Sixth Mississippi State Conference on Differential Equations and Computational Simulations, Electronic Journal of Differential Equations, Conference 15 (2007), pp. 41–50. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu (login: ftp)

YOUNG MEASURE MINIMIZERS IN THE ASYMPTOTIC ANALYSIS OF THIN FILMS

MARIAN BOCEA

Dedicated to Klaus Schmitt on the occasion of his 65th birthday

Abstract. An integral representation for a relaxed functional arising in the membrane theory is obtained in terms of Young measures generated by sequences \{(∇_{α}u_{ε_{n}} | \frac{1}{ε_{n}} ∇_{3}u_{ε_{n}})\} of scaled gradients.

1. Introduction

Let ω ⊂ ℝ^{2} be an open bounded domain with Lipschitz boundary, and consider a thin three dimensional domain Ω_{ε} := ω × (−ε, ε) filled with an elastic material with elastic energy density W_{ε} and subject to dead loading body forces of densities f_{ε} ∈ L^{p′}(Ω_{ε}, ℝ^{3}) where 1 < p < +∞, and p′ stands for the conjugate exponent of p, i.e. \frac{1}{p} + \frac{1}{p′} = 1. We assume that for fixed ε > 0, in order to reach equilibrium, u_{ε} seeks to minimize

E_{ε}(u) := ∫_{Ω_{ε}} W_{ε}(x, ∇u(x)) dx − ∫_{Ω_{ε}} f_{ε}(x) · u(x) dx,

among all kinematically admissible fields u.

To study the effective behavior of a very thin film, we consider a sequence {ε_{n}} of positive real numbers (half-thickness) converging to zero and we recast energy functionals over varying domains Ω_{ε_{n}} into functionals with a fixed domain of integration Ω := ω × (−1, 1) by means of a reformulation of the problem through a \frac{1}{ε_{n}} -dilation in the transverse direction x_{3}. With x = (x_{1}, x_{2}, x_{3}), set

W^{(ε_{n})}(x, ·) := W_{ε_{n}}(x_{1}, x_{2}, ε_{n}x_{3}; ·),
f^{(ε_{n})}(x) := f_{ε_{n}}(x_{1}, x_{2}, ε_{n}x_{3}),
v_{n}(x) := u_{ε_{n}}(x_{1}, x_{2}, ε_{n}x_{3}).

After an appropriate rescaling, v_{n} seeks to minimize

E^{(ε_{n})}(v) := ∫_{Ω} W^{(ε_{n})} (x, (∇_{α}v|\frac{1}{ε_{n}} ∇_{3}v) (x)) dx − ∫_{Ω} f^{(ε_{n})}(x) · v(x) dx,

2000 Mathematics Subject Classification. 49J45, 74B20, 74G10, 74K15, 74K35.

Key words and phrases. Equi-integrability; concentrations; oscillations; relaxation;

Young measure.

\begin{matrix}c \\ ◯ \end{matrix}2007 Texas State University - San Marcos.

Published February 28, 2007.

among all kinematically admissible fields v = (v_{1}, v_{2}, v_{3}) on Ω, where ∇_{α}v stands for the 3 × 2 matrix of partial derivatives \frac{∂v_{i}}{∂x_{α}} , i ∈ \{1, 2, 3\}, α ∈ \{1, 2\}, ∇_{3}v is the three-dimensional vector of partial derivatives \frac{∂v_{i}}{∂x_{3}} , i ∈ \{1, 2, 3\}, and (A|a) denotes a 3 × 3 matrix whose first two columns are those of the 3 × 2 matrix A and the last column is the vector a ∈ ℝ^{3}. We assume that the rescaled energy density W^{(ε_{n})} does not explicitly depend on ε_{n}. Precisely, W^{(ε_{n})} = W where W : Ω × 𝕄^{3×3} → ℝ is a Carathéodory integrand (see Definition 2.6) satisfying for some 1 < p < +∞ the p-growth and coercivity condition

\frac{1}{C} |A|^{p} − C ≤ W (x, A) ≤ C(1 + |A|^{p})  (1.1)

for L^{3}-a.e x ∈ Ω and for all A ∈ 𝕄^{3×3}, where C > 0 is a real constant and 𝕄^{3×3} denotes the space of real 3 × 3 matrices endowed with the usual Euclidean norm |A| := \sqrt{tr (A^{T} A)}. Assuming, moreover, that the rescaled body force density f^{(ε_{n})} is independent of n (see e.g. [17]), the study of the effective energy of the limiting system is hinged on the understanding of the asymptotic behavior of the energies

I_{n}(v_{n}) := ∫_{Ω} W (x, (∇_{α}v_{n}|\frac{1}{ε_{n}} ∇_{3}v_{n}) (x)) dx.

An extensive literature in this direction (see [2, 6, 7, 11, 12, 13, 17, 23, 24], among others) is usually formulated in the natural mathematical setting of Γ-convergence, and this approach gives rise to the so-called membrane theory. In view of the a priori bound

sup_{n∈ℕ} ∫_{Ω} | (∇_{α}v_{n}|\frac{1}{ε_{n}} ∇_{3}v_{n}) (x)|^{p}dx < +∞

for energy bounded sequences, and derived from (1.1), in this paper we obtain an integral representation of the relaxed energy functional W : W^{1,p}(ω; ℝ^{3}) × L^{p}(Ω; ℝ^{3}) → ℝ defined by

W(v, c) := inf \{ lim inf_{n→+∞} ∫_{Ω} W (x, (∇_{α}v_{n}|\frac{1}{ε_{n}} ∇_{3}v_{n}) (x)) dx : ε_{n} → 0^{+},

v_{n} ⇀ v weakly in W^{1,p}(Ω; ℝ^{3}),

\frac{1}{ε_{n}} ∇_{3}v_{n} ⇀ c weakly in L^{p}(Ω; ℝ^{3})\},

(1.2)

in terms of scaled gradient p-Young measures, which are essentially Young measures generated by sequences of scaled gradients \{(∇_{α}v_{n}|\frac{1}{ε_{n}} ∇_{3}v_{n})\} (see Definitions 2.1 and 2.5).

Definition 1.1. Let Ω := ω × (−1, 1), where ω ⊂ ℝ^{2} is an open domain, and let 1 ≤ p ≤ +∞. A Young measure μ on Ω × ℝ^{9} is called a scaled gradient p-Young measure (scaled gradient Young measure if p = +∞) if there exist sequences ε_{n} → 0^{+} and {v_{n}} ⊂ W^{1,p}(Ω; ℝ^{3}) such that

(i) {v_{n}} is weakly (weakly * if p = +∞) convergent in W^{1,p}(Ω; ℝ^{3}), (ii) \{\frac{1}{ε_{n}} ∇_{3}v_{n}\} is weakly (weakly * if p = +∞) convergent in L^{p}(Ω; ℝ^{3}), (iii) E_{(∇_{α}v_{n}|\frac{1}{εn} ∇_{3}v_{n})} ⇀ μ weakly * in C_{0}(Ω × ℝ^{9})′.

The weak (weak * if p = +∞) limit of v_{n} in W^{1,p}(Ω; ℝ^{3}) is called an underlying deformation for μ while the weak (weak * if p = +∞) limit of \frac{1}{ε_{n}} ∇_{3}v_{n} in L^{p}(Ω; ℝ^{3}) is called a Cosserat vector associated to μ. For 1 ≤ p < +∞, we set

Y_{v,c}^{p} := {ν ∈ Y (Ω × ℝ^{9}) : ν is a scaled gradient p-Young measure with

underlying deformation v and associated Cosserat vector c}.

Our relaxation result is the following.

Theorem 1.2. Let 1 < p < +∞, Ω := ω × (−1, 1), where ω ⊂ ℝ^{2} is an open, bounded Lipschitz domain, and let W : Ω × 𝕄^{3×3} → ℝ be a Carathéodory integrand satisfying (1.1). Let v ∈ W^{1,p}(ω; ℝ^{3}) and c ∈ L^{p}(Ω; ℝ^{3}) be given, and define the relaxed functional W : W^{1,p}(ω; ℝ^{3}) × L^{p}(Ω; ℝ^{3}) → ℝ by (1.2). Then

W(v, c) = inf_{ν∈Y_{v,c}^{p}} ∫_{Ω×𝕄^{3×3}} W dν.  (1.3)

Moreover, the infimum on the right hand side is attained, i.e. there exists a scaled gradient p-Young measure μ_{0} ∈ Y_{v,c}^{p} such that

inf_{ν∈Y_{v,c}^{p}} ∫_{Ω×𝕄^{3×3}} W dν = ∫_{Ω×𝕄^{3×3}} W dμ_{0}.  (1.4)

A challenging open problem is the identification of algebraic and analytical conditions on parametrized probability measures both necessary and sufficient to guarantee that they belong to Y_{v,c}^{p}. The corresponding program for probability measures generated by gradients bounded in L^{p} was carried out by Kinderlehrer and Pedregal (see [19, 20]) and subsequently generalized to the realm of A-quasiconvexity by Fonseca and Müller in [15]. In the case of Young measures generated by sequences of scaled gradients, recent progress was made by Bocea and Fonseca [9], where the slightly broader class of bending Young measures has been completely characterized.

The crucial ingredient needed to prove Theorem 1.2 is the following decomposition result whose proof may be found in [8].

Theorem 1.3. Let Ω := ω × (−1, 1), where ω ⊂ ℝ^{2} is an open, bounded Lipschitz domain, let {ε_{n}} be a sequence of positive real numbers converging to zero, and consider a sequence {v_{n}} bounded in W^{1,p}(Ω; ℝ^{3}) (1 < p < +∞) and satisfying

sup_{n∈ℕ} ∫_{Ω} | (∇_{α}v_{n}|\frac{1}{ε_{n}} ∇_{3}v_{n}) (x)|^{p}dx < +∞.

Suppose further that v_{n} ⇀ v in W^{1,p}(Ω; ℝ^{3}) and \frac{1}{ε_{n}} ∇_{3}v_{n} ⇀ c in L^{p}(Ω; ℝ^{3}). Then there exists a subsequence {v_{n_{k}}} of {v_{n}} and a sequence {w_{k}} ⊂ W^{1,p}(Ω; ℝ^{3}) such that

lim_{k→∞} L^{3} ({x ∈ Ω : w_{k}(x) ≠ v_{n_{k}}(x)} ∪ {x ∈ Ω : ∇w_{k}(x) ≠ ∇v_{n_{k}}(x)}) = 0, (1.5)
\{(∇_{α}w_{k}|\frac{1}{ε_{n_{k}}} ∇_{3}w_{k})\} is p-equi-integrable,  (1.6)
w_{k} ⇀ v weakly in W^{1,p}(Ω; ℝ^{3}),  (1.7)
\frac{1}{ε_{n_{k}}} ∇_{3}w_{k} ⇀ c weakly in L^{p}(Ω; ℝ^{3}).  (1.8)

This theorem allows us to decompose (up to a subsequence) \{(∇_{α}v_{n}|\frac{1}{ε_{n}} ∇_{3}v_{n})\} as a sum of a sequence \{(∇_{α}w_{n}|\frac{1}{ε_{n}} ∇_{3}w_{n})\} whose p-th power is equi-integrable and a remainder converging to zero in measure. We may say that \{(∇_{α}w_{n}|\frac{1}{ε_{n}} ∇_{3}w_{n})\} carries the oscillations, while the remainder accounts for the concentration effects.

In the next section we present the basic facts about Young measures needed for the proof of our relaxation theorem. The proof of Theorem 1.2 is the subject of Section 3 of the paper.

2. Young measures

The characterization of (oscillatory) limits of nonlinear quantities in the Calculus of Variations has been successfully analyzed in several contexts by means of Young measures. Young measures were originally introduced in Optimal Control Theory by L.C. Young in connection to nonconvex problems, thus providing the appropriate framework for the description of generalized minimizers in the Calculus of Variations (see [29, 30]). Later, Tartar introduced the use of Young measures in the PDE framework (see [26, 27, 28]). For more details regarding the study of Young measures we refer the reader to [3, 4, 5, 16, 19, 20, 21, 22, 25], among others.

In this section we recall the definition and the relevant results about Young measures that we will need in the sequel; we follow closely the approach of Kristensen (see [22]). Let D be an open subset of ℝ^{l}(l ≥ 1), C(D) be the space of real-valued continuous functions on D and define

C_{0}(D) := {φ ∈ C(D) : for every ε > 0 there exists a compact set K ⊂ D
such that |φ(x)| ≤ ε if x ∈ D \ K}.

Endowed with the supremum norm, C_{0}(D) is a separable Banach space. In view of Riesz’ Theorem the dual space C_{0}(D)′ can be identified with the space of bounded Radon measures on D with the norm ∥μ∥ := |μ|(D), via the duality pairing

⟨μ, φ⟩ = ∫_{D} φ(x) · \frac{dμ}{d|μ|}(x)d|μ|(x),

where |μ| stands for the total variation of μ and is a non-negative, finite Radon measure on D.

Definition 2.1. (i) A non-negative Radon measure μ on Ω × ℝ^{d} with the property

μ(B × ℝ^{d}) = L^{N}(B) for all Borel subsets of Ω,

is called a Young measure. The set of Young measures on Ω×ℝ^{d} is denoted by Y (Ω × ℝ^{d}).

(ii) A Young measure μ for which there exists a L^{N}-measurable mapping V : Ω → ℝ^{d} such that

∫_{Ω×ℝ^{d}} fdμ = ∫_{Ω} f(x, V (x))dx, for all f ∈ C_{0}(Ω × ℝ^{d}),

is called an elementary Young measure. We write

μ = E_{V} := ∫_{Ω} δ_{x} ⊗ δ_{V (x)}dx,

where δ_{x} and δ_{V (x)} are the Dirac measures on Ω concentrated at x and on ℝ^{d} concentrated at V (x), respectively.

(iii) A product measure (L^{N}⌊Ω)⊗μ̃ on Ω×ℝ^{d}, where μ̃ is a probability measure on ℝ^{d}, is called a homogeneous Young measure.

Remark 2.2. The definition of Young measures in Definition 2.1 (i) follows that of Berliocchi and Lasry (see [5]). It can be shown (cf. [22]) to be equivalent to the original definition of L.C. Young [29] and the ones used in literature (e.g., [3, 4, 25]).

Proposition 2.3. Let μ ∈ Y (Ω × ℝ^{d}). Then there exists a mapping x ↦ μ_{x} from Ω into the set of non-negative, finite Radon measures on ℝ^{d}, such that

(i) μ = ∫_{Ω} δ_{x} ⊗ μ_{x}dx, i.e. for any Borel function f : Ω × ℝ^{d} → [0, +∞] the

function x ↦ ∫_{ℝ^{d}} f(x, A)dμ_{x}(A) is L^{N}-measurable, and
∫_{Ω×ℝ^{d}} fdμ = ∫_{Ω} ∫_{ℝ^{d}} f(x, A)dμ_{x}(A)dx;  (2.1)
(ii) μ_{x}(ℝ^{d}) = 1, for L^{N}-a.e. x ∈ Ω.

Moreover, if x ↦ ν_{x} is another such mapping then ν_{x} = μ_{x} for L^{N}-a.e. x ∈ Ω.

Remark 2.4. Proposition 2.3 is a special case of a result in [10] (Proposition 13, pp. 39-40). See also [1].

Consider a sequence {V_{n}} of measurable mappings of Ω into ℝ^{d}. The corresponding sequence {E_{V_{n}}} of elementary Young measures is bounded in C_{0}(Ω × ℝ^{d})′ and thus, by virtue of Banach-Alaoglu’s Theorem, there exists a subsequence {V_{n_{k}}} and a measure μ ∈ C_{0}(Ω × ℝ^{d})′ such that

E_{V_{n_{k}}} ⇀ μ weakly ∗ in C_{0}(Ω × ℝ^{d})′.  (2.2)

A necessary and sufficient condition for μ to be a Young measure is that

lim_{R→∞} sup_{k∈ℕ} L^{N} ({x ∈ Ω : |V_{n_{k}}(x)| ≥ R}) = 0,  (2.3)

or, equivalently (see [18, 21]): There exists a Borel function g : ℝ^{d} → [0, +∞] such that lim_{|A|→+∞} g(A) = +∞, and

sup_{k∈ℕ} ∫_{Ω} g(V_{n_{k}}(x))dx < +∞.

Definition 2.5. If (2.2) and (2.3) hold, then we say that the Young measure μ is generated by the sequence {V_{n_{k}}}.

Definition 2.6. (i) A function f : Ω × ℝ^{d} → ℝ ∪ {+∞} is called a normal integrand if f is Borel measurable and f(x, ·) : ℝ^{d} → ℝ ∪ {+∞} is lower semicontinuous for every x ∈ Ω.

(ii) A real-valued function f : Ω × ℝ^{d} → ℝ is called a Carathéodory integrand if both f and −f are normal integrands.

Set f^{−} := − min{f, 0}. The following result is well-known (see [3, 4, 5, 14, 20, 21, 22, 25]).

Lemma 2.7. Let {v_{n}} be a sequence of measurable mappings from Ω into ℝ^{d} which generates the Young measure μ.

(i) If f : Ω × ℝ^{d} → ℝ ∪ {+∞} is a normal integrand and if {f^{−}(·, v_{n})} is equi-integrable then

∫_{Ω×ℝ^{d}} fdμ ≤ lim inf_{n→∞} ∫_{Ω} f(x, v_{n}(x))dx.

Moreover, if f is a Carathéodory integrand then {f(·, v_{n})} is equi-integrable if and only if

∫_{Ω×ℝ^{d}} fdμ = lim_{n→∞} ∫_{Ω} f(x, v_{n}(x))dx.

(ii) If {w_{n}} is a sequence of measurable mappings from Ω into ℝ^{d} such that v_{n} − w_{n} → 0 in measure then {w_{n}} also generates μ.

3. Proof of Theorem 1.2

We will identify ℝ^{9} with the space of real 3 × 3 matrices 𝕄^{3×3}. To prove that

W(v, c) ≥ inf_{ν∈Y_{v,c}^{p}} ∫_{Ω×𝕄^{3×3}} W dν,  (3.1)

let ε_{n} → 0^{+} and {v_{n}} ⊂ W^{1,p}(Ω; ℝ^{3}) be such that v_{n} ⇀ v weakly in W^{1,p}(Ω; ℝ^{3}) and \frac{1}{ε_{n}} ∇_{3}v_{n} ⇀ c weakly in L^{p}(Ω; ℝ^{3}). Extract a subsequence (not relabelled) so that

lim inf_{n→∞} ∫_{Ω} W (x, (∇_{α}v_{n}|\frac{1}{ε_{n}} ∇_{3}v_{n}) (x)) dx

= lim_{n→∞} ∫_{Ω} W (x, (∇_{α}v_{n}|\frac{1}{ε_{n}} ∇_{3}v_{n}) (x)) dx < +∞,

(3.2)

where the last inequality follows by (1.1). By Banach-Alaoglu’s Theorem, there exists a subsequence \{(∇_{α}v_{n_{j}} |\frac{1}{ε_{n_{j}}} ∇_{3}v_{n_{j}})\} of \{(∇_{α}v_{n}|\frac{1}{ε_{n}} ∇_{3}v_{n})\} such that

E_{(∇_{α}v_{n_{j}} |_{\frac{1}{εn_{j}}} ∇_{3}v_{n_{j}} )} ⇀ μ weakly * in C_{0}(Ω × ℝ^{9})′,

for some Radon measure μ on Ω × ℝ^{9}. Since

sup_{j∈ℕ} ∫_{Ω} | (∇_{α}v_{n_{j}} |\frac{1}{ε_{n_{j}}} ∇_{3}v_{n_{j}}) (x)|^{p}dx < +∞,

it follows that (2.3) holds, with k replaced by j, and V_{n_{j}} by (∇_{α}v_{n_{j}} |\frac{1}{ε_{n_{j}}} ∇_{3}v_{n_{j}}). Thus, μ is a Young measure. It is clear that we actually have μ ∈ Y_{v,c}^{p}. By (1.1), W (x, A) + C ≥ 0 for L^{3}-a.e. x ∈ Ω and all A ∈ 𝕄^{3×3}. Thus, we can apply Lemma 2.7 (i) (take f = W + C) and we obtain

∫_{Ω×𝕄^{3×3}}(W + C)dμ ≤ lim inf_{j→∞} ∫_{Ω} (W (x, (∇_{α}v_{n_{j}} |\frac{1}{ε_{n_{j}}} ∇_{3}v_{n_{j}}) (x)) + C) dx

= lim_{j→∞} ∫_{Ω} W (x, (∇_{α}v_{n_{j}} |\frac{1}{ε_{n_{j}}} ∇_{3}v_{n_{j}}) (x)) dx + CL^{3}(Ω)

= lim_{n→∞} ∫_{Ω} W (x, (∇_{α}v_{n}|\frac{1}{ε_{n}} ∇_{3}v_{n}) (x)) dx + CL^{3}(Ω),

where we have used (3.2). By Proposition 2.3 we have

∫_{Ω×𝕄^{3×3}} Cdμ = ∫_{Ω} ∫_{𝕄^{3×3}} Cdμ_{x}dx = C ∫_{Ω} μ_{x}(𝕄^{3×3})dx = CL^{3}(Ω),

and in view of the previous equation we obtain

∫_{Ω×𝕄^{3×3}} W dμ ≤ lim inf_{n→∞} ∫_{Ω} W (x, (∇_{α}v_{n}|\frac{1}{ε_{n}} ∇_{3}v_{n}) (x)) dx.

Thus,

inf_{ν∈Y_{v,c}^{p}} ∫_{Ω×𝕄^{3×3}} W dν ≤ lim inf_{n→∞} ∫_{Ω} W (x, (∇_{α}v_{n}|\frac{1}{ε_{n}} ∇_{3}v_{n}) (x)) dx.

By the arbitrariness of {ε_{n}} and {v_{n}} satisfying the admissibility conditions we assert (3.1).

Conversely, let ν ∈ Y_{v,c}^{p}. There exist sequences ε_{n} → 0^{+} and {v_{n}} ⊂ W^{1,p}(Ω; ℝ^{3}) such that v_{n} ⇀ v weakly in W^{1,p}(Ω; ℝ^{3}), \frac{1}{ε_{n}} ∇_{3}v_{n} ⇀ c weakly in L^{p}(Ω; ℝ^{3}), and the sequence \{(∇_{α}v_{n}|\frac{1}{ε_{n}} ∇_{3}v_{n})\} generates ν.

By Theorem 1.3 there exist a subsequence {v_{n_{k}}} of {v_{n}} and a sequence {w_{k}} ⊂ W^{1,p}(Ω; ℝ^{3}) such that (1.5)-(1.8) hold. In view of (1.5), the sequence

\{(∇_{α}w_{k}|\frac{1}{ε_{n_{k}}} ∇_{3}w_{k}) − (∇_{α}v_{n_{k}} |\frac{1}{ε_{n_{k}}} ∇_{3}v_{n_{k}})\}

converges to zero in measure and thus, by Lemma 2.7 (ii), \{(∇_{α}w_{k}|\frac{1}{ε_{n_{k}}} ∇_{3}w_{k})\} also

generates ν. Since by (1.1) and (1.6) the sequence \{W (·, (∇_{α}w_{k}|\frac{1}{ε_{n_{k}}} ∇_{3}w_{k}))\} is equi-integrable, we deduce by (i) of Lemma 2.7 that

∫_{Ω×𝕄^{3×3}} W dν = lim_{k→∞} ∫_{Ω} W (x, (∇_{α}w_{k}|\frac{1}{ε_{n_{k}}} ∇_{3}w_{k}) (x)) dx ≥ W(v, c),

where the last inequality follows by (1.7) and (1.8). Passing to the infimum over all ν ∈ Y_{v,c}^{p} we have that

inf_{ν∈Y_{v,c}^{p}} ∫_{Ω×𝕄^{3×3}} W dν ≥ W(v, c).

Taking into account (3.1), we obtain that (1.3) holds. It remains to prove (1.4).

Claim: There exist sequences ε_{n} → 0^{+} and {v_{n}} ⊂ W^{1,p}(Ω; ℝ^{3}) such that

v_{n} ⇀ v weakly in W^{1,p}(Ω; ℝ^{3}),

\frac{1}{ε_{n}} ∇_{3}v_{n} ⇀ c weakly in L^{p}(Ω; ℝ^{3}),

lim_{n→∞} ∫_{Ω} W (x, (∇_{α}v_{n}|\frac{1}{ε_{n}} ∇_{3}v_{n}) (x)) dx = W(v, c).

(3.3)

Assuming that the claim holds, let μ_{0} be the scaled gradient p-Young measure generated by a subsequence (not relabelled) of \{(∇_{α}v_{n}|\frac{1}{ε_{n}} ∇_{3}v_{n})\}. Let \{n_{k}\} ⊂ \{n\} and {w_{k}} ⊂ W^{1,p}(Ω; ℝ^{3}) be the sequences provided by Theorem 1.3. Taking into account (1.5)-(1.8), (3.3), and making use of Lemma 2.7, we have

W(v, c) ≤ lim inf_{k→∞} ∫_{Ω} W (x, (∇_{α}w_{k}|\frac{1}{ε_{n_{k}}} ∇_{3}w_{k}) (x)) dx

= ∫_{Ω×𝕄^{3×3}} W dμ_{0}

≤ lim_{n→∞} ∫_{Ω} W (x, (∇_{α}v_{n}|\frac{1}{ε_{n}} ∇_{3}v_{n}) (x)) dx

= W(v, c).

Thus,

∫_{Ω×𝕄^{3×3}} W dμ_{0} = W(v, c),

and in view of (1.3), we deduce (1.4).

Proof of Claim: For any n ∈ ℕ, let {ε_{k,n}} ⊂ (0, +∞) and {v_{k,n}} ⊂ W^{1,p}(Ω; ℝ^{3}) be such that lim_{k→∞} ε_{k,n} = 0, v_{k,n} ⇀ v weakly in W^{1,p}(Ω; ℝ^{3}), \frac{1}{ε_{k,n}} ∇_{3}v_{k,n} ⇀ c weakly in L^{p}(Ω; ℝ^{3}) as k → ∞ and, in addition,

W(v, c) ≤ lim inf_{k→∞} ∫_{Ω} W (x, (∇_{α}v_{k,n}|\frac{1}{ε_{k,n}} ∇_{3}v_{k,n}) (x)) dx ≤ W(v, c) + \frac{1}{n}.

Extract an increasing subsequence {k(j, n)}_{j} of {k} so that

lim inf_{k→∞} ∫_{Ω} W (x, (∇_{α}v_{k,n}|\frac{1}{ε_{k,n}} ∇_{3}v_{k,n}) (x)) dx

= lim_{j→∞} ∫_{Ω} W (x, (∇_{α}v_{k(j,n),n}|\frac{1}{ε_{k(j,n),n}} ∇_{3}v_{k(j,n),n}) (x)) dx,

and put \overline{ε}_{j,n} := ε_{k(j,n),n} and \overline{v}_{j,n} := v_{k(j,n),n}. Thus,

lim_{j→∞} \overline{ε}_{j,n} = 0,  (3.4)
lim_{n→∞} lim_{j→∞} ∫_{Ω} W (x, (∇_{α}\overline{v}_{j,n}|\frac{1}{\overline{ε}_{j,n}} ∇_{3}\overline{v}_{j,n}) (x)) dx = W(v, c),  (3.5)

\overline{v}_{j,n} ⇀ v weakly in W^{1,p}(Ω; ℝ^{3}) as j → ∞, and

\frac{1}{\overline{ε}_{j,n}} ∇_{3}\overline{v}_{j,n} ⇀ c weakly in L^{p}(Ω; ℝ^{3}) as j → ∞.

Consider a countable family \{φ_{i}\}_{i∈ℕ} dense in L^{p′}(Ω). The weak convergence of \overline{v}_{j,n} to v in W^{1,p}(Ω; ℝ^{3}) and that of \frac{1}{\overline{ε}_{j,n}} ∇_{3}\overline{v}_{j,n} to c in L^{p}(Ω; ℝ^{3}) imply that for each i, n ∈ ℕ we have

lim_{j→∞} ∫_{Ω} φ_{i}(x)\overline{v}_{j,n}(x)dx = ∫_{Ω} φ_{i}(x)v(x)dx,  (3.6)
lim_{j→∞} ∫_{Ω} φ_{i}(x)(\frac{1}{\overline{ε}_{j,n}} ∇_{3}\overline{v}_{j,n}(x))dx = ∫_{Ω} φ_{i}(x)c(x)dx,  (3.7)
lim_{j→∞} ∫_{Ω} φ_{i}(x)∇\overline{v}_{j,n}(x)dx = ∫_{Ω} φ_{i}(x)∇v(x)dx.  (3.8)

Taking into account (3.4), (3.5), (3.6), (3.7), and (3.8), a diagonalization process allows us to find an increasing subsequence {j(n)} of {j} such that, after denoting ε_{n} := \overline{ε}_{j(n),n} and v_{n} := \overline{v}_{j(n),n} we have

ε_{n} → 0^{+}, v_{n} ⇀ v weakly in W^{1,p}(Ω; ℝ^{3}),

\frac{1}{ε_{n}} ∇_{3}v_{n} ⇀ c weakly in L^{p}(Ω; ℝ^{3}),

and (3.3) holds. □

Acknowledgement. Part of this work has been written while the author was supported by a Burgess Assistant Professorship at the University of Utah.

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Marian Bocea Department of Mathematics, 300 Minard Hall, North Dakota State University, Fargo, ND 58105-5075, USA

E-mail address: marian.bocea@ndsu.edu