincaré’s inequality

∫_{L_{ε}} |u_{ε} − u_{0} − ū_{ε}|^{p} dx ≤ Cε^{p} ∫_{L_{ε}} |∇ (u_{ε} − a ⊗ e_{N})|^{p} dx  (6.8)

due to the fact that (u_{ε} − u_{0} − ū_{ε}) (x′, z_{ε}) ≡ 0, and using the identity |ξ − ∇u_{0}|^{p} = (|ξ′|^{2} + |ξ_{N} ± a|^{2})^{p/2}. In addition,

\frac{1}{ε} ∫_{L_{ε}} W (∇v_{ε}) dx ≤ \frac{1}{ε} ∫_{L_{ε}∩\{|∇v_{ε}−∇u_{0}|<ρ\}} C |∇v_{ε} − a ⊗ e_{N}|^{p} dx + \frac{1}{ε} ∫_{L_{ε}∩\{|∇v_{ε}−∇u_{0}|≥ρ\}} C (1 + |∇v_{ε}|^{p}) dx

≤ \frac{C}{ε} ∫_{L_{ε}} |∇v_{ε} − a ⊗ e_{N}|^{p} dx,

where we have used (H_{2})′′, (H_{4}), and the fact that

L^{N} (L_{ε} ∩ \{|∇v_{ε} − a ⊗ e_{N}| ≥ ρ\}) ≤ \frac{1}{ρ^{p}} ∫_{L_{ε}} |∇v_{ε} − a ⊗ e_{N}|^{p} dx.

By (ii) we easily deduce (iii).

Finally,

∫_{L_{ε}} ε |∇^{2}v_{ε}|^{2} dx ≤C ∫_{L_{ε}} ε |∇_{x′}^{2}u_{ε} (x′, z_{ε})|^{2} + ε |∇^{2}u_{ε}|^{2} + \frac{1}{ε} |∇u_{ε} − a ⊗ e_{N}|^{2}

+ \frac{1}{ε} |∇_{x′}u_{ε} (x′, z_{ε})|^{2} + \frac{1}{ε^{3}} |u_{ε} − u_{0} − ū_{ε}|^{2} dx

≤C \{ε^{2} ∫_{Q′} |∇_{x′}^{2}u_{ε} (x′, z_{ε})|^{2} dx′ + ε ∫_{L_{ε}} |∇^{2}u_{ε}|^{2} dx + ∫_{L_{ε}} \frac{1}{ε} |∇u_{ε} − a ⊗ e_{N}|^{2} dx

+ ∫_{Q′} |∇_{x′}u_{ε} (x′, z_{ε})|^{2} dx′ + \frac{1}{ε^{3}} ε^{(p−2)/p} (∫_{L_{ε}} |u_{ε} − u_{0} − ū_{ε}|^{p} dx)^{2/p}\}

≤C \{ε^{2} ∫_{Q′} |∇_{x′}^{2}u_{ε} (x′, z_{ε})|^{2} dx′ + ε ∫_{L_{ε}} |∇^{2}u_{ε}|^{2} dx + (\frac{1}{ε} ∫_{L_{ε}} |∇u_{ε} − a ⊗ e_{N}|^{p} dx)^{2/p}

+ (∫_{Q′} |∇u_{ε} (x′, z_{ε}) − a ⊗ e_{N}|^{p} dx′)^{2/p} dx′} → 0

by (6.5), (6.6), Hölder’s inequality, and (6.7), (6.8).

Second matching: Set

w_{ε} (x) := u_{0} + c_{ε} + ψ (x_{N}) (ū_{ε} − c_{ε}) ,

where, by (6.6),

c_{ε} := ∫_{Q′} ū_{ε} (x′, z_{ε}) dx′ → 0,

and ψ is a smooth cut-off function such that ψ = 1 nearby x_{N} = θ_{ε} (< \frac{1}{3}) , ψ = 0 nearby x_{N} = \frac{1}{2}, and

∥ψ′∥_{∞} ≤ c, ∥ψ′′∥_{∞} ≤ c.

We claim that

(i) ∫_{Q′×(θ_{ε}, \frac{1}{2})} |w_{ε} − u_{0}|^{p} dx → 0;

(ii) ∫_{Q′×(θ_{ε}, \frac{1}{2})} \frac{1}{ε}W (∇w_{ε}) dx → 0 or, as seen before, \frac{1}{ε} ∫_{Q′×(θ_{ε}, \frac{1}{2})} |∇w_{ε} − a ⊗ e_{N}|^{p} dx → 0;

(iii) ∫_{Q′×(θ_{ε}, \frac{1}{2})} ε |∇^{2}w_{ε}|^{2} dx → 0.

It is clear that (i) is a consequence of (6.6). To prove (ii) and (iii) we notice that

\frac{1}{ε} ∫_{Q′×(θ_{ε}, \frac{1}{2})} |∇w_{ε} − a ⊗ e_{N}|^{p} dx ≤ C \frac{1}{ε} ∫_{Q′×(θ_{ε}, \frac{1}{2})} |∇_{x′}u_{ε} (x′, z_{ε})|^{p} + |ū_{ε} − c_{ε}|^{p} dx

≤ C \frac{1}{ε} ∫_{Q′} |∇_{x′}u_{ε} (x′, z_{ε})|^{p} dx′

≤ C ∫_{Q′} \frac{1}{ε}W (∇u_{ε} (x′, z_{ε})) dx′ → 0

by (6.6), and where we have used Poincaré-Friedrichs’s inequality and Remark 6.1. Furthermore, also by (6.6), and using Hölder’s inequality

∫_{Q′×(θ_{ε}, \frac{1}{2})} ε |∇^{2}w_{ε}|^{2} dx ≤ C ∫_{Q′×(θ_{ε}, \frac{1}{2})} ε |∇_{x′}^{2} ū_{ε} (x′, z_{ε})|^{2} + ε |∇_{x′}u_{ε} (x′, z_{ε})|^{2} dx

≤ Cε {∫_{Q′} |∇_{x′}^{2}u_{ε} (x′, z_{ε})|^{2} dx′ + (∫_{Q′} |∇u_{ε} (x′, z_{ε}) − a ⊗ e_{N}|^{p} dx′)^{2/p}} → 0.

To conclude the proof, note that the sequence

U_{ε} := \{ \begin{matrix}u_{ε} & if x_{N} < θ_{ε} − [\frac{1}{ε}]^{−1} \frac{1}{6}, \\ v_{ε} & if θ_{ε} − [\frac{1}{ε}]^{−1} \frac{1}{6} ≤ x_{N} ≤ θ_{ε}, \\ w_{ε} & if x_{N} > θ_{ε}, \end{matrix}

satisfies condition (6.4) with

lim sup_{ε→0^{+}} ∫_{Q} \frac{1}{ε}W (∇U_{ε}) + ε |∇^{2}U_{ε}|^{2} dx ≤ lim_{ε→0^{+}} ∫_{Q} \frac{1}{ε}W (∇u_{ε}) + ε |∇^{2}u_{ε}|^{2} dx = K^{∗}.

This procedure pins down the boundary conditions at x_{N} = 0, and nearby x_{N} = \frac{1}{2} we have U_{ε} = u_{0} + c_{ε}. Now we repeat the argument in Q′ × (−\frac{1}{2}, 0) with the obvious adaptations, in order to change U_{ε} on the bottom half of the cylinder so that the new field V_{ε} is equal to u_{0} + c_{ε}^{+} on top and to u_{0} + c_{ε}^{−} on bottom. It suffices to set z_{ε} := V_{ε} − C_{ε}, with C_{ε} := c_{ε}^{+} − c_{ε}^{−} and to invoke Theorem 4.1.

Proposition 6.3 (Transversal Periodicity) Assume that W satisfies conditions (H_{1}) , (H_{2})′′, (H_{4}) and (H_{5}) . Then there exist sequences {ε_{n}} ⊂ ℝ_{+}, {u_{n}} ⊂ W^{2,∞} (Q; ℝ^{d}) , such that ε_{n} → 0^{+}, u_{n} → |x_{N}| a in L^{1} (Q; ℝ^{d}) , ∇u_{n} = ±a ⊗ e_{N} nearby x_{N} = ±\frac{1}{2} (resp.), u_{n} is periodic of period one in x′, and

lim inf_{n→+∞} ∫_{Q} \frac{1}{ε_{n}} W (∇u_{n}) + ε_{n} |∇^{2}u_{n}|^{2} dx = K^{∗}.

Proof. We claim that we may find sequences {ε_{n}} ⊂ ℝ_{+}, {v_{n}} ⊂ W_{loc}^{2,∞} (ℝ^{N}; ℝ^{d}) , such that ε_{n} → 0^{+}, v_{n} (·, x_{N}) is 2Q′-periodic for all x_{N} ∈ ℝ, ∇v_{n} = ±a ⊗ e_{N} nearby x_{N} = ±\frac{1}{2} (resp.), and

lim_{n→∞} ∫_{2Q′×(− \frac{1}{2} , \frac{1}{2})} \frac{1}{ε_{n}} W (∇v_{n}) + ε_{n} |∇^{2}v_{n}|^{2} dx = 2^{N−1}K^{∗},

lim_{n→∞} ∫_{2Q′×(− \frac{1}{2} , \frac{1}{2})} |v_{n}(x) − |x_{N}|a| dx = 0.

If the claim holds, then extend v_{n} linearly to 2Q and define u_{n} (x) := \frac{1}{2}v_{n} (2x) for x ∈ 2Q. Then \{u_{n}\} ⊂ W_{loc}^{2,∞} (ℝ^{N}; ℝ^{d}) , u_{n} (·, x_{N}) is Q′-periodic for all x_{N} ∈ ℝ, ∇u_{n} = ±a ⊗ e_{N} nearby x_{N} = ±\frac{1}{2} (resp.), and

lim_{n→∞} ∫_{Q} \frac{2}{ε_{n}} W (∇u_{n}) + \frac{ε_{n}}{2} |∇^{2}u_{n}|^{2} dx = K^{∗},

lim_{n→∞} ∫_{Q} |u_{n}(x) − |x_{N}|a| dx = 0,

thus completing the proof.

We divide the proof of the claim in two steps, where, as before, for simplicity of notation we write ε := ε_{n}. Step 1: The two dimensional case N = 2. In view of Lemma 4.2, consider sequences {ε} ⊂ ℝ_{+}, {u_{ε}} ⊂ W^{2,2} (Q; ℝ^{d}) , such that ε → 0^{+}, u_{ε} → u_{0} := |x_{2}| a in L^{1} (Q; ℝ^{d}) , and

lim_{ε→0^{+}} ∫_{Q} \frac{1}{ε}W (∇u_{ε}) + ε |∇^{2}u_{ε}|^{2} dx = K^{∗}.

By (H_{2})′′ we may assume, without loss of generality, that {u_{ε}} ⊂ W^{2,2} (Q; ℝ^{d}) ∩ C^{2} (Q; ℝ^{d}) , and by Proposition 6.2 that ∇u_{ε} (x) = ± a ⊗ e_{N} nearby x_{2} = ±\frac{1}{2} (resp.). By Theorem 4.1 we have

K^{∗} = lim_{ε→0^{+}} ∫_{Q} \frac{1}{ε}W (∇u_{ε}) + ε |∇^{2}u_{ε}|^{2} dx ≥ lim inf_{ε→0^{+}} ∫_{Q\I_{m}} \frac{1}{ε}W (∇u_{ε}) + ε |∇^{2}u_{ε}|^{2} dx

≥ K^{∗}H^{1} ((−\frac{1}{2} + \frac{1}{m}, \frac{1}{2} − \frac{1}{m}) × \{0\}) = K^{∗} (1 − \frac{2}{m}) ,

where I_{m} := ((−\frac{1}{2}, −\frac{1}{2} + \frac{1}{m}) ∪ (\frac{1}{2} − \frac{1}{m}, \frac{1}{2})) × (−\frac{1}{2}, \frac{1}{2}), and so

lim sup_{ε→0^{+}} ∫_{I_{m}} \frac{1}{ε}W (∇u_{ε}) + ε |∇^{2}u_{ε}|^{2} dx ≤ K^{∗} \frac{2}{m}.

Divide (−\frac{1}{2}, −\frac{1}{2} + \frac{1}{m})×(−\frac{1}{2}, \frac{1}{2}) into [\frac{1}{ε}] vertical strips of horizontal width \frac{1}{m} [\frac{1}{ε}]^{−1} , and proceed symmetrically in (\frac{1}{2} − \frac{1}{m}, \frac{1}{2})×(−\frac{1}{2}, \frac{1}{2}) . Order these strips in pairs (R_{ε,m,i}^{−}, R_{ε,m,i}^{+}) with R_{ε,m,i}^{+} ⊂ (\frac{1}{2} − \frac{1}{m}, \frac{1}{2})×(−\frac{1}{2}, \frac{1}{2}) , R_{ε,m,i}^{−} ⊂ (−\frac{1}{2}, −\frac{1}{2} + \frac{1}{m}) × (−\frac{1}{2}, \frac{1}{2}). Then for all ε > 0 sufficiently small we have

\begin{matrix}[ & \frac{1}{ε} & ] \\ ∑ \\ i=1 \end{matrix} ∫_{R_{ε,m,i}^{+}∪R_{ε,m,i}^{−}} \frac{1}{ε}W (∇u_{ε}) + ε |∇^{2}u_{ε}|^{2} + m^{p} |∇u_{ε} − ∇u_{0}|^{p} + |u_{ε} − u_{0}| dx ≤ K^{∗} \frac{3}{m},  (6.9)

where we have used the fact that u_{ε} → u_{0} in W^{1,p} (Q; ℝ^{d}) (see Theorem 3.1 and Remark 3.2 (ii)). Choose one pair (R_{ε,m,i}^{−}, R_{ε,m,i}^{+}) , with i = i (ε, m) , such that

∫_{R_{ε,m,i}^{+}∪R_{ε,m,i}^{−}} \frac{1}{ε}W (∇u_{ε}) + ε |∇^{2}u_{ε}|^{2} + m^{p} |∇u_{ε} − ∇u_{0}|^{p} + |u_{ε} − u_{0}| dx ≤ K^{∗} \frac{3}{m} [\frac{1}{ε}]^{−1} .  (6.10)

For simplicity, from now on we denote R_{ε,m,i}^{+} =: R_{ε,m}^{+} = (b_{ε,m}, c_{ε,m}) × (−\frac{1}{2}, \frac{1}{2}) and R_{ε,m,i}^{−} =: R_{ε,m}^{−} = (−c_{ε,m}, −b_{ε,m}) × (−\frac{1}{2}, \frac{1}{2}) . Since

∫ c_{ε,m}_{\frac{bε,m+cε,m}{2}} ∫− \frac{1}{2}^{\frac{1}{2}} [\frac{1}{ε}W (∇u_{ε} (x)) + ε |∇^{2}u_{ε} (x)|^{2} + m^{p} |∇ (u_{ε} − u_{0})|^{p} (x) + |(u_{ε} − u_{0}) (x)|

+ \frac{1}{ε}W (∇u_{ε} (−x_{1}, x_{2})) + ε |∇^{2}u_{ε} (−x_{1}, x_{2})|^{2} + m^{p} |∇ (u_{ε} − u_{0}) (−x_{1}, x_{2})|^{p}

+ |(u_{ε} − u_{0}) (−x_{1}, x_{2})| ] dx ≤ K^{∗} \frac{3}{m} [\frac{1}{ε}]^{−1} ,

with c_{ε,m} − \frac{b_{ε,m}+c_{ε,m}}{2} = \frac{1}{2} \frac{1}{m} [\frac{1}{ε}]^{−1} , there exists a_{ε,m} ∈ (\frac{b_{ε,m}+c_{ε,m}}{2} , c_{ε,m}) such that

∫− \frac{1}{2}^{\frac{1}{2}} [\frac{1}{ε}W (∇u_{ε} (a_{ε,m}, x_{2})) + ε |∇^{2}u_{ε} (a_{ε,m}, x_{2})|^{2} + |(u_{ε} − u_{0}) (a_{ε,m}, x_{2})|  (6.11)

+ m^{p} |∇ (u_{ε} − u_{0}) (a_{ε,m}, x_{2})|^{p} + m^{p} |∇ (u_{ε} − u_{0}) (−a_{ε,m}, x_{2})|^{p}

+ \frac{1}{ε}W (∇u_{ε} (−a_{ε,m}, x_{2})) + ε |∇^{2}u_{ε} (−a_{ε,m}, x_{2})|^{2} + |(u_{ε} − u_{0}) (−a_{ε,m}, x_{2})|] dx_{2} ≤ 6K^{∗}.

Now (−a_{ε,m}, −a_{ε,m} + \frac{ε}{2m}) ⊂ (−c_{ε,m}, −b_{ε,m}) because a_{ε,m} ∈ (\frac{b_{ε,m}+c_{ε,m}}{2} , c_{ε,m}) and \frac{b_{ε,m}+c_{ε,m}}{2} − b_{ε,m} =

\frac{1}{2} \frac{1}{m} [\frac{1}{ε}]^{−1} . We will now modify u_{ε} on (−a_{ε,m}, −b_{ε,m}) × (−\frac{1}{2}, \frac{1}{2}) so that the new sequence will match u_{ε} near x_{1} = −a_{ε,m} + \frac{ε}{2m}, hence near −b_{ε,m}, and will coincide with u_{ε} (−a_{ε,m}, ·) near x_{1} = −a_{ε,m}. Let φ_{ε,m} be a smooth cut-off function such that φ_{ε,m} = 1 if x_{1} > −a_{ε,m} + \frac{ε}{2m}, φ_{ε,m} = 0 if x_{1} < −a_{ε,m}, and

∥φ′_{ε,m}∥_{∞} ≤ \frac{cm}{ε}, ∥φ′′_{ε,m}∥_{∞} ≤ \frac{cm^{2}}{ε^{2}} .

Define

w_{ε,m} (x) := φ_{ε,m} (x_{1}) u_{ε} (x) + (1 − φ_{ε,m} (x_{1})) u_{ε} (−a_{ε,m}, x_{2}) .

Then \{w_{ε,m}\} ⊂ W^{2,∞} (Q; ℝ^{d}) , and ∇w_{ε,m} (x) = ± a ⊗ e_{N} nearby x_{2} = ±\frac{1}{2} (resp.). We show that

lim sup_{m→∞} lim sup_{ε→0^{+}} ∫_{Q} \frac{1}{ε}W (∇w_{ε,m}) + ε |∇^{2}w_{ε,m}|^{2} dx ≤ K^{∗}, lim sup_{m→∞} lim sup_{ε→0^{+}} ∫_{Q} |w_{ε,m} − u_{0}| dx = 0. (6.12)

If (6.12) holds, and repeating the argument now nearby x_{1} = a_{ε,m}, then after a diagonalization procedure and invoking Theorem 4.1, we can find a subsequence ε_{m} → 0^{+} and a sequence {w_{m}} ⊂ W^{2,∞} (Q; ℝ^{d}) such that w_{m} = w_{m} (±a_{m}, x_{2}) nearby x_{1} = ±\frac{1}{2} resp, ∇w_{m} (x) = ± a ⊗ e_{N} nearby x_{2} = ±\frac{1}{2} (resp.), and

lim_{m→∞} ∫_{Q} \frac{1}{ε_{m}} W (∇w_{m}) + ε_{m} |∇^{2}w_{m}|^{2} dx = K^{∗}, lim_{m→∞} ∫_{Q} |w_{m} − u_{0}| dx = 0.

Construct by reflection about x_{1} = \frac{1}{2} a new function, still denoted w_{m}, x_{1}-periodic with period 2. Precisely, for x_{1} ∈ (\frac{1}{2}, \frac{3}{2}) , x_{2} ∈ (−\frac{1}{2}, \frac{1}{2}), set w_{m}(x_{1}, x_{2}) := w_{m}(1 − x_{1}, x_{2}). Since the problem is translation invariant, for simplicity of notation in what follows we identify w with its translation (x_{1}, x_{2}) ↦ w(x_{1} − 1/2, x_{2}), and in this way we work with periodic functions with period (−1, 1) × (−\frac{1}{2}, \frac{1}{2}), such that

lim_{m→∞} ∫_{(−1,1)×(− \frac{1}{2} , \frac{1}{2})} \frac{1}{ε_{m}} W (∇w_{m}) + ε_{m} |∇^{2}w_{m}|^{2} dx = 2K^{∗}, lim_{m→∞} ∫_{(−1,1)×(− \frac{1}{2} , \frac{1}{2})} |w_{m} − u_{0}| dx = 0.

Note that here we have used condition (H_{5}), and note also that the new function w_{m} extended to ℝ×(−\frac{1}{2}, \frac{1}{2}) is still in W_{loc}^{2,∞} (ℝ × (−\frac{1}{2}, \frac{1}{2}); ℝ^{d}) because w_{m} does not depend on x_{1} nearby the axis of reflection x_{1} = \frac{1}{2}.

The remainder of Step 1 is devoted to the proof of (6.12), where we use the notation I_{m}^{−} := (−1/2, −b_{m})× (−1/2, 1/2). We have

∫_{Q} \frac{1}{ε}W (∇w_{ε,m}) + ε |∇^{2}w_{ε,m}|^{2} dx = ∫_{Q\I_{m}^{−}} \frac{1}{ε}W (∇u_{ε}) + ε |∇^{2}u_{ε}|^{2} dx + ∫_{R_{ε,m}^{−}} \frac{1}{ε}W (∇w_{ε,m}) + ε |∇^{2}w_{ε,m}|^{2} dx

+ ∫_{I_{m}^{−}\R_{ε,m}^{−}} \frac{1}{ε}W (0, \frac{∂u_{ε}}{∂x_{2}} (−a_{ε,m}, x_{2})) + ε |\frac{∂^{2}u_{ε}}{∂x_{2}^{2}} (−a_{ε,m}, x_{2})|^{2} dx

≤ ∫_{Q} \frac{1}{ε}W (∇u_{ε}) + ε |∇^{2}u_{ε}|^{2} dx + ∫_{R_{ε,m}^{−}} \frac{1}{ε}W (∇w_{ε,m}) + ε |∇^{2}w_{ε,m}|^{2} dx

+ \frac{1}{m} ∫− \frac{1}{2}^{\frac{1}{2}} \frac{1}{ε}W (0, \frac{∂u_{ε}}{∂x_{2}} (−a_{ε,m}, x_{2})) + ε |\frac{∂^{2}u_{ε}}{∂x_{2}^{2}} (−a_{ε,m}, x_{2})|^{2} dx.

By Remark 6.1 we have

W (0, \frac{∂u_{ε}}{∂x_{2}} (−a_{ε,m}, x_{2})) ≤ CW (∇u_{ε} (−a_{ε,m}, x_{2})) + C |\frac{∂u_{ε}}{∂x_{1}} (−a_{ε,m}, x_{2})|^{p} ≤ CW (∇u_{ε} (−a_{ε,m}, x_{2})) , (6.13)

hence, by (6.11)

\frac{1}{m} ∫− \frac{1}{2}^{\frac{1}{2}} \frac{1}{ε}W (0, \frac{∂u_{ε}}{∂x_{2}} (−a_{ε,m}, x_{2})) + ε |\frac{∂^{2}u_{ε}}{∂x_{2}^{2}} (−a_{ε,m}, x_{2})|^{2} dx

≤ \frac{C}{m} ∫− \frac{1}{2}^{\frac{1}{2}} \frac{1}{ε}W (∇u_{ε} (−a_{ε,m}, x_{2})) + ε |\frac{∂^{2}u_{ε}}{∂x_{2}^{2}} (−a_{ε,m}, x_{2})|^{2} dx ≤ \frac{C}{m},

and, in turn,

∫_{Q} \frac{1}{ε}W (∇w_{ε,m}) + ε |∇^{2}w_{ε,m}|^{2} dx ≤ ∫_{Q} \frac{1}{ε}W (∇u_{ε}) + ε |∇^{2}u_{ε}|^{2} dx  (6.14)

+ ∫_{R_{ε,m}^{−}} \frac{1}{ε}W (∇w_{ε,m}) + ε |∇^{2}w_{ε,m}|^{2} dx + \frac{C}{m}.

Similarly, again by (6.11),

∫_{Q} |w_{ε,m} − u_{0}| dx ≤ ∫_{Q\I_{m}^{−}} |u_{ε} − u_{0}| dx + ∫_{R_{ε,m}^{−}} |w_{ε,m} − u_{0}| dx

+ \frac{1}{m} ∫− \frac{1}{2}^{\frac{1}{2}} |u_{ε} (−a_{ε,m}, x_{2}) − |x_{2}| a| dx

≤ ∫_{Q\I_{m}^{−}} |u_{ε} − u_{0}| dx + ∫_{R_{ε,m}^{−}} |w_{ε,m} − u_{0}| dx + \frac{C}{m},

and thus, also by (6.14), to prove (6.12) it is sufficient to show that:

(i) lim_{m→∞} lim_{ε→0^{+}} ∫_{R_{ε,m}^{−}} \frac{1}{ε}W (∇w_{ε,m}) dx = 0;

(ii) lim_{m→∞} lim_{ε→0^{+}} ∫_{R_{ε,m}^{−}} ε |∇^{2}w_{ε,m}|^{2} dx = 0;

(iii) lim_{m→∞} lim_{ε→0^{+}} ∫_{R_{ε,m}^{−}} |w_{ε,m} − u_{ε}| dx = 0.

Now

∇w_{ε,m} (x) = φ_{ε,m} (x_{1}) ∇u_{ε} (x) + (1 − φ_{ε,m} (x_{1})) (0 |\frac{∂u_{ε}}{∂x_{2}} (−a_{ε,m}, x_{2}))  (6.15)
+ (u_{ε} (x_{1}, x_{2}) − u_{ε} (−a_{ε,m}, x_{2})) ⊗ φ′_{ε,m} (x_{1}) e_{1}.

By (H_{2})′′ we have

W (ξ) ≤ C (1 + |ξ|^{p}) ≤ C (1 + 2^{p−1} |∇u_{0}|^{p} + 2^{p−1} |ξ − ∇u_{0}|^{p}) ,

and so, using (6.15),

\frac{1}{ε} ∫_{R_{ε,m}^{−}}W (∇w_{ε,m}) dx ≤ C \frac{1}{ε} ∫_{R_{ε,m}^{−}} (1 + |∇u_{0}|^{p} + |∇w_{ε,m} − ∇u_{0}|^{p}) dx

≤C \frac{1}{εm} [\frac{1}{ε}]^{−1} (1 + |a ⊗ e_{2}|^{p})

+ C \frac{1}{ε} ∫_{R_{ε,m}^{−}} |∇u_{ε} − ∇u_{0}|^{p} + |∇u_{0} − (0, \frac{∂u_{ε}}{∂x_{2}} (−a_{ε,m}, x_{2}))|^{p} dx

+ C \frac{1}{ε} ∫_{R_{ε,m}^{−}} (\frac{m}{ε})^{p} |u_{ε} (x) − u_{ε} (−a_{ε,m}, x_{2})|^{p} dx

≤C (\frac{1}{m} + \frac{1}{ε} ∫_{R_{ε,m}^{−}} |∇u_{ε} − ∇u_{0}|^{p} dx + \frac{1}{m} [\frac{1}{ε}]^{−1} \frac{1}{ε} ∫− \frac{1}{2}^{\frac{1}{2}} |∇ (u_{0} − u_{ε}) (−a_{ε,m}, x_{2})|^{p} dx_{2})

+ C \frac{m^{p}}{ε^{p+1}} ∫_{R_{ε,m}^{−}} |u_{ε} (x) − u_{ε} (−a_{ε,m}, x_{2})|^{p} dx

≤C (\frac{1}{m} + \frac{1}{ε} \frac{3K^{∗}}{m^{p+1}} [\frac{1}{ε}]^{−1} + \frac{6K^{∗}}{m^{p+1}} [\frac{1}{ε}]^{−1} \frac{1}{ε} + \frac{m^{p}}{ε^{p+1}} ∫_{R_{ε,m}^{−}} |u_{ε} (x) − u_{ε} (−a_{ε,m}, x_{2})|^{p} dx) ,

where we have used (6.9), (6.10), (6.11). Thus to prove (i) it remains to show that

lim_{m→∞} lim_{ε→0^{+}} \frac{m^{p}}{ε^{p+1}} ∫_{R_{ε,m}^{−}} |u_{ε} (x) − u_{ε} (−a_{ε,m}, x_{2})|^{p} dx = 0.

Indeed, by Hölder’s inequality

|u_{ε} (x_{1}, x_{2}) − u_{ε} (−a_{ε,m}, x_{2})|^{p} ≤ (∫_{−b_{ε,m}}^{−c_{ε,m}} |\frac{∂u_{ε}}{∂x_{1}} (s, x_{2})| ds)^{p}

≤ C (\frac{ε}{m})^{p/p′} ∫_{−b_{ε,m}}^{−c_{ε,m}} |\frac{∂u_{ε}}{∂x_{1}} (s, x_{2})|^{p} ds

≤ C (\frac{ε}{m})^{p−1} ∫_{−b_{ε,m}}^{−c_{ε,m}} |∇ (u_{ε} − u_{0}) (s, x_{2})|^{p} ds,

and thus by (6.10)

∫_{R_{ε,m}^{−}} |u_{ε} (x_{1}, x_{2}) − u_{ε} (−a_{ε,m}, x_{2})|^{p} dx ≤ C ∫_{R_{ε,m}^{−}} (\frac{ε}{m})^{p−1} ∫_{−b_{ε,m}}^{−c_{ε,m}} |∇ (u_{ε} − u_{0}) (s, x_{2})|^{p} dsdx  (6.16)

≤ C [\frac{1}{ε}]^{−1} \frac{ε^{p−1}}{m^{p}} ∫_{R_{ε,m}^{−}} |∇u_{ε} − ∇u_{0}|^{p} dx ≤ C \frac{ε^{p+1}}{m^{2p+1}}

and the (6.12) holds.

To show (ii) note that for x_{1} ∈ (−b_{ε,m}, −c_{ε,m})

|∇^{2}w_{ε,m} (x)| ≤ φ_{ε,m} (x_{1}) |∇^{2}u_{ε} (x)| + (1 − φ_{ε,m} (x_{1})) |\frac{∂^{2}u_{ε}}{∂x_{2}^{2}} (−a_{ε,m}, x_{2})|

+ |u_{ε} (x_{1}, x_{2}) − u_{ε} (−a_{ε,m}, x_{2})| |φ′′_{ε,m} (x_{1})| + 2 |φ′_{ε,m} (x_{1})| |∇u_{ε} (x) − (0 |\frac{∂u_{ε}}{∂x_{2}} (−a_{ε,m}, x_{2}))| ,

and so

|∇^{2}w_{ε,m}|^{2} ≤C (|∇^{2}u_{ε}|^{2} + |\frac{∂^{2}u_{ε}}{∂x_{2}^{2}} (−a_{ε,m}, x_{2})|^{2}  (6.17)

+ (\frac{m}{ε})^{4} |u_{ε} − u_{ε} (−a_{ε,m}, x_{2})|^{2} + (\frac{m}{ε})^{2} |∇u_{ε} − (0 |\frac{∂u_{ε}}{∂x_{2}} (−a_{ε,m}, x_{2}))|^{2}) .

We now estimate the two terms on the right hand side of (6.17). By (6.16) and Hölder’s inequality

∫_{R_{ε,m}^{−}} |u_{ε} (x_{1}, x_{2}) − u_{ε} (−a_{ε,m}, x_{2})|^{2} dx ≤ |R_{ε,m}^{−}|^{(p−2)/p} (∫_{R_{ε,m}^{−}} |u_{ε} (x_{1}, x_{2}) − u_{ε} (−a_{ε,m}, x_{2})|^{p} dx)^{2/p} (6.18)

≤ (\frac{1}{m} [\frac{1}{ε}]^{−1})^{(p−2)/p} (C \frac{ε^{p+1}}{m^{2p+1}} )^{2/p} ≤ C \frac{ε^{3}}{m^{5}} ,

while

|∇u_{ε} (x) − (0 |\frac{∂u_{ε}}{∂x_{2}} (−a_{ε,m}, x_{2}))|^{2} ≤ 2 |\frac{∂u_{ε}}{∂x_{1}} (x)|^{2} + 2 |\frac{∂u_{ε}}{∂x_{2}} (x) − \frac{∂u_{ε}}{∂x_{2}} (−a_{ε,m}, x_{2})|^{2}

≤ 2 |∇u_{ε} − ∇u_{0}|^{2} + 2 (∫_{−c_{ε,m}}^{−b_{ε,m}} |\frac{∂^{2}u_{ε}}{∂x_{2}∂x_{1}} (s, x_{2})| ds)^{2}

≤ 2 |∇u_{ε} − ∇u_{0}|^{2} + \frac{3ε}{m} ∫_{−c_{ε,m}}^{−b_{ε,m}} |∇^{2}u_{ε} (s, x_{2})|^{2} ds.

Hence, by (6.10) and Hölder’s inequality,

∫_{R_{ε,m}^{−}} |∇u_{ε} (x) − (0 |\frac{∂u_{ε}}{∂x_{2}} (−a_{ε,m}, x_{2}))|^{2} dx ≤2 ∫_{R_{ε,m}^{−}} |∇u_{ε} − ∇u_{0}|^{2} dx  (6.19)

+ \frac{3ε}{m} ∫_{R_{ε,m}^{−}} ∫_{−c_{ε,m}}^{−b_{ε,m}} |∇^{2}u_{ε} (s, x_{2})|^{2} dsdx

≤2 |R_{ε,m}^{−}|^{(p−2)/p} (∫_{R_{ε,m}^{−}} |∇u_{ε} − ∇u_{0}|^{p} dx)^{2/p}

+ \frac{3}{m^{2}} [\frac{1}{ε}]^{−1} ε ∫_{R_{ε,m}^{−}} |∇^{2}u_{ε}|^{2} dx

≤C \frac{ε}{m^{3}} + C \frac{ε^{2}}{m^{3}} .

By (6.17), (6.18) and (6.19) we conclude that

ε ∫_{R_{ε,m}^{−}} |∇^{2}w_{ε,m}|^{2} dx ≤ Cε ∫_{R_{ε,m}^{−}} |∇^{2}u_{ε}|^{2} dx + C \frac{ε^{2}}{m} ∫− \frac{1}{2}^{\frac{1}{2}} |∇^{2}u_{ε} (−a_{ε,m}, x_{2})|^{2} dx_{2}

+ C \frac{m^{2}}{ε} ∫_{R_{ε,m}^{−}} |∇u_{ε} − (0 |\frac{∂u_{ε}}{∂x_{2}} (−a_{ε,m}, x_{2}))|^{2} dx

+ C \frac{m^{4}}{ε^{3}} ∫_{R_{ε,m}^{−}} |u_{ε} − u_{ε} (−a_{ε,m}, x_{2})|^{2} dx

≤ C (K^{∗} \frac{3}{m} [\frac{1}{ε}]^{−1} + 6K^{∗} \frac{ε}{m} + \frac{1}{m} + \frac{ε}{m} + \frac{1}{m}) → 0

as ε → 0^{+} and m → ∞.

Finally, note that by (6.18)

∫_{R_{ε,m}^{−}} |w_{ε,m} − u_{ε}| dx ≤ C (\frac{ε}{m} ∫_{R_{ε,m}^{−}} |u_{ε} − u_{ε} (−a_{ε,m}, x_{2})|^{2} dx)^{1/2} → 0

as ε → 0^{+}. Step 2: The N-dimensional case. Take sequences {ε_{n}} ⊂ ℝ_{+}, {u_{n}} ⊂ W^{2,2} (Q; ℝ^{d}) , such that ε_{n} → 0^{+}, u_{n} → u_{0} := |x_{N}| a in L^{1} (Q; ℝ^{d}) and

lim_{ε→0^{+}} ∫_{Q} \frac{1}{ε_{n}} W (∇u_{n}) + ε_{n} |∇^{2}u_{n}|^{2} dx = K^{∗}.

By Step 1 there exist a subsequence {ε_{m}} ⊂ {ε_{n}} such that the corresponding fields u_{m} may be modified so as to obtain a new sequence \{w_{m}^{(1)}\} ⊂ W_{loc}^{2,∞} (ℝ × (−\frac{1}{2}, \frac{1}{2})^{N−1} ; ℝ^{d}) , x_{1}-periodic with period 2, such that ∇w_{m}^{(1)} (x) = ± a ⊗ e_{N} nearby x_{N} = ±\frac{1}{2} (resp.) and

lim_{m→∞} ∫_{(−1,1)×(− \frac{1}{2} , \frac{1}{2})^{N−1}} \frac{1}{ε_{m}} W (∇w_{m}^{(1)}) + ε_{m} |∇^{2}w_{m}^{(1)}|^{2} dx = 2K^{∗},

lim_{m→∞} ∫_{(−1,1)×(− \frac{1}{2} , \frac{1}{2})^{N−1}} |w_{m}^{(1)} − u_{0}| dx = 0.

We treat x_{2} just as above. Starting from the w_{m}^{(1)} above we construct

w_{k}^{(2)} (x) := φ_{k} (x_{2}) w_{m_{k}}^{(1)} (x) + (1 − φ_{k} (x_{2})) w_{m_{k}}^{(1)} (x_{1,} ± b_{k}, x_{3}, ⋯, x_{N}) .

with b_{k} → \frac{1}{2}^{−}, such that

lim_{k→∞} ∫_{(−1,1)×(− \frac{1}{2} , \frac{1}{2})^{N−1}} \frac{1}{ε_{m_{k}}} W (∇w_{k}^{(2)}) + ε_{m_{k}} |∇^{2}w_{k}^{(2)}|^{2} dx = 2K^{∗},

lim_{k→∞} ∫_{(−1,1)×(− \frac{1}{2} , \frac{1}{2})^{N−1}} |w_{k}^{(2)} − u_{0}| dx = 0.

Note that w_{k}^{(2)} (x) is still periodic in x_{1} with period 2 and ∇w_{k}^{(2)} (x) = ± a ⊗ e_{N} nearby x_{N} = ±\frac{1}{2} (resp.). After reflection about x_{2} = \frac{1}{2} we obtain a sequence w_{k}^{(2)} (x) periodic in x_{1} and in x_{2}, of period 2, such that ∇w_{k}^{(2)} (x) = ± a ⊗ e_{N} nearby x_{N} = ±\frac{1}{2} (resp.), and

lim_{k→∞} ∫_{(−1,1)×(−1,1)×(− \frac{1}{2} , \frac{1}{2})^{N−2}} \frac{1}{ε_{m_{k}}} W (∇w_{k}^{(2)}) + ε_{m_{k}} |∇^{2}w_{k}^{(2)}|^{2} dx = 4K^{∗},

lim_{k→∞} ∫_{(−1,1)×(−1,1)×(− \frac{1}{2} , \frac{1}{2})^{N−2}} |w_{k}^{(2)} − u_{0}| dx = 0.

By repeating this process for all remaining variables x_{3}, ⋯, x_{N−1}, we obtain sequences fulfilling the claim.

Define

K_{per} := inf \{∫_{Q} L W (∇v) + \frac{1}{L} |∇^{2}v|^{2} dx : L > 0, v ∈ W^{2,∞} (Q; ℝ^{d}) ,
∇v = ±a ⊗ e_{N} nearby x_{N} = ±\frac{1}{2}, v periodic of period one in x′\} .

Proposition 6.4 Assume that W satisfies conditions (H_{1}) , (H_{2})′′, (H_{4}) and (H_{5}) . Then

K^{∗} = K_{per}.

Proof. By Proposition 6.3 K^{∗} ≥ K_{per}. To prove the opposite inequality, fix δ > 0 and let L > 0, v ∈ W_{loc}^{2,∞} (Q; ℝ^{d}) , v (·, x_{N}) Q′-periodic for all x_{N} ∈ ℝ, such that ∇v = ±a ⊗ e_{N} nearby x_{N} = ±\frac{1}{2} (resp.) and

∫_{Q} L W (∇v) + \frac{1}{L} |∇^{2}v|^{2} dx ≤ K_{per} + δ.

Let {ε_{n}} be a sequence converging to 0^{+}, and writing ε := ε_{n} define

z_{ε} (x) := \{ \begin{matrix}εLv (\frac{x′}{εL}, \frac{1}{2}) + a (x_{N} − \frac{εL}{2}) & if x_{N} > \frac{εL}{2}, \\ εLv (\frac{x}{εL}) & if |x_{N}| ≤ \frac{εL}{2}, \\ εLv (\frac{x′}{εL}, −\frac{1}{2}) − a (x_{N} + \frac{εL}{2}) & if x_{N} < −\frac{εL}{2}, \end{matrix}  (6.20)

so that

∇z_{ε} (x) = \{ \begin{matrix}a ⊗ e_{N} & if x_{N} > \frac{εL}{2}, \\ ∇v (\frac{x}{εL}) & if |x_{N}| ≤ \frac{εL}{2}, \\ −a ⊗ e_{N} & if x_{N} < −\frac{εL}{2}. \end{matrix}  (6.21)

Then

∫_{Q^{+}} |∇z_{ε} − ∇u_{0}|^{p} dx = ∫_{Q′} ∫0^{\frac{εL}{2}} |∇v (\frac{x}{εL}) − a ⊗ e_{N} |^{p} dx′dx_{N}

= εL ∫_{Q′} ∫− \frac{1}{2}^{\frac{1}{2}} |∇v (\frac{x′}{εL}, t) − a ⊗ e_{N} |^{p} dx′dx_{N} → 0

since v ∈ W^{2,∞} (Q; ℝ^{d}) , and where u_{0} := |x_{N}| a, Q^{+} := Q′ × (0, \frac{1}{2}) . A similar conclusion holds in Q^{−} := Q′ × (−\frac{1}{2}, 0) and so ∇z_{ε} → ∇u_{0} in L^{p} (Q; ℝ^{d}) . Moreover, by the Riemann-Lebesgue Lemma

∫_{Q} \frac{1}{ε}W (∇z_{ε}) + ε |∇^{2}z_{ε}|^{2} dx = ∫− \frac{εL}{2}^{\frac{εL}{2}} ∫_{Q′} \frac{1}{ε}W (∇v (\frac{x}{εL})) + \frac{1}{εL^{2}} |∇^{2}v (\frac{x}{εL})|^{2} dx′dx_{N}

= ∫− \frac{1}{2}^{\frac{1}{2}} ∫_{Q′} LW (∇v (\frac{x′}{εL}, t)) + \frac{1}{L} |∇^{2}v (\frac{x′}{εL}, t)|^{2} dx′dt → ∫_{Q} LW (∇v) + \frac{1}{L} |∇^{2}v|^{2} dx,

and so

K^{∗} ≤ ∫_{Q} LW (∇v) + \frac{1}{L} |∇^{2}v|^{2} dx ≤ K_{per} + δ.

It now suffices to let δ → 0^{+}.

Remark 6.5 If Ω = ω × (−h, h) , where ω ⊂ ℝ^{N−1} is a bounded open set, and if we consider the sequence defined in (6.20) , then we have by the Riemann-Lebesgue Lemma

∫ω×(−h,h) \frac{1}{ε}W (∇z_{ε}) + ε |∇^{2}z_{ε}|^{2} dx = ∫− \frac{εL}{2}^{\frac{εL}{2}} ∫_{ω} \frac{1}{ε}W (∇v (\frac{x}{εL})) + \frac{1}{εL^{2}} |∇^{2}v (\frac{x}{εL})|^{2} dx′dx_{N}

= ∫− \frac{1}{2}^{\frac{1}{2}} ∫_{ω} LW (∇v (\frac{x′}{εL}, t)) + \frac{1}{L} |∇^{2}v (\frac{x′}{εL}, t)|^{2} dx′dt → H^{N−1} (ω) ∫_{Q} LW (∇v) + \frac{1}{L} |∇^{2}v|^{2} dx.

6.2 x′-connected domains

Theorem 6.6 Let Ω ⊂ ℝ^{N} be an open, bounded, simply connected domain with Lipschitz boundary. Assume (5.14) and let W satisfy the conditions (H_{1}) , (H_{2})′′, (H_{4}) and (H_{5}) . Let u ∈ W^{1,1} (Ω; ℝ^{d}) , with ∇u ∈ BV (Ω; {A, B}) . Then

Γ − lim_{ε→0^{+}} I_{ε} (u; Ω) = K^{∗} Per_{Ω}(E),

where ∇u(x) = (1 − χ_{E} (x)) A + χ_{E} (x) B for L^{N} a.e. x ∈ Ω.

Proof. The proof is very similar to that of Theorem 5.6. Step 1: One interface. We assume first that u has the form

u (x) = |x_{N}| a a.e. in Ω.

We proceed as in Step 1 of Theorem 5.6 and in place of the function g we consider a function v ∈ W^{2,∞} (Q; ℝ^{d}) admissible for K_{per}, and, consequently, we define

u_{ε,δ} (x) := ψ_{δ} (x) z_{ε} (x) + (1 − ψ_{δ} (x)) u (x)

where z_{ε} is now defined as in (6.20). The estimate (5.18) should be replaced by

∫_{\{ψ_{δ}=1\}} \frac{1}{ε}W (∇z_{ε}) + ε |∇^{2}z_{ε}|^{2} dx = ∫− \frac{1}{2}^{\frac{1}{2}} ∫_{ω_{δ}^{+}} LW (∇v (\frac{x′}{εL}, t)) + \frac{1}{L} |∇^{2}v (\frac{x′}{εL}, t)|^{2} dx′dt  (6.22)

→ H^{N−1} (ω_{δ}^{+}) (∫_{Q} LW (∇v) + \frac{1}{L} |∇^{2}v|^{2} dx)

as ε → 0^{+}, by Remark 6.5. By (6.20) and (6.21) the bound (5.24) continues to hold, while (5.25) should be replaced by

|∇ (z_{ε} − u)| ≤ |∇v (\frac{x}{εL})| + ∥∇u∥_{∞} ≤ C, |∇^{2}z_{ε}| ≤ \frac{1}{ε} |∇v (\frac{x}{εL})| ≤ \frac{C}{ε} ,  (6.23)

for |x_{N}| ≤ εL. We can continue essentially as before, using (6.22) in the right hand side of the new formula corresponding to (5.28). We omit the details. Steps 2 and 3: In the cases of finitely many and countably many interfaces we may proceed, respectively, as in Steps 2 and 3 of Theorem 5.5. We omit the details.

6.3 General domains

In this section we remove the condition (5.14) .

Theorem 6.7 Assume that W satisfies the conditions (H_{1}) , (H_{2})′′, (H_{4}) and (H_{5}) . Let u ∈ W^{1,1} (Ω; ℝ^{d}) , with ∇u ∈ BV (Ω; {A, B}) . Then

Γ − lim_{ε→0^{+}} I_{ε} (u; Ω) = K^{∗} Per_{Ω}(E),

where ∇u(x) = (1 − χ_{E} (x)) A + χ_{E} (x) B for L^{N} a.e. x ∈ Ω.

Proof. The proof follows closely that of Subsection 5.4, with the only differences that in (5.34) of Proposition 5.7 the function z_{ε} is now defined as in (6.20) and, in turn, the estimate (5.25) should be replaced by (6.23).

7 Condition (H_{4})′

In this section we weaken the condition (H_{4}) on the bounds of W near the wells. Theorem 7.1 All the results of the previous section continue to hold if condition (H_{4}) is replaced by (H_{4})′ .

The next lemma ensures that the function g introduced in (H_{4})′ to control the behavior of W near the wells may be extended to a function G still satisfying the doubling condition, and such that G (|· − A|) and G (|· − B|) may be compared with W in the whole ℝ^{d×N}. As before, in what follows we are assuming without loss of generality that A and B satisfy (2.1). Lemma 7.2 Let g : [0, ∞) → [0, ∞) be a convex function, with g (s) = 0 if and only if s = 0, such that

g (2t) ≤ Cg (t)  (7.1)

for all 0 ≤ t ≤ ρ,

g (|ξ − A|) ≤ W (ξ) ≤ Cg (|ξ − A|)  (7.2)

for all ξ ∈ ℝ^{d×N} with |ξ − A| ≤ ρ, and

g (|ξ − B|) ≤ W (ξ) ≤ Cg (|ξ − B|)  (7.3)

for all ξ ∈ ℝ^{d×N} with |ξ − B| ≤ ρ, for some constant C = C (ρ) > 0. Then there exists a convex function G : [0, ∞) → [0, ∞) such that G (t) = g (t) for all t ∈ [0, ρ] ,

G (s + t) ≤ C_{1} (G (s) + G (t))  (7.4)

for all s, t ≥ 0 and for some constant C_{1} > 0,

lim_{t→∞} \frac{G (t)}{t^{p}} = 1,  (7.5)
\frac{1}{C_{2}} G (|ξ′|) ≤ \frac{1}{C_{2}} min \{G (|ξ − A|) , G (|ξ − B|)\} ≤ W (ξ) ≤ C_{2} min \{G (|ξ − A|) , G (|ξ − B|)\}  (7.6)

for all ξ ∈ ℝ^{d×N} and for some constant C_{2} > 0,

W (ξ) ≤ C_{3} (W (η) + G (|ξ − η|))  (7.7)

for all ξ, η ∈ ℝ^{d×N} and for some constant C_{3} > 0,

C_{4}G (|ξ − A|) ≤ W (ξ)  (7.8)

for all ξ ∈ ℝ^{d×N} such that |ξ − A| , |ξ − B| ≥ ρ and for some constant C_{4} > 0.

Proof. Let a > ρ be any Lebesgue point for g′ (recall that, since g is convex, g′ is a function of bounded variation, precisely g′ ∈ BV_{loc} ([0, ∞))), and define

G (t) = \{ \begin{matrix}g (t) & for 0 ≤ t ≤ a, \\ t^{p} + (g′ (a) − pa^{p−1}) t + g (a) − g′ (a) a + (p − 1) a^{p} & for t > a. \end{matrix}  (7.9)

We claim that G is convex. Assume first that g ∈ C^{2} ([0, ∞)) . Then

G′ (t) = \{ \begin{matrix}g′ (t) & for 0 ≤ t < a, \\ pt^{p−1} + (g′ (a) − pa^{p−1}) & for t > a, \end{matrix}

and

G′′ (t) = \{ \begin{matrix}g′′ (t) & for 0 ≤ t < a, \\ p (p − 1) t^{p−2} & for t > a. \end{matrix}

Hence G′ is continuous and nondecreasing, since G′′ (t) ≥ 0 for all t ≠ a, and, as G is continuous, this implies that G is convex. In the general case, consider g_{ε} := η_{ε} ∗ g and let G_{ε} be the corresponding convex functions defined as in (7.9). Since g_{ε} → g pointwise and g′_{ε} (t) → g′ (t) for every Lebesgue point t of g′, we obtain that G_{ε} → G pointwise, and thus G is convex. Condition (7.5) is now immediate. To prove (7.4) we first show that

G (2t) ≤ C_{1}G (t)  (7.10)

for all t ≥ 0 and for some constant C_{1} > 0. It t ≤ ρ this follows from (7.1). Let ρ_{1} > a be so large that

2t^{p} > − (2^{p+1} − 2) (g′ (a) − 2a) t − (2^{p+1} − 1) (g (a) − g′ (a) a + (p − 1) a^{p})

for all t ≥ ρ_{1}. Then

G (2t) = 2^{p}t^{p} + (g′ (a) − pa^{p−1}) 2t + g (a) − g′ (a) a + (p − 1) a^{p}
= 2^{p+1}G (t) − 2t^{p} − (2^{p+1} − 2) (g′ (a) − 2a) t − (2^{p+1} − 1) (g (a) − g′ (a) a + (p − 1) a^{p})
< 2^{p+1}G (t)

for all t ≥ ρ_{1}. Thus (7.10) holds for t ≤ ρ and t ≥ ρ_{1} taking as a constant max {C, 2^{p+1}} . For t ∈ [ρ, ρ_{1}] we have

G (2t) ≤ \frac{max_{[2ρ,2ρ_{1}]} G}{min_{[ρ,ρ_{1}]} G}G (t)

and thus (7.10) holds for all t ≥ 0 with

C_{1} := max \{C, 2^{p+1}, \frac{max_{[2ρ,2ρ_{1}]} G}{min_{[ρ,ρ_{1}]} G}\} .

To prove that (7.10) implies (7.4) is standard, note that by convexity and (7.10)

G (s + t) = G (\frac{2 (s + t)}{2}) ≤ \frac{1}{2} (G (2s) + G (2t)) ≤ \frac{1}{2}C_{1} (G (s) + G (t)) .

By (7.2) and (7.3) condition (7.6) holds if either |ξ − A| ≤ ρ or |ξ − B| ≤ ρ. For k > 1 set E_{k} := {ξ ∈ ℝ^{d×N} : |ξ − A| , |ξ − B| ≥ ρ, |ξ| ≤ k} . For ξ ∈ E_{k}

\frac{min_{E_{k}} W}{max_{E_{k}} G (|· − A|)} G (|ξ − A|) ≤ W (ξ) ≤ \frac{max_{E_{k}} W}{min_{E_{k}} G (|· − A|)} G (|ξ − A|) ,

and a similar inequality holds when the G (|ξ − A|) is replaced by G (|ξ − B|) . Thus it is sufficient to prove (7.6) and (7.8) for |ξ| ≥ k where k > 1 remains to be chosen. This is an obvious consequence of (H_{2})′′ and (7.5).

Remark 7.3 In light of Lemma 7.2, and in spite of the fact that the qualitative properties of g are only given nearby zero, in the remaining of this section, and without loss of generality, we will assume that g satisfies (7.4)-(7.6) and (7.7)-(7.8) with g in place of G.

The next result has been proved by Bhattacharya and Leonetti [13] in the case where Ω is convex and S = Ω, and a generalized version for for open, bounded domains with the cone property may be found in the Appendix.

Proposition 7.4 Let Ω ⊂ ℝ^{N} be an open bounded set, starshaped with respect to a set S ⊂ Ω, with |S| > 0. Let g : [0, ∞) → [0, ∞) be a convex function, with g (0) = 0. Let u ∈ W^{1,1} (Ω; ℝ^{d}) be such that g (|∇u|) ∈ L^{1} (Ω) . Then

∫_{Ω} g (\frac{|u (x) − u_{S}|}{d}) dx ≤ (\frac{α_{N} d^{N}}{|Ω|} )^{1− \frac{1}{N}} \frac{|Ω|}{|S|} ∫_{Ω} g (|∇u|) dx,

where u_{S} := \frac{1}{|S|} ∫_{S} u dx, d is any number greater or equal than the diameter of Ω, and α_{N} is the volume of the unit ball in ℝ^{N}.

Proof of Theorem 7.1. Condition (H_{4}) was used only in the proof of Propositions 6.2 and 6.3. Thus it remains to show that these propositions continue to work under the weaker hypothesis (H_{4})′. We begin with Proposition 6.2. Vertical matching– first matching: The proof of the first matching continues to work up to (6.8). By (7.6) and since g is increasing

\frac{1}{ε} ∫_{L_{ε}} W (∇v_{ε}) dx ≤ \frac{1}{ε} ∫_{L_{ε}} Cg (|∇v_{ε} − a ⊗ e_{N}|) dx  (7.11)

≤ \frac{1}{ε} ∫_{L_{ε}} g (C (|∇_{x′}u_{ε} (x′, z_{ε})| + |∇u_{ε} − a ⊗ e_{N}| + \frac{1}{ε} |u_{ε} − u_{0} − ū_{ε}|)) dx

≤ \frac{C}{ε} ∫_{L_{ε}} g (|∇_{x′}u_{ε} (x′, z_{ε})|) + g (|∇u_{ε} − a ⊗ e_{N}|) + g (\frac{1}{ε} |u_{ε} − u_{0} − ū_{ε}|) dx

where we have used (7.4). We now estimate the three terms on the right hand side of (7.11). By (7.6)

\frac{1}{ε} ∫_{L_{ε}} g (|∇_{x′}u_{ε} (x′, z_{ε})|) dx = [\frac{1}{ε}]^{−1} \frac{1}{6ε} ∫_{Q′} g (|∇_{x′}u_{ε} (x′, z_{ε})|) dx′  (7.12)
≤ C ∫_{Q′} W (∇u_{ε} (x′, z_{ε})) dx′ → 0

as ε → 0^{+} by (6.6). If |∇u_{ε} − a ⊗ e_{N}| ≤ ρ or |∇u_{ε} − a ⊗ e_{N}| , |∇u_{ε} + a ⊗ e_{N}| ≥ ρ then by (7.8)

g (|∇u_{ε} − a ⊗ e_{N}|) ≤ CW (∇u_{ε}) ,

while if |∇u_{ε} − a ⊗ e_{N}| ≥ ρ and |∇u_{ε} + a ⊗ e_{N}| ≤ ρ then

g (|∇u_{ε} − a ⊗ e_{N}|) ≤ g (ρ + 2 |a ⊗ e_{N}|) ≤ g (ρ + 2 |a ⊗ e_{N}|) \frac{|∇u_{ε} − a ⊗ e_{N}|^{p}}{ρ^{p}} .

Hence

\frac{C}{ε} ∫_{L_{ε}} g (|∇u_{ε} − a ⊗ e_{N}|) dx ≤ \frac{C}{ε} ∫_{L_{ε}} W (∇u_{ε}) + |∇u_{ε} − a ⊗ e_{N}|^{p} dx → 0  (7.13)

as ε → 0^{+} by (6.5). Finally, by Jensen’s inequality

\frac{C}{ε} ∫_{L_{ε}} g (\frac{1}{ε} |u_{ε} − u_{0} − ū_{ε}|) dx ≤ \frac{C}{ε} ∫_{L_{ε}} g (\frac{1}{ε} ∫_{θ_{ε}−[\frac{1}{ε}]^{−1} \frac{1}{6}}^{θ_{ε}} |∇u_{ε} (x′, t) − a ⊗ e_{N}| dt) dx  (7.14)

≤ \frac{C}{ε} ∫_{L_{ε}} \frac{1}{ε} ∫_{θ_{ε}−[\frac{1}{ε}]^{−1} \frac{1}{6}}^{θ_{ε}} g (|∇u_{ε} (x′, t) − a ⊗ e_{N}|) dtdx

= \frac{C}{ε} ∫_{L_{ε}} g (|∇u_{ε} − a ⊗ e_{N}|) dx → 0

as ε → 0^{+} by (6.5). Thus, by (7.11)-(7.14) the claim (iii) holds as before.

Vertical matching–second matching: To prove (ii) in the second matching, note that by (7.6) and since g is increasing,

\frac{1}{ε} ∫_{Q′×(θ_{ε}, \frac{1}{2})} W (∇v_{ε}) dx ≤ \frac{1}{ε} ∫_{Q′×(θ_{ε}, \frac{1}{2})} Cg (|∇v_{ε} − a ⊗ e_{N}|) dx

≤ \frac{C}{ε} ∫_{Q′×(θ_{ε}, \frac{1}{2})} g (|∇_{x′}u_{ε} (x′, z_{ε})|) + g (|ū_{ε} − c_{ε}|) dx

≤ \frac{C}{ε} ∫_{Q′×(θ_{ε}, \frac{1}{2})} W (∇u_{ε}) + g (|ū_{ε} − c_{ε}|) dx,

where we have used (7.4). Thus by (6.5), it suffices to prove that

\frac{C}{ε} ∫_{Q′×(θ_{ε}, \frac{1}{2})} g (|ū_{ε} − c_{ε}|) dx → 0.

By Proposition 7.4

\frac{C}{ε} ∫_{Q′×(θ_{ε}, \frac{1}{2})} g (|ū_{ε} − c_{ε}|) dx = \frac{C}{ε} ∫θ_{ε}^{\frac{1}{2}} ∫_{Q′} g (|ū_{ε} − \frac{1}{|Q′|} ∫_{Q′} ū_{ε} (x′, z_{ε}) dx′|) dx′dx_{N}

≤ \frac{C}{ε} ∫θ_{ε}^{\frac{1}{2}} ∫_{Q′} g (|∇_{x′} ū_{ε}|) dx′dx_{N}

≤ \frac{C}{ε} ∫_{Q′} g (|∇_{x′}u_{ε} (x′, z_{ε})|) dx′

≤ \frac{C}{ε} ∫_{Q′} W (∇u_{ε} (x′, z_{ε})) dx′ → 0

as ε → 0^{+} by (6.6), and where we have used (7.6).

It remains to ensure that Proposition 6.3 still holds. Transversal periodicity: The proof of Proposition 6.3 continues to work. The only difference is on the estimate (6.13) which continues to hold since, by (7.6) and the fact that g is increasing, we have

W (0, \frac{∂u_{ε}}{∂x_{2}} (−a_{ε,m}, x_{2})) ≤ C_{2} min \{g (|\frac{∂u_{ε}}{∂x_{2}} (−a_{ε,m}, x_{2}) − a|) , g (|\frac{∂u_{ε}}{∂x_{2}} (−a_{ε,m}, x_{2}) + a|)\}
≤ C_{2} min {g (|∇u_{ε} (−a_{ε,m}, x_{2}) − a ⊗ e_{2}|) , g (|∇u_{ε} (−a_{ε,m}, x_{2}) + a ⊗ e_{2}|)}
≤ C_{2}^{2}W (∇u_{ε} (−a_{ε,m}, x_{2})) ,

and we can now proceed as before.

8 Example of a non one-dimensional interface

In this section we show that when (H_{5}) holds but (H_{3}) fails, the asymptotic limiting problem may not have a one dimensional character, namely,

K_{per} < K.

Consider the case where N = 2, d = 1, so that, with x = (x_{1}, x_{2}) , we have

K_{per} := inf \{∫_{Q} L W (∇v) + \frac{1}{L} |∇^{2}v|^{2} dx_{1}dx_{2} : L > 0, v ∈ W^{2,∞} (Q; ℝ) ,
∇v = ±e_{2} nearby x_{2} = ±\frac{1}{2}, v is periodic of period one in x_{1}\} .

In what follows we say that v ∈ W_{loc}^{2,∞} ((−\frac{1}{2}, \frac{1}{2}) × ℝ; ℝ) is such that ∇v = ±e_{2} nearby x_{2} = ±∞ if there exists a constant M > 0 such that ∇v = ±e_{2} for all x ∈ (−\frac{1}{2}, \frac{1}{2}) × ℝ (resp.) with x_{2} ≥ M (resp. x_{2} ≤ −M).

Proposition 8.1

K_{per} = inf \{∫_{−∞}^{∞} ∫− \frac{1}{2}^{\frac{1}{2}} L W (∇v) + \frac{1}{L} |∇^{2}v|^{2} dx_{1}dx_{2} : L > 0, v ∈ W_{loc}^{2,∞} ((−\frac{1}{2}, \frac{1}{2}) × ℝ; ℝ) ,
∇v = ±e_{2} nearby x_{2} = ±∞, v is periodic of period one in x_{1}} =: K_{∞}.

Proof. By linear continuation it is easy to see that K_{∞} ≤ K_{per}. To prove the converse inequality, fix δ > 0 and let v ∈ W^{2,∞} ((−\frac{1}{2}, \frac{1}{2}) × ℝ; ℝ) be an admissible function for K_{∞} such that, for some L > 0,

∫_{−∞}^{∞} ∫− \frac{1}{2}^{\frac{1}{2}} L W (∇v) + \frac{1}{L} |∇^{2}v|^{2} dx_{1}dx_{2} ≤ K_{∞} + δ.

Since ∇v = ±e_{2} nearby x_{2} = ±∞ we may find a positive integer m such that

K_{∞} + δ ≥ ∫_{−∞}^{∞} ∫− \frac{1}{2}^{\frac{1}{2}} L W (∇v) + \frac{1}{L} |∇^{2}v|^{2} dx_{1}dx_{2} = ∫_{−m− \frac{1}{2}}^{m+ \frac{1}{2}} ∫− \frac{1}{2}^{\frac{1}{2}} L W (∇v) + \frac{1}{L} |∇^{2}v|^{2} dx_{1}dx_{2}

and ∇v = ±e_{2} for ±x_{2} ≥ m. Due to the periodicity of v with respect to x_{1}, we have

K_{∞} + δ ≥ ∫_{−m− \frac{1}{2}}^{m+ \frac{1}{2}} ∫− \frac{1}{2}^{\frac{1}{2}} L W (∇v (x)) + \frac{1}{L} |∇^{2}v (x)|^{2} dx_{1}dx_{2}

= ∫_{−m− \frac{1}{2}}^{m+ \frac{1}{2}} \frac{1}{2m + 1} ∑_{k=−m}^{m} ∫− \frac{1}{2}^{\frac{1}{2}} L W (∇v (x_{1} + k, x_{2})) + \frac{1}{L} |∇^{2}v (x_{1} + k, x_{2})|^{2} dx_{1}dx_{2}

= ∫_{−m− \frac{1}{2}}^{m+ \frac{1}{2}} \frac{1}{2m + 1} ∑_{k=−m}^{m} ∫_{k− \frac{1}{2}}^{k+ \frac{1}{2}} L W (∇v (x_{1}, x_{2})) + \frac{1}{L} |∇^{2}v (x_{1}, x_{2})|^{2} dx_{1}dx_{2}

= \frac{1}{2m + 1} ∫_{−m− \frac{1}{2}}^{m+ \frac{1}{2}} ∫_{−m− \frac{1}{2}}^{m+ \frac{1}{2}} L W (∇v (x_{1}, x_{2})) + \frac{1}{L} |∇^{2}v (x_{1}, x_{2})|^{2} dx_{1}dx_{2}

= \frac{1}{2m + 1} ∫_{(2m+1)Q} L W (∇v (x)) + \frac{1}{L} |∇^{2}v (x)|^{2} dx.

Via the change of variables x := (2m + 1) y we have

K_{∞} + δ ≥ (2m + 1) ∫_{Q} LW (∇v ((2m + 1) y)) + \frac{1}{L} |∇^{2}v ((2m + 1) y)|^{2} dy

= ∫_{Q} (2m + 1) L W (∇z (y)) + \frac{1}{(2m + 1) L} |∇^{2}z (y)|^{2} dy,

where we have set z (y) := \frac{1}{2m+1}v ((2m + 1) y) . Note that

z (y_{1} + 1, y_{2}) = \frac{1}{2m + 1}v ((2m + 1) y_{1} + (2m + 1) , (2m + 1) y_{2})

= \frac{1}{2m + 1}v ((2m + 1) y_{1}, (2m + 1) y_{2})

= z (y_{1}, y_{2}) ,

since v periodic of period one in x_{1}, and ∇z (y) = ±e_{2} if y_{2} is nearby ±\frac{1}{2}, resp. Hence z is admissible for K_{per}, and so

K_{∞} + δ ≥ \frac{1}{2m + 1} ∫_{(2m+1)Q} L W (∇v (x)) + \frac{1}{L} |∇^{2}v (x)|^{2} dx

= (2m + 1) ∫_{Q} L W (∇v ((2m + 1) y)) + \frac{1}{L} |∇^{2}v ((2m + 1) y)|^{2} dy

= ∫_{Q} (2m + 1) L W (∇z (y)) + \frac{1}{(2m + 1) L} |∇^{2}z (y)|^{2} dy ≥ K_{per}.

Figure 9: The left panel shows W (ξ_{1}, ξ_{2}), as defined in Eq. (8.1), for ξ_{1} ∈ (−\frac{1}{2}, \frac{1}{2}) and ξ_{2} ∈ (−1, \frac{3}{2}). The darker line connecting the two wells represents the trace W (0, ξ_{2}) and it can be seen easily in the picture that for |ξ_{2}| < 1 these values are not energetically favourable as compared with W (ξ_{1}, ξ_{2}), and thus (H_{3}) fails. The right panel shows the gradient of one of the test functions (8.3).

It now suffices to let δ → 0^{+}.

Next we exhibit an example of an energy density satisfying (H_{1}) , (H_{2})′′, (H_{4})′ and (H_{5}) for which K_{per} < K. Define

W (ξ) = W (ξ_{1}, ξ_{2}) := (1 − αξ_{1}^{2} − ξ_{2}^{2})^{2} + ξ_{1}^{2},  (8.1)

where α > 0 (see Figure 9). The potential W (ξ) is a natural perturbation of the one used by Jin and Kohn [30]. Then W (ξ) = 0 if and only if ξ ∈ {(0, 1) , (0, −1)}. By Lemma 3.5 in [26] we have that

K = inf {∫_{−∞}^{∞} W (0, g (s)) + |g′ (s)|^{2} ds : g piecewise C^{1}, g(−∞) = −1, g(∞) = 1}  (8.2)
= inf {∫_{−∞}^{∞} (1 − g^{2} (s))^{2} + |g′ (s)|^{2} ds : g piecewise C^{1}, g(−∞) = −1, g(∞) = 1} .

It is not difficult to see that K is realized by the unique solution of the boundary value problem

\{ \begin{matrix}g′′ + 2g − 2g^{3} = 0 \\ g(−∞) = −1, g(∞) = 1, \end{matrix}

which is given by g (s) := tanh s. Define ū (t) := ∫_{0}^{t} g (s) ds = ln cosh t.

Proposition 8.2 If α is sufficiently large then we have K_{per} < K.

Proof. Set

v (x) = v (x_{1}, x_{2}) := ū (x_{2}) + λψ (x_{1}, x_{2}) ,  (8.3)

where ψ (x_{1}, x_{2}) := sin (2πx_{1}) f (x_{2}) and f is a smooth nonnegative function with compact support (see Figure 9). With L = 1 we have

∫_{−∞}^{∞} ∫− \frac{1}{2}^{\frac{1}{2}} W (∇v) + |∇^{2}v|^{2} dx_{1}dx_{2} = ∫_{−∞}^{∞} (1 − g^{2} (s))^{2} + |g′ (s)|^{2} ds

+ λ^{2} ∫_{−∞}^{∞} ∫− \frac{1}{2}^{\frac{1}{2}} [|∇^{2}ψ|^{2} + 2 (3g^{2} (x_{2}) − 1) sin^{2} (2πx_{1}) (f′ (x_{2}))^{2} + 4π^{2} cos^{2}(2πx_{1})f^{2} (x_{2})

−8π^{2}α (1 − g^{2} (x_{2})) cos^{2} (2πx_{1}) f^{2} (x_{2})] dx_{1}dx_{2}

+ λ^{4} ∫_{−∞}^{∞} ∫− \frac{1}{2}^{\frac{1}{2}} [sin^{2} (2πx_{1}) (f′ (x_{2}))^{2} + 4π^{2}α cos^{2} (2πx_{1}) f^{2} (x_{2})]^{2} dx_{1}dx_{2}

=: I_{1} + λ^{2}I_{2} (α) + λ^{4}I_{3} (α) .

We now choose α > 0 so large that I_{2} (α) < 0, and then λ so small that λ^{2}I_{2} (α) + λ^{4}I_{3} (α) < 0. In view of (8.2)

∫_{−∞}^{∞} ∫− \frac{1}{2}^{\frac{1}{2}} W (∇v) + |∇^{2}v|^{2} dx_{1}dx_{2} < ∫_{−∞}^{∞} (1 − g^{2} (s))^{2} + |g′ (s)|^{2} ds = K.

Let {u_{n}} be a sequence of smooth functions converging to ū strongly in W_{loc}^{2,p} (ℝ) for all p ≥ 1, and such that u′_{n} (x_{2}) = ±e_{2} (resp.) for all x_{2} ≥ n (resp. x_{2} ≤ −n). Define

v_{n} (x) := u_{n} (x_{2}) + λψ (x_{1}, x_{2}) .

Then v_{n} are admissible for K_{∞}, and so

K_{∞} ≤ lim_{n→∞} ∫_{−∞}^{∞} ∫− \frac{1}{2}^{\frac{1}{2}} W (∇v_{n}) + |∇^{2}v_{n}|^{2} dx_{1}dx_{2}

= ∫_{−∞}^{∞} ∫− \frac{1}{2}^{\frac{1}{2}} W (∇v) + |∇^{2}v|^{2} dx_{1}dx_{2} < K.

This, together with Proposition 8.1, concludes the proof.

9 Appendix

In the Appendix we generalize Poincaré’s inequality to Orlicz-Sobolev spaces (see Proposition 9.2). Although it is probably known to experts, we have not been able to find it in the literature. The proof follows that of Maz’ja [33] for the case g (s) = |s|^{p} . A first version has been proved by Bhattacharya and Leonetti [13] in the case where Ω is convex and S = Ω.

We recall that an open set Ω ⊂ ℝ^{N} is starshaped with respect to a set S ⊂ Ω if Ω is starshaped with respect to each point of S, i.e. if x ∈ Ω and s ∈ S then θx + (1 − θ) s ∈ Ω for all θ ∈ (0, 1) .

Proposition 9.1 Let Ω ⊂ ℝ^{N} be an open bounded set, starshaped with respect to a set S ⊂ Ω, with |S| > 0. Let g : [0, ∞) → [0, ∞) be a convex function, with g (0) = 0. Let u ∈ W^{1,1} (Ω; ℝ^{d}) be such that g (|∇u|) ∈ L^{1} (Ω) . Then

∫_{Ω} g (\frac{|u (x) − u_{S}|}{d}) dx ≤ (\frac{α_{N} d^{N}}{|Ω|} )^{1− \frac{1}{N}} \frac{|Ω|}{|S|} ∫_{Ω} g (|∇u|) dx,

where u_{S} := \frac{1}{|S|} ∫_{S} u dx, d is any number greater or equal than the diameter of Ω, and α_{N} is the volume of the unit ball in ℝ^{N}.

Proof. We follow Lemma 7.16 in Gilbarg and Trudinger [27]. Assume first that u ∈ W^{1,1} (Ω; ℝ^{d}) ∩ C^{1} (Ω; ℝ^{d}) . Since Ω is starshaped with respect to S ⊂ Ω, for x ∈ Ω and y ∈ S we have

u (x) − u (y) = − ∫_{0}^{|x−y|} D_{r}u (x + rω) dr, ω = \frac{y − x}{|y − x|}.

Averaging with respect to y over S yields

u (x) − u_{S} = −\frac{1}{|S|} ∫_{S} dy ∫_{0}^{|x−y|} D_{r}u (x + rω) dr.

Since |x − y| ≤ d we have

\frac{|u (x) − u_{S}|}{d} ≤ \frac{1}{|S|} ∫_{S} \frac{1}{|x − y|} ∫_{0}^{|x−y|} |D_{r}u (x + rω)| drdy.

As g is convex, it now follows from applying twice Jensen’s inequality that

g (\frac{|u (x) − u_{S}|}{d}) ≤ \frac{1}{|S|} ∫_{S} \frac{1}{|x − y|} ∫_{0}^{|x−y|} g (|D_{r}u (x + rω)|) drdy.

Defining

V (x) = \{ \begin{matrix}|∇u (x)| & x ∈ Ω, \\ 0 & x ∉ Ω, \end{matrix}

and, as g is increasing, we have

g (\frac{|u (x) − u_{S}|}{d}) ≤ \frac{1}{|S|} ∫_{\{y:|x−y|<d\}} \frac{1}{|x − y|} ∫_{0}^{∞} g (V (x + rω)) drdy

= \frac{1}{|S|} ∫_{0}^{∞} ∫_{|ω|=1} ∫_{0}^{d} g (V (x + rω)) ρ^{N−2}dρdωdr

= \frac{d^{N−1}}{(N − 1) |S|} ∫_{0}^{∞} ∫_{|ω|=1} g (V (x + rω)) dωdr

= \frac{d^{N−1}}{(N − 1) |S|} ∫_{Ω} |x − y|^{1−N} g (|∇u (y)|) dy,

where we have used the fact that g (0) = 0. The theory of Riesz potentials (Lemma 7.12 in Gilbarg and Trudinger [27]) now yields

∫_{Ω} g (\frac{|u (x) − u_{S}|}{d}) dx ≤ \frac{1}{N} (α_{N})^{1− \frac{1}{N}} |Ω|^{\frac{1}{N}} \frac{d^{N−1}}{(N − 1) |S|} ∫_{Ω} g (|∇u (x)|) dx

and the proof is complete.

Proposition 9.2 Let Ω ⊂ ℝ^{N} be an open bounded domain having the cone property, let g : [0, ∞) → [0, ∞) be a convex function satisfying the doubling condition, with g (0) = 0. Let u ∈ W^{1,1} (Ω; ℝ^{d}) be such that g (|∇u|) ∈ L^{1} (Ω) . Then

∫_{Ω} g (|u (x) − u_{B}|) dx ≤ C ∫_{Ω} g (|∇u|) dx,

where

u_{B} := \frac{1}{|B|} ∫_{B} u (y) dy,

B is any fixed ball whose closure is contained in Ω, and C is a positive constant depending only on Ω and on the ball B.

Proof. Since Ω has the cone property, it is the union of a finite number of domains starshaped with respect to a ball. Let d be a number greater than the diameter of all these domains, and let A be any of these subdomains with D being the corresponding ball. Construct a finite family of balls B_{0}, ⋯, B_{M} contained in Ω and such that B_{0} = D, B_{i} ∩ B_{i+1} ≠ ∅, B_{M} = B. Since A is starshaped with respect to any fixed ball B̃ contained in B_{0} ∩ B_{1}, by Proposition 7.4 we obtain

∫_{A} g (\frac{|u (x) − uB̃|}{d}) dx ≤ (\frac{α_{N} d^{N}}{|A|} )^{1− \frac{1}{N}} \frac{|A|}{|B̃|} ∫_{A} g (|∇u|) dx.

By Remark 7.3 and (7.4)

∫_{A} g (\frac{|u (x)|}{d}) dx ≤ C |A| g (\frac{|uB̃|}{d}) + C (\frac{α_{N} d^{N}}{|A|} )^{1− \frac{1}{N}} \frac{|A|}{|B̃|} ∫_{A} g (|∇u|) dx

≤ C \frac{|A|}{|B̃|} ∫B̃ g (\frac{|u (x)|}{d}) dx + C (\frac{α_{N} d^{N}}{|A|} )^{1− \frac{1}{N}} \frac{|A|}{|B̃|} ∫_{A} g (|∇u|) dx,

where we have used Jensen’s inequality. Hence

∫_{A} g (\frac{|u (x)|}{d}) dx ≤ C \frac{|A|}{|B̃|} ∫_{B_{0}∩B_{1}} g (\frac{|u (x)|}{d}) dx + C (\frac{α_{N} d^{N}}{|A|} )^{1− \frac{1}{N}} \frac{|A|}{|B̃|} ∫_{A} g (|∇u|) dx.

Similarly, since for i = 1, ⋯, M − 1 the ball B_{i} is starshaped with respect to any fixed ball B̃_{i} contained in B_{i} ∩ B_{i+1} ≠ ∅, we obtain

∫_{B_{i}} g (\frac{|u (x)|}{d}) dx ≤ C \frac{|B_{i}|}{|B̃_{i}|} ∫_{B_{i}∩B_{i+1}} g (\frac{|u (x)|}{d}) dy + C (\frac{α_{N} d^{N}}{|B_{i}|} )^{1− \frac{1}{N}} \frac{|B_{i}|}{|B̃_{i}|} ∫_{B_{i}} g (|∇u|) dx.

Therefore

∫_{A} g (\frac{|u (x)|}{d}) dx ≤ C (∫_{B} g (\frac{|u (x)|}{d}) dx + ∫_{Ω} g (|∇u|) dx) .

Summing over all A gives

∫_{Ω} g (\frac{|u (x)|}{d}) dx ≤ C (∫_{B} g (\frac{|u (x)|}{d}) dx + ∫_{Ω} g (|∇u|) dx) .  (9.1)

Since B is convex, by Proposition 7.4

∫_{B} g (\frac{|u (x) − u_{B}|}{d}) dx ≤ (\frac{α_{N} d^{N}}{|B|} )^{1− \frac{1}{N}} ∫_{B} g (|∇u|) dx,

where u_{B} := \frac{1}{|B|} ∫_{B} u dx. Replacing u by u − u_{B} in (9.1) we obtain

∫_{Ω} g (\frac{|u (x) − u_{B}|}{d}) dx ≤ C (∫_{B} g (\frac{|u (x) − u_{B}|}{d}) + ∫_{Ω} g (|∇u|) dx)
≤ C ∫_{Ω} g (|∇u|) dx.

Applying the latter inequality to du in place of u yields

∫_{Ω} g (|u (x) − u_{B}|) dx ≤ C ∫_{Ω} g (d |∇u|) dx ≤ C_{1} ∫_{Ω} g (|∇u|) dx,

where we have used the fact that g (dz) ≤const. g (z) for all z ≥ 0 (see Remark 7.3 and (7.4)). This concludes the proof.

Acknowledgements

The study of the compactness and the characterization of Γ−liminf (see Sections 3 and 4) were obtained with Luc Tartar in 1989, and very many discussions on the subject unfolded with him since then, for which the authors are profoundly indebted.

Also, stimulating conversations with Giovanni Alberti, Georg Dolzmann and Stefan Müller are gratefully acknowledged.

The research of S. Conti was partially supported by the EU TMR network “Phase Transitions in Crystalline Solids”, contract No. FMRX-CT98-0229. The research of I. Fonseca was partially supported by the National Science Foundation under Grant No. DMS–9731957. The research of G. Leoni was partially supported by MURST, by the Italian CNR, through the strategic project “Metodi e modelli per la Matematica e l’Ingegneria”, and by GNAFA.

The authors wish to thank the Center for Nonlinear Analysis (NSF Grant No. DMS–9803791) for its support during the preparation of this paper.

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