3D-2D ASYMPTOTIC ANALYSIS FOR INHOMOGENEOUS THIN FILMS

ANDREA BRAIDES, IRENE FONSECA, AND GILLES FRANCFORT

Abstract. A dimension reduction analysis is undertaken using Γ-convergence techniques within a relaxation theory for 3D nonlinear elastic thin domains of the form

Ω_{ε} := {(x_{1}, x_{2}, x_{3}) : (x_{1}, x_{2}) ∈ ω, |x_{3}| < εf_{ε}(x_{1}, x_{2})},

where ω is a bounded domain of ℝ^{2} and f_{ε} is an ε-dependent profile. An abstract representation of the effective 2D energy is obtained, and specific characterizations are found for nonhomogeneous plate models, periodic profiles, and within the context of optimal design for thin films.

Keywords : dimension reduction, Γ-convergence, plate models, periodicity, relaxation. 1998 Mathematics Subject Classification: 35E99, 35M10, 49J45, 74B20, 74K15, 74K20, 74K35.

Contents

1. Introduction 1 2. A compactness result in a general setting 4 3. First application – Nonhomogeneous plate models 11 4. Second application –The periodic case 16 5. Third application – Optimal design of a thin film 19 6. Final Remarks 25 7. Appendix 25 References 26

1. Introduction

Dimensional reduction through asymptotic analysis is by now a well established theory in a linear setting. Specifically, the work of Ciarlet et al. [13], [15], has paved the way for a variety of studies ranging from linearly elastic plates [12], [13], to various beam models [21], [22], or shells [14], and also spanning various constitutive behaviors [8].

There have, however, been comparatively few studies in a nonlinear setting (other than the semi-linear setting of [13]). To our knowledge, a quasi-exhaustive list can be readily drawn: in [24], fully nonlinear homogeneous elastic plate models are obtained, thereby providing a rigorous mathematical framework for prior work [20]. Note the absence, in that work, of the well thought of requirement that the energy density become infinite as the jacobian of the transformation tends to 0. The only attempt in that direction is to be found in [7]. In [1], fully nonlinear beam models are obtained while in [25] a very thorough investigation of the monotone, albeit not necessarily variational, case is undertaken, again in a 3D-1D setting. Finally, a general study of Γ-convergence and dimensional reduction is proposed in [5].

As emphasized in [17], thin film technology has drastically improved as of late and a precise control over thickness as well as material composition of a film is possible. This motivated in part the study in [17] of the optimal(ly worst!) design of a two-phase nonlinearly elastic thin film. In a different direction, “optimal” stiffeners for a linearly elastic plate with fixed average thickness are analyzed in [23] under directional restrictions on the stiffeners, and the existence of a Kirchhofflike plate model is established (at least formally) as limit of 3D domains with (locally) periodic profiles, i.e., profiles of the form {|x_{3}| < εf(x, x/ε^{τ})}, where ε is the thickness of the domain, 0 < τ < ∞ determines the period of the oscillations, and f(x, .) is periodic.

In the present paper, we propose, in the context of fully nonlinear elasticity, a general approach that allows for material heterogeneity as well as rapidly varying profiles. We show in Theorem 2.5 that membrane-type models of the form firstly derived in [24] are generic; of course, Theorem 2.5 is a mere abstract existence result and a more precise determination of the membrane energy density in the spirit of [23] is unfeasible with such a degree of generality. We then proceed in the remainder of the paper to specialize the obtained energy density to more specific settings. Section 3 is devoted to revisiting the model obtained in [24] for transversally inhomogeneous thin domains. Section 4 examines a typical homogenization type problem, namely that in which both microstructure and profile periodically oscillate on a scale that is comparable to that of the thickness of the domain. Finally, Section 5 investigates the optimal problem discussed in [17] without the restriction that the mixtures be of cylindrical type, that is allowing for any kind of two-phase mixture, provided of course that the resulting volume fraction of each material be independent of the transverse variable x_{3}, a must if one is to hope for a plate-like behavior.

It is worthwhile at this point to be somewhat more specific, so as to achieve a better understanding of the scope and limitations of the model. A profiled 3D thin domain Ω(ε) is considered; it is of the form

Ω(ε) := {(x_{1}, x_{2}, x_{3}) : (x_{1}, x_{2}) ∈ ω and |x_{3}| < εf_{ε}(x_{1}, x_{2})},

where ω is a bounded domain of ℝ^{2} and f_{ε}(x_{1}, x_{2}) determines the ε-dependent profile x_{3} = ±f_{ε}(x_{1}, x_{2}). This domain is filled with an elastic material with elastic energy density W (ε)(x_{1}, x_{2}, x_{3}; ·). Let us assume, for the sake of illustration, that Ω(ε) is clamped on its lateral boundary and subject to body loads F (ε)(x_{1}, x_{2}, x_{3}), so that, for fixed ε, in order to reach equilibrium the transformation field u(ε)(x_{1}, x_{2}, x_{3}) seeks to minimize

w ↦ ∫_{Ω(ε)} W (ε)(x_{1}, x_{2}, x_{3}; Dw) dx − ∫_{Ω(ε)} F (ε) · w dx,

among all kinematically admissible fields w. It is tempting to reformulate this problem on a “fixed” domain through a 1/ε-dilation in the transverse direction x_{3}. Set

\begin{aligned}Ω & := & ω × (−1, 1), \\ Ω_{ε} & := & \{(x_{1}, x_{2}, x_{3}) : (x_{1}, x_{2}, εx_{3}) ∈ Ω(ε)\}, \\ u_{ε}(x_{1}, x_{2}, x_{3}) & := & u(ε)(x_{1}, x_{2}, εx_{3}), \\ W_{ε}(x_{1}, x_{2}, x_{3}; ·) & := & W (ε)(x_{1}, x_{2}, εx_{3}; ·), \\ F_{ε}(x_{1}, x_{2}, x_{3}) & := & F (ε)(x_{1}, x_{2}, εx_{3}). \end{aligned}

Equivalently, u_{ε} wants to minimize

v ↦ ∫_{Ω_{ε}} W_{ε} (x_{1}, x_{2}, x_{3}; D_{1}v|D_{2}v|\frac{1}{ε}D_{3}v) dx − ∫_{Ω_{ε}} F_{ε} · v dx

among all kinematically admissible fields v on Ω_{ε}, where (ξ_{1}|ξ_{2}|ξ_{3}), with ξ_{i} ∈ ℝ^{3}, i = 1, 2, 3, stands for the 3 × 3 matrix with columns ξ_{1}, ξ_{2}, ξ_{3}. Under appropriate coercivity assumptions on W_{ε} (or W (ε)), it is easily checked (cf. Remark 2.3) that, for a subsequence {ε_{k}} of {ε}, there exists u ∈ W^{1,p}(ω; ℝ^{3}) such that (u_{ε_{k}} − u)χ_{Ω_{ε_{k}}} → 0 strongly in L^{p}(ℝ^{3}), where χ_{Ω_{ε_{k}}} denotes the characteristic function of Ω_{ε_{k}}, provided that {F_{ε}χ_{Ω_{ε}}} (resp. {F (ε)χ_{Ω(ε)}}) is bounded in L^{p}(ℝ^{3}), with p > 1. It then makes sense to investigate the Γ(L^{p})–limit of the functionals

v ↦ E_{ε}(v; ω) := ∫_{Ω_{ε}} W_{ε} (x_{1}, x_{2}, x_{3}; D_{1}v|D_{2}v|\frac{1}{ε}D_{3}v) dx

since minimizers of E_{ε} – if they exist – will L^{p}-converge to minimizers of that Γ(L^{p})– limit, and thus a characterization of the latter will entail an asymptotic effective energy for equilibria states of Ω_{ε}. This is what the present paper undertakes.

This approach depends on the adopted scaling in a non trivial way. Indeed, a different kind of estimate on the loads – or, as the language of asymptotics would have it, a different scaling on the loads – will render the subsequent analysis obsolete. In particular, note that the usual scaling of linearized elasticity, that is loads such that \frac{1}{ε}F (ε)_{3} is of the same order as F (ε)_{1}, F (ε)_{2}, is not amenable to the proposed setting; a rescaling of u(ε)_{3} as

u_{ε_{3}}(x_{1}, x_{2}, x_{3}) := εu(ε)_{3}(x_{1}, x_{2}, εx_{3})

is that proposed in linearized elasticity (cf. e.g. [13]). It can be shown, however, to prohibit local models in the limit [18].

We now close this introduction with a few remarks of a mathematical nature. Firstly, it should be noted that there is nothing in the analysis that precludes a higher (or lower) number of horizontal and vertical directions, the setting being then of mappings from ℝ^{N} into ℝ^{d} with N, d ∈ ℕ arbitrary, although the physical meaning becomes dubious. The reader’s attention should be drawn to the pervading problem of the explicit appearance of the parameter ε in the functional. This is a source of numerous difficulties and it prompts extreme caution when extracting subsequences (see e.g. the extraction of the subsequence {ε^{R}} in the proof of Theorem 2.5). We also have to appeal to both Γ–limits and Γ–liminfs. Let us recall that if \{E_{n}\} is a sequence of functions from a Banach space X into \overline{ℝ} and E is a function from X into \overline{ℝ}, then Γ1. E is the Γ(X)– lim inf of E_{n} if, for any x in X,

E(x) = inf_{{x_{n}}}{lim inf_{n→0^{+}} E_{n}(x_{n}) : x_{n} → x in X},

Γ2. E is the Γ(X)– lim sup of E_{ε} if, for every x in X,

E(x) = inf_{{x_{n}}}{lim sup_{n→0^{+}} E_{n}(x_{n}) : x_{n} → x in X}.

Also Γ3. if Γ(X)– lim inf E_{n} = Γ(X)– lim sup E_{n} then the common value is called the

Γ(X)-limit of E_{n}.

Therefore, E(u) = Γ– lim E_{n}(u) if and only if

i) whenever u_{n} → u in X then

E(u) ≤ lim inf_{n→+∞} E_{n}(u_{n}),

ii) there exists a sequence {u_{n}} such that u_{n} → u in X and

E(u) = lim inf_{n→+∞} E_{n}(u_{n}).

Moreover, given a family of maps E_{ε} : X → \overline{ℝ}, ε > 0, and if u ∈ X then we say that Γ4. Γ(X)– lim E_{ε}(u) = E(u) if E(u) = Γ(X)– lim E_{ε_{n}}(u) for every sequence ε_{n} →

0^{+}.

Hence, it can be shown that Γ(X)– lim E_{ε}(u) = E(u) if and only if

i) for every sequences {u_{n}} and {ε_{n}} such that u_{n} → u in X and ε_{n} → 0^{+}

then

E(u) ≤ lim inf_{n→+∞} E_{ε_{n}}(u_{n}),

ii) for every sequence {ε_{n}} converging to 0^{+} there exists a sequence {u_{n}} such that u_{n} → u in X and

E(u) = lim_{n→+∞} E_{ε_{n}}(u_{n}).

Finally, we adopt the following notation: Greek letters will always run from 1 to 2 when taken as indices. Thus coordinates will be denoted by x_{α}, x_{3}. The notation (F_{α}|F_{3}) will refer to the 3 × 3 matrix with column elements F_{1}, F_{2}, F_{3} (3 vectors in ℝ^{3}). We will identify W^{1,p}(Ω) ∩ \{u : \frac{∂u}{∂x_{3}} = 0\} with W^{1,p}(ω) (Ω := ω × (−1, 1)). Also, attention will be paid to the order in which limits are taken. As a last point, → will always denote strong convergence whereas ⇀ (resp. ⇀^{∗}) will denote weak (resp. weak-*) convergence.

2. A compactness result in a general setting

In all that follows, {ε} is any decreasing sequence of real numbers with limit 0. We assume that {W_{ε}(x; F)}_{ε} is a sequence of Carathéodory functions on Ω × ℝ^{3×3} such that, for a.e. x in Ω, and any F in ℝ^{3×3},

β′|F |^{p} ≤ W_{ε}(x; F) ≤ β(1 + |F |^{p}), 0 < β′ ≤ β < ∞, 1 ≤ p < ∞. (2.1)

For each ε let f_{ε}(x_{α}) be a continuous function on ω such that, for some γ > 0 independent of ε,

0 < γ ≤ f_{ε}(x_{α}) ≤ 1, for all x_{α} ∈ ω, (2.2)

and set, for any open subset A of ω,

A_{ε} := {(x_{α}, x_{3}) : x_{α} ∈ A, |x_{3}| < f_{ε}(x_{α})},

and

∂_{t}A_{ε} := {(x_{α}, x_{3}) : |x_{3}| < f_{ε}(x_{α}), x_{α} ∈ ∂A}.

Note that ω_{ε} = Ω_{ε}. Define, for any u in L^{p}(Ω; ℝ^{3}),

E_{ε}(v; A) := \begin{cases}∫_{A_{ε}} W_{ε} (x_{α}, x_{3}; D_{α}v|\frac{1}{ε}D_{3}v) dx_{α}dx_{3}, & if v ∈ W^{1,p}(A_{ε}; ℝ^{3}), \\ +∞, & otherwise, \end{cases}

and, for any u in L^{p}(Ω; ℝ^{3}),

(2.3) J_{{ε}}(u; A) := inf_{{v_{ε}}} { lim inf_{ε→0^{+}} E_{ε}(v_{ε}; A) : v_{ε} ∈ W^{1,p}(A_{ε}; ℝ^{3}) and
(v_{ε} − u)χ_{A_{ε}} → 0 in L^{p}(Ω; ℝ^{3})}.

Remark 2.1. J_{{ε}}(u; ·) is an increasing function on open subsets of ω.

Remark 2.2. If u ∈ W^{1,p}(ω; ℝ^{3}), i.e., u does not depend upon x_{3}, then (2.1) implies that J_{{ε}}(u; ω) < ∞, as immediately seen upon inserting v_{ε} = u in the definition (2.3) of J_{{ε}}.

Remark 2.3. Assume that p > 1. We claim that energy bounded sequences are compact in L^{p} in the sense of (2.4) below, and with limit in W^{1,p}(ω; ℝ^{3}). Indeed, let {v_{ε}} be a sequence in W^{1,p}(Ω_{ε}; ℝ^{3}) with, say, v_{ε} = 0 on ∂_{t}ω_{ε} and

sup_{ε} ∫_{Ω_{ε}} W_{ε} (x_{α}, x_{3}; D_{α}v_{ε}|\frac{1}{ε}D_{3}v_{ε}) dx_{α}dx_{3} < ∞

Note that f_{ε} must be such that the trace of v_{ε} is meaningful on ∂_{t}ω_{ε}.

We must show that there exist u in W_{0}^{1,p}(ω; ℝ^{3}) and a subsequence {ε_{k}} of {ε} such that

(v_{ε_{k}} − u)χ_{Ω_{ε_{k}}} → 0 in L^{p}(Ω; ℝ^{3}). (2.4)

In view of (2.1), (2.2),

∫_{ω×(−γ,γ)} (|D_{α}v_{ε}|^{p} + \frac{1}{ε^{p}} |D_{3}v_{ε}|^{p}) dx_{α}dx_{3} (2.5)

≤ ∫_{ω_{ε}} (|D_{α}v_{ε}|^{p} + \frac{1}{ε^{p}} |D_{3}v_{ε}|^{p}) dx_{α}dx_{3} < ∞,

so that Poincaré’s inequality and Rellich’s theorem imply the existence of an element u in W^{1,p}(ω × (−γ, γ); ℝ^{3}) and of a subsequence {ε_{k}} of {ε} such that

\begin{cases}v_{ε_{k}} ⇀ u & in W^{1,p}(ω × (−γ, γ); ℝ^{3}), \\ v_{ε_{k}}(x_{α}, ±γ) → u(x_{α}, ±γ) & in L^{p}(ω; ℝ^{3}). \end{cases} (2.6)

Further, (2.5) implies that D_{3}u = 0, i.e., that u lies in W_{0}^{1,p}(ω; ℝ^{3}). Finally,

∫_{ω_{ε}} |v_{ε_{k}} − u|^{p} dx_{α}dx_{3} = ∫_{ω×(−γ,γ)} |v_{ε_{k}} − u|^{p} dx_{α}dx_{3}

+ ∫_{ω} ∫_{γ}^{f_{ε}(x_{α})} |v_{ε_{k}} − u|^{p} dx_{α}dx_{3} + ∫_{ω} ∫_{−f_{ε}(x_{α})}^{−γ} |v_{ε_{k}} − u|^{p} dx_{α}dx_{3}
= ∫_{ω×(−γ,γ)} |v_{ε_{k}} − u|^{p} dx_{α}dx_{3}

+ ∫_{ω} ∫_{γ}^{f_{ε}(x_{α})} |∫_{γ}^{x_{3}} D_{3}v_{ε_{k}}(x_{α}, s) ds + v_{ε_{k}}(x_{α}, γ) − u(x_{α})|^{p} dx_{α}dx_{3}

+ ∫_{ω} ∫_{−f_{ε}(x_{α})}^{−γ} |∫_{−γ}^{x_{3}} D_{3}v_{ε_{k}}(x_{α}, s) ds + v_{ε_{k}}(x_{α}, −γ) − u(x_{α})|^{p} dx_{α}dx_{3}

≤ ∫_{ω×(−γ,γ)} |v_{ε_{k}} − u|^{p} dx_{α}dx_{3}
+C {∫_{ω} |v_{ε_{k}}(x_{α}, ±γ) − u(x_{α})|^{p} dx_{α} + ∫_{ω_{ε}} |D_{3}v_{ε_{k}}|^{p} dx_{α}dx_{3}} ,

so that (2.5) and (2.6) imply (2.4).

Remark 2.4. Note that given a function u ∈ W^{1,p}(ω; ℝ^{3}) then J_{{ε}}(u; Ω) < ∞, and, conversely, if (u_{ε} − u)χ_{Ω_{ε}} → 0 in L^{1} and {u_{ε}} is an energy-bounded sequence, then we may assume that u ∈ W^{1,p}(ω; ℝ^{3}), where we have used the fact that the sequence {f_{ε}} is uniformly bounded away from zero (see (2.2)).

Introduce a countable collection C of subsets of ω such that, for any δ > 0 and any open subset A of ω, there exists a finite union C_{A} of disjoint elements of C satisfying

\begin{cases}\overline{C}_{A} ⊂ A, \\ L^{2}(A) ≤ L^{2}(C_{A}) + δ. \end{cases}

Denote by R the countable collection of all finite unions of elements of C, i.e.,

R := {∪_{i=1}^{k}C_{i} : k ∈ ℕ, C_{i} ∈ C} .

A diagonalization argument, together with a simple argument of Γ-convergence based on the separable and metrizable character of L^{p}(Ω; ℝ^{3}) – see Proposition 7.9 in [9] – permits to assert the existence, for any sequence {ε} ↘ 0^{+}, of a subsequence {ε^{R}} such that, upon setting

(2.7) J_{{ε^{R}}}(u; A) := inf_{{v_{ε^{R}} }} {lim inf_{ε^{R}→0^{+}} E_{ε^{R}}(v_{ε^{R}}; A) : v_{ε^{R}} ∈ W^{1,p}(A_{ε^{R}}; ℝ^{3}) and
(v_{ε^{R}} − u)χ_{A_{ε^{R}}} → 0 in L^{p}(Ω; ℝ^{3})},

then, for each u in L^{p}(Ω; ℝ^{3}) and each C in R, there exists a sequence {v_{ε^{R}}^{C}} in W^{1,p}(C_{ε^{R}}; ℝ^{3}) such that

\begin{cases}(v_{ε^{R}} − u)χ_{ε^{R}} → 0 in L^{p}(Ω; ℝ^{3}) \\ J_{\{ε^{R}\}}(u; C) = lim_{ε^{R}→0+} E_{ε^{R}}(v_{ε^{R}}; C). \end{cases} (2.8)

In other words, J_{{ε^{R}}}(·; C) is the Γ(L^{p})–limit of E_{ε^{R}}(·; C), for every C ∈ R. We then prove the following

Theorem 2.5. For any u ∈ W^{1,p}(ω; ℝ^{3}), any open subset A of ω, and any decreasing sequence {ε} ↘ 0^{+}, J_{{ε^{R}}}(·; A) defined in (2.7) is the Γ(L^{p})–limit of E_{ε^{R}}(·; A).

Furthermore, there exists a Carathéodory function W_{{ε^{R}}} : ℝ^{2} × ℝ^{3×2} → ℝ such that

J_{{ε^{R}}}(u; A) = 2 ∫_{A} W_{{ε^{R}}}(x_{α}; D_{α}u) dx_{α}. (2.9)

Proof. We extend to the present framework the so-called direct methods of the theory of Γ-convergence (see [9] Part II).

The proof is divided into four steps. A first step is devoted to a lemma which will be used in the sequel. The second step establishes the claim that J_{{ε^{R}}}(u; A) is the Γ(L^{p})–limit of E_{ε^{R}}(u; A). The third step ensures that J_{{ε^{R}}}(u; ·) is a finite nonnegative Radon measure. The fourth, and final step, is a mere application of a result in [11] (see Theorem 4.3.2) ensuring the integral representation (2.9).

Step 1. In this step we observe that approximating sequences may as well take the value u on the lateral boundary of A_{ε}.

Lemma 2.6. Let u ∈ W^{1,p}(A; ℝ^{3}) where A is an open subset of ω. If \{\overline{ε}\} ⊂ \{ε^{R}\} and v\overline{ε} ∈ W^{1,p}(A\overline{ε}; ℝ^{3}) are such that

\begin{cases}(v\overline{ε} − u)χ_{A\overline{ε}} → 0 in L^{p}(Ω; ℝ^{3}), \\ J_{\{ε^{R}\}}(u; A) = lim_{\overline{ε}→0^{+}} E\overline{ε}(v\overline{ε}; A), \end{cases}

then there exists a sequence \{w\overline{ε}\} ⊂ W^{1,p}(A\overline{ε}; ℝ^{3}) which satisfies (2.10) and is such that

w\overline{ε} = u in \{(x_{α}, x_{3}) : x_{α} ∈ A \ K\overline{ε} and |x_{3}| < f\overline{ε}(x_{α})\}

for some compact set K\overline{ε} ⊂ A.

Proof. The proof relies on De Giorgi’s slicing argument, and on the possibility of considering cut-off functions which are independent of the variable x_{3}. Set

C := sup_{\overline{ε}} ∫_{A\overline{ε}} (1 + |D_{α}v\overline{ε}|^{p} + \frac{1}{\overline{ε}^{p}} |D_{3}v\overline{ε}|^{p}) dx_{α}dx_{3},

and note that C < ∞ by virtue of (2.1). Define

K(\overline{ε}) := |[\frac{1}{||v\overline{ε} − u||_{L^{p}(A\overline{ε})}^{1/2}} ]|, (2.10)

where |[a]| stands for the integer part of the number a, and M(\overline{ε}) := |[\sqrt{K(\overline{ε})}]|; set also

A(\overline{ε}) := \{x_{α} ∈ A : dist(x_{α}, ∂A) < \frac{M(\overline{ε})}{K(\overline{ε})}\} .

Note that, in view of (2.10), K(\overline{ε}) ↗ ∞ while L^{2}(A(\overline{ε})) ↘ 0^{+} as \overline{ε} ↘ 0^{+}. Subdivide A(\overline{ε}) into M(\overline{ε}) disjoint subsets,

A\overline{ε} ; i := \{x_{α} ∈ A : dist(x_{α}, ∂A) ∈ [\frac{i}{K(\overline{ε})}, \frac{i + 1}{K(\overline{ε})})\} , i = 0, …, M(\overline{ε}) − 1.

Then, there exists i(\overline{ε}) ∈ \{0, ...., M(\overline{ε}) − 1\} such that

∫_{(A\overline{ε} ; i)\overline{ε}} (1 + |D_{α}v\overline{ε}|^{p} + \frac{1}{\overline{ε}^{p}} |D_{3}v\overline{ε}|^{p}) dx_{α}dx_{3} ≤ \frac{C}{M(\overline{ε})}, (2.11)

where (A\overline{ε} ; i(\overline{ε}))\overline{ε} := \{(x_{α}, x_{3}) : x_{α} ∈ A\overline{ε} ; i, |x_{3}| < f\overline{ε}(x_{α})\}. Consider ϕ(\overline{ε}) ∈ C_{0}^{∞}(A)

such that 0 ≤ ϕ(\overline{ε}) ≤ 1, ||D_{α}ϕ(\overline{ε})||_{L^{∞}} ≤ 2K(\overline{ε}),

ϕ(\overline{ε}) = \begin{cases}1, & if dist(x_{α}, ∂A) > \frac{i(\overline{ε})+1}{K(\overline{ε})}, \\ ϕ(\overline{ε}) = 0, & if dist(x_{α}, ∂A) ≤ \frac{i(\overline{ε})}{K(\overline{ε})}, \end{cases} (2.12)

and set

w\overline{ε} := ϕ(\overline{ε})v\overline{ε} + (1 − ϕ(\overline{ε}))u. (2.13)

Note that w\overline{ε} = u in \{(x_{α}, x_{3}) : x_{α} ∈ A \ K\overline{ε}, |x_{3}| < f\overline{ε}(x_{α})\}, with K\overline{ε} := \{x_{α} ∈ A : dist(x_{α}, ∂A) ≥ \frac{i(\overline{ε})}{K(\overline{ε})}\} , and that w\overline{ε} ∈ W^{1,p}(A\overline{ε}; ℝ^{3}). Furthermore, in view of (2.8),

(w\overline{ε} − u)χ_{A\overline{ε}} → 0 in L^{p}(Ω; ℝ^{3}). (2.14)

Then, by virtue of the bound from above in (2.1), together with (2.12), (2.13),

(2.15)

J_{{ε^{R}}}(u; A) ≥

lim sup_{\overline{ε}→0^{+}} ∫_{A\overline{ε}∩[\{x_{α}: dist(x_{α},∂A)> \frac{i(\overline{ε})+1}{K(\overline{ε})}\}×(−1,1)]}W\overline{ε} (x_{α}, x_{3}; D_{α}v\overline{ε}|\frac{1}{\overline{ε}}D_{3}v\overline{ε}) dx_{α}dx_{3}

≥ lim sup_{\overline{ε}→0^{+}} \{∫_{A\overline{ε}} W\overline{ε} (x_{α}, x_{3}; D_{α}w\overline{ε}|\frac{1}{\overline{ε}}D_{3}w\overline{ε}) dx_{α}dx_{3}

−C ∫_{A\overline{ε}∩[\{x_{α}: dist(x_{α},∂A)< \frac{i(\overline{ε})}{K(\overline{ε})}\}×(−1,1)]}(1 + |D_{α}u|^{p}) dx_{α}dx_{3}

−β ∫_{(A\overline{ε} ; i(\overline{ε}))\overline{ε}} (1 + |D_{α}v\overline{ε}|^{p} + \frac{1}{\overline{ε}^{p}} |D_{3}v\overline{ε}|^{p}) dx_{α}dx_{3}

−C|K(\overline{ε})|^{p} ∫_{(A\overline{ε} ; i(\overline{ε}))\overline{ε}} |v\overline{ε} − u|^{p} dx_{α}dx_{3}\}

≥ lim sup_{\overline{ε}→0^{+}} E\overline{ε}(w\overline{ε}; A) − C lim inf_{\overline{ε}→0^{+}} L^{2}(A(\overline{ε})) − Cβ lim inf_{\overline{ε}→0^{+}} \frac{1}{M(\overline{ε})}

−β lim inf_{\overline{ε}→0^{+}} ||v\overline{ε} − u||_{L^{p}(A\overline{ε})}^{p/2} = lim sup_{\overline{ε}→0^{+}} E\overline{ε}(w\overline{ε}; A),

where (2.10) and (2.11) have been used in deriving the last inequality in (2.15). But the very definition (2.7) of J_{{ε^{R}}}(u; A), together with (2.14), imply that

J_{\{ε^{R}\}}(u; A) ≤ lim inf_{\overline{ε}→0^{+}} E\overline{ε}(w\overline{ε}; A),

which, in view of (2.15), yields the desired result.

Step 2. Let u ∈ W^{1,p}(A; ℝ^{3}), let A be an open subset of ω. In order to prove that J_{{ε^{R}}}(u; A) is the Γ–limit of E_{ε^{R}}(v_{ε^{R}}^{C^{δ}}; A) it suffices, in view of its definition (2.7), to prove that property Γ3ii) (see the Introduction) holds. To this end, fix δ > 0 and choose a subset C^{δ} of A in R such that

\begin{cases}\overline{C^{δ}} ⊂ A, \\ ∫_{A\C^{δ}}(1 + |Du_{α}|) dx_{α} < \frac{δ}{2β} . \end{cases}

Consider a sequence v_{ε^{R}}^{C^{δ}} satisfying

lim_{ε^{R}→0^{+}} E_{ε^{R}}(v_{ε^{R}}^{C^{δ}}; C^{δ}) = J_{{ε^{R}}}(u; C^{δ}).

In view of Lemma 2.6, we may extend v_{ε^{R}}^{C^{δ}} as u outside C_{ε^{R}}^{δ} so as to belong to W^{1,p}(A_{ε^{R}}; ℝ^{3}), and since J_{{ε^{R}}}(u; C^{δ}) ≤ J_{{ε^{R}}}(u; A) for all δ > 0, we have

lim sup_{δ→0^{+}} lim sup_{ε^{R}→0^{+}} E_{ε^{R}}(v_{ε^{R}}^{C^{δ}}; A)

≤ lim sup_{δ→0^{+}} lim_{ε^{R}→0^{+}} {E_{ε^{R}}(v_{ε^{R}}^{C^{δ}}; C^{δ}) + 2β ∫_{A\C^{δ}}(1 + |D_{α}u|^{p}) dx_{α}dx_{3}}

= lim sup_{δ→0^{+}} J_{{ε^{R}}}(u; C^{δ})

≤ J_{{ε^{R}}}(u; A)

≤ lim inf_{δ→0^{+}} lim inf_{ε^{R}→0^{+}} E_{ε^{R}}(v_{ε^{R}}^{C^{δ}}; A).

Lemma 7.1 in the Appendix permits to conclude the existence of a decreasing sequence {δ(ε^{R})} ↘ 0^{+} such that

\begin{cases}(v_{ε^{R}}^{C^{δ(ε^{R})}} − u)χ_{A_{ε^{R}}} → 0 in L^{p}(Ω; ℝ^{3}) \\ J_{\{ε^{R}\}}(u; A) = lim_{ε^{R}→0^{+}} E_{ε^{R}} (v_{ε^{R}}^{C^{δ(ε^{R})}}; A) , \end{cases}

which, together with (2.7), asserts that J_{{ε^{R}}}(·; A) is the Γ(L^{p})–limit of E_{ε^{R}}(·; A).

Step 3. Let u be an element of W^{1,p}(ω; ℝ^{3}). Implicit in the proof of Step 2 above is the inner regularity of J_{{ε^{R}}}(u; A), namely, for any δ > 0 there exists C^{δ} ∈ R such that

\begin{cases}\overline{C}^{δ} ⊂ A, \\ J_{\{ε^{R}\}}(u; A) ≤ J_{\{ε^{R}\}}(u; C^{δ}) + δ. \end{cases} (2.16)

Remark 2.7. Note that (2.16), together with the trivial inequality

J_{\{ε^{R}\}}(u; A) ≥ J_{\{ε^{R}\}}(u; A \ \overline{C}^{δ}) + J_{\{ε^{R}\}}(u; C^{δ}),

immediately implies that

J_{\{ε^{R}\}}(u; A \ \overline{C}^{δ}) ≤ δ. (2.17)

Remark also that (2.17) is obtained simply upon choosing u as a test function in the definition (2.7) of J_{\{ε^{R}\}}(u; A \ \overline{C}^{δ}).

We now show that J_{{ε^{R}}} is subadditive, that is that for every open subsets C, B, A of ω with C ⊂⊂ B ⊂ A,

J_{\{ε^{R}\}}(u; A) ≤ J_{\{ε^{R}\}}(u; B) + J_{\{ε^{R}\}}(u; A \ \overline{C}). (2.18)

To this effect we consider, for any small enough δ > 0, B^{δ}, D^{δ} two elements of R with B^{δ} ⊂ B, D^{δ} ⊂ A \ \overline{C}, such that

∫_{A\(B^{δ}∪D^{δ})}(1 + |D_{α}u|)^{p} dx_{α} < δ. (2.19)

Note that a small enough δ ensures that B^{δ} ∩ D^{δ} ≠ ∅. Then, there exist two sequences {v_{ε^{R}}^{B^{δ}}} , {v_{ε^{R}}^{D^{δ}}} such that (2.8) is satisfied for B^{δ} and D^{δ}, respectively, and (see Lemma 2.6)

v_{ε^{R}}^{B^{δ}} = u on ∂_{t}B_{ε^{R}}^{δ}, v_{ε^{R}}^{D^{δ}} = u on ∂_{t}D_{ε^{R}}^{δ}.

Consider the sequence of Radon measures

λ_{ε^{R}} := {1 + |D_{α}v_{ε^{R}}^{B^{δ}} |^{p} + |D_{α}v_{ε^{R}}^{D^{δ}} |^{p}

+ (\frac{1}{ε^{R}} )^{p} (|D_{3}v_{ε^{R}}^{B^{δ}} |^{p} + |D_{3}v_{ε^{R}}^{D^{δ}} |^{p}) \}χ_{(B^{δ}∩D^{δ})_{ε^{R}}} L^{3}

where, as usual, (B^{δ} ∩D^{δ})_{ε^{R}} = {x_{α}, x_{3}; x_{α} ∈ B^{δ} ∩D^{δ}, |x_{3}| < f_{ε^{R}}(x_{α})}. By virtue of the coercivity hypothesis in (2.1), {λ_{ε^{R}}} is a bounded sequence of finite nonnegative Radon measures on ℝ^{3}, hence there exists a finite nonnegative Radon measure λ such that a subsequence of \{λ_{ε^{R}}\} – denoted by \{λ\overline{ε}\} – satisfies

(2.20) λ\overline{ε} ⇀^{∗} λ weakly-* in the sense of measures.

Set λ̂(X) := λ(X × [−1, 1]) for any Borel subset X of ω. Define, for 0 < η < 1,

S_{η}^{δ} := {x ∈ B^{δ} ∩ D^{δ} : dist(x_{α}, ∂B^{δ}) = η}.

The family {S_{η}^{δ}}_{η} is made up of pairwise disjoint elements, thus there exists η_{0} ∈ (0, 1) such that

λ̂(S_{η_{0}}^{δ}) = 0. (2.21)

For L_{ζ}^{δ}, a layer of thickness ζ around S_{η_{0}}^{δ}, i.e.,

L_{ζ}^{δ} := {x_{α} ∈ B^{δ} ∩ D^{δ} : dist(x_{α}, S_{η_{0}}^{δ}) ≤ ζ},

consider a smooth cut-off function ϕ_{ζ}^{δ} ∈ C_{0}^{∞}(ℝ^{2}) such that

||ϕ_{ζ}^{δ}||_{L^{∞}} ≤ 1, ||D_{α}ϕ_{ζ}^{δ}||_{L^{∞}} ≤ C/ζ,

ϕ_{ζ}^{δ} = \{ \begin{matrix}0 & if x_{α} ∈ B^{δ} and dist(x_{α}, ∂B^{δ}) ≥ η_{0} + ζ, \\ 1 & if x_{α} ∉ B^{δ} or dist(x_{α}, ∂B^{δ}) ≤ η_{0} − ζ. \end{matrix} (2.22)

Setting

v_{ζ,\overline{ε}} := ϕ_{ζ}^{δ}vD^{δ} ; \overline{ε} + (1 − ϕ_{ζ}^{δ})vB^{δ} ; \overline{ε} + χ_{(A\(B^{δ}∪D^{δ}))\overline{ε}}u,

then v_{ζ,\overline{ε}}^{δ} ∈ W^{1,p}(A\overline{ε}; ℝ^{3}), and

(v_{ζ,\overline{ε}} − u)χ_{A\overline{ε}} → 0 in L^{p}(Ω; ℝ^{3}).

Thus, by the very definition (2.7) and in view of (2.22),

J_{\{ε^{R}\}}(u; A) ≤ lim inf_{\overline{ε}→0^{+}} E\overline{ε}(v_{ζ,\overline{ε}}; A) (2.23)

≤ J_{{ε^{R}}}(u; B^{δ}) + J_{{ε^{R}}}(u; D^{δ})

+2β ∫_{A\(B^{δ}∪D^{δ})}(1 + |D_{α}u|) dx_{α}

+β \{lim sup_{\overline{ε}→0^{+}} λ\overline{ε}(L_{ζ}^{δ} × (−1, 1)) + \frac{C}{ζ^{p}} lim sup_{\overline{ε}→0^{+}} ∫_{(L_{ζ}^{δ})\overline{ε}} |vB^{δ} ; \overline{ε} − vD^{δ} ; \overline{ε} |^{p} dx_{α}dx_{3}\} ,

where (L_{ζ}^{δ})\overline{ε} := \{(x_{α}, x_{3}) : x_{α} ∈ L_{ζ}^{δ} and |x_{3}| < f\overline{ε}(x_{α})\}. Since |vB^{δ} ; \overline{ε} − vD^{δ} ; \overline{ε} | χ_{(L_{ζ}^{δ})\overline{ε}} ≤

|vB^{δ} ; \overline{ε} − u| χ_{B^{δ}} \overline{ε} + |vD^{δ} ; \overline{ε} − u| χ_{D^{δ}} \overline{ε}, the last term in the last expression in (2.23) is 0, while, by virtue of (2.20),

lim sup_{\overline{ε}→0^{+}} λ\overline{ε}(L_{ζ} × (−1, 1)) ≤ λ̂(\overline{L}_{ζ}).

But, as ζ tends to 0, λ̂(\overline{L_{ζ}^{δ}}) goes to λ̂(S_{η_{0}}^{δ}) = 0 (cf. (2.21)), therefore, upon letting ζ tend to 0 in (2.23), and by (2.19), we obtain

J_{{ε^{R}}}(u; A) ≤ lim inf_{δ→0^{+}} [J_{{ε^{R}}}(u; B^{δ}) + J_{{ε^{R}}}(u; D^{δ}) + βδ]
≤ J_{\{ε^{R}\}}(u; B) + J_{\{ε^{R}\}}(u; A \ \overline{C}),

and this proves (2.18).

Finally, the definition (2.7) of J_{{ε^{R}}}(u; ω) implies the existence of a subsequence \{\overline{ε}\} of \{ε^{R}\} and of an associated subsequence \{v\overline{ε}\} in W^{1,p}(ω\overline{ε}; ℝ^{3}) such that

\begin{cases}(v\overline{ε} − u)χ_{ω\overline{ε}} → 0 in L^{p}(Ω; ℝ^{3}), \\ J_{\{ε^{R}\}}(u; ω) = lim_{\overline{ε}→0^{+}} E\overline{ε}(v\overline{ε}; ω). \end{cases} (2.24)

For a well chosen subsequence of \{\overline{ε}\}, still denoted by \{\overline{ε}\}, there exists a finite Radon measure μ such that

(2.25) W\overline{ε}(x_{α}, x_{3}; D_{α}v\overline{ε}|1/\overline{ε}D_{3}v\overline{ε})χ_{ω\overline{ε}}L^{3} ⇀^{∗} μ weakly-* in the sense of measures .

Set, for any Borel subset X ⊂ ℝ^{3}, μ̂(X) := μ(X × [−1, 1]). Then, by virtue of (2.24), (2.25),

J_{{ε^{R}}}(u; ω) ≥ μ̂(ℝ^{2}), (2.26)

while, clearly for all open subsets A ⊂ ω,

(2.27) J_{\{ε^{R}\}}(u; A) ≤ lim inf_{\overline{ε}→0^{+}} E\overline{ε}(v\overline{ε}; A)
= lim inf_{\overline{ε}→0^{+}} ∫_{A\overline{ε}} W\overline{ε} (x_{α}, x_{3}; D_{α}v\overline{ε}|\frac{1}{\overline{ε}}D_{3}v\overline{ε}) dx_{α}dx_{3}
≤ μ(\overline{A} × [−1, 1]) = μ̂(\overline{A}).

In view of (2.17), (2.18), (2.26), (2.27), Lemma 7.3 in the Appendix allows us to conclude that J_{{ε^{R}}}(u; ·) is the trace on the open subsets of ω of a finite nonnegative Radon measure. The bound from above in (2.1) immediately implies that it is absolutely continuous with respect to L^{2}⌊ω.

Step 4. In view of the preceding considerations we are now in a position to apply Theorem 4.3.2 in [11], guaranteeing the existence of an energy density W_{{ε^{R}}} satisfying (2.9). Indeed, J_{{ε^{R}}} maps any pair (u, A), u ∈ W^{1,p}(ω; ℝ^{3}), A an open subset of ω, into ℝ, and, furthermore,

(i) J_{{ε^{R}}}(u; A) = J_{{ε^{R}}}(v; A) whenever u = v, a.e. on ℝ^{2}, (ii) J_{{ε^{R}}}(u; ·) is a finite nonnegative Radon measure, (iii) J_{{ε^{R}}}(u; A) ≤ 2β ∫_{A}(1 + |D_{α}u|^{p})dx_{α}, (iv) J_{{ε^{R}}}(u + c; A) = J_{{ε^{R}}}(u; A), c ∈ ℝ. The proof of Theorem 2.5 is complete.

Remark 2.8. It follows immediately from the growth condition (2.1) and the lower-semicontinuity of the L^{p}-norm that the density function W_{{ε^{R}}} in Theorem 2.5 still satisfies (2.1).

Remark 2.9. The conclusions of Theorem 2.5 are valid for more general domains Ω_{ε}, since their particular form is not used in the course of the proof. Namely, we may choose in place of Ω_{ε} any open set Ω′_{ε} ⊂ ω × (−1, 1), and consider the set

A_{ε} := (A × (−1, 1)) ∩ Ω′_{ε}

in the definition of E_{ε}(u; A). Of course, the price to pay for such a degree of generality may be reflected in the possible degeneracy of the limit energy. In fact, Remarks 2.3, 2.4, and 2.8 do not hold true in general; hence, (2.9) may fail to describe fully the Γ–limit of E_{{ε^{R}}}, which may be finite also outside W^{1,p}(ω; ℝ^{3}).

On one end of the spectrum of this degeneracy we have the case where Ω′_{ε} := ∅, for which the Γ–limit reduces to 0 on the whole L^{p}(Ω; ℝ^{3}). The same conclusion holds if we take Ω′_{ε} := ω × (−r_{ε}, r_{ε}) with lim_{ε} r_{ε} = 0.

Another type of degeneracy may be found when Ω′_{ε} is not connected. As an example, take Ω′_{ε} := ω × ((−1, −1/2) ∪ (1/2, 1)). It is clear that the Γ–limit is given by a functional defined on pairs of functions in W^{1,p}(ω; ℝ^{3}), the necessary changes in the statement and proof of the corresponding Theorem 2.5 being straightforward.

Finally, it may also be possible that, even though Ω′_{ε} is connected for all ε, the domain of the Γ–limit is all of W^{1,p}(Ω; ℝ^{3}). An example of this phenomenon, obtained by taking Ω′_{ε} to be a domain with a periodical array of cracks, has been studied in detail by Batthacharya and Braides [6].

3. First application – Nonhomogeneous plate models

In [24], a nonlinear plate model is derived from a 3D domain of the form ω × (−ε, ε) occupied by a nonlinearly elastic material upon letting the thickness 2ε tend to 0. Specifically, under the assumption that the elastic energy density W is homogeneous and satisfies

β′|F |^{p} ≤ W (F) ≤ β(1 + |F |^{p}), 0 < β′ ≤ β < ∞, 1 ≤ p < ∞, (3.1)

it is shown that, for any u ∈ W^{1,p}(ω; ℝ^{3}), any A open subset of ω, and any sequence {ε} ↘ 0^{+},

J_{\{ε\}}(u; A) := inf_{\{v_{ε}\}} \{lim inf_{ε→0^{+}} ∫_{A×(−1,1)} W (D_{α}v_{ε}|\frac{1}{ε}D_{3}v_{ε}) dx_{α}dx_{3} :

v_{ε} ∈ W^{1,p}(A × (−1, 1); ℝ^{3}), v_{ε} → u in L^{p}(A × (−1, 1); ℝ^{3})},

is given by

J_{\{ε\}}(u; A) = 2 ∫_{A} Q\overline{W} (D_{α}u)dx_{α},

where

\overline{W} (\overline{F}) := inf_{z∈ℝ^{3}} W (\overline{F} |z), \overline{F} ∈ ℝ^{3×2},

and

Q\overline{W} (\overline{F}) := inf_{ϕ∈W_{0}^{1,p}(Q′;ℝ^{3})} ∫_{Q′} \overline{W} (\overline{F} + D_{α}ϕ) dx_{α},

where Q′ is the unit cube (0, 1)^{2} in ℝ^{2}, and Q\overline{W} is the 2D quasiconvexification of \overline{W}. Here we propose to extend this result to the nonhomogeneous case where W is also function of x_{3}.

We thus assume that W (x_{3}; F) is a Carathéodory function on (−1, 1) × ℝ^{3×3} such that

β′|F |^{p} ≤ W (x_{3}; F) ≤ β(1 + |F |^{p}), 0 < β′ ≤ β < ∞, for a.e. x ∈ (−1, 1),

or, in other words, that W_{ε} defined in Section 2 is independent of ε, and that f_{ε}(x_{α}) ≡ 1, x_{α} ∈ ω.

Direct application of Theorem 2.5 permits to assert the existence, for any sequence {ε} ↘ 0^{+}, of a subsequence {ε^{R}} ↘ 0^{+} such that J_{{ε^{R}}}(u; A) defined in (2.7) is given by

J_{{ε^{R}}}(u; A) = 2 ∫_{A} W_{{ε^{R}}}(x_{α}; D_{α}u)dx_{α}. (3.2)

It remains to identify W_{\{ε^{R}\}}. To this effect, we define, for any \overline{F} ∈ ℝ^{3×2},

(3.3) \underline{W} (\overline{F}) := inf_{λ>0} inf_{ϕ} \{\frac{1}{2} ∫_{Q′×(−1,1)} W (x_{3}; \overline{F} + D_{α}ϕ|λD_{3}ϕ) dx_{α}dx_{3} :

ϕ ∈ W^{1,p}(Q′ × (−1, 1); ℝ^{3}), ϕ = 0 on ∂Q′ × (−1, 1)}.

Then, the following theorem holds true:

Theorem 3.1. For almost any x_{α} ∈ ω and for all \overline{F} ∈ ℝ, W_{\{ε^{R}\}}(x_{α}; \overline{F}) = \underline{W} (\overline{F}). Consequently, for all u ∈ W^{1,p}(ω; ℝ^{3}), any A open subset of ω,

Γ(L^{p})– lim E_{ε}(u; A) = J_{\{ε\}}(u; A) = 2 ∫_{A} \underline{W} (D_{α}u)dx_{α}.

Proof. Consider any sequence {ε} ↘ 0^{+} and let {ε^{R}} be as (2.7), (2.8). Fix \overline{F} ∈ ℝ^{3×2} and let x_{0} be a Lebesgue point for W_{\{ε^{R}\}}(·; \overline{F}). Then,

W_{\{ε^{R}\}}(x_{0}; \overline{F}) = lim_{q→∞} q^{2} ∫_{Q′(x_{0};1/q)} W_{\{ε^{R}\}}(x_{α}; \overline{F}) dx_{α}, (3.4)

where Q′(x_{0}; 1/q) is the cube of ℝ^{2} of center x_{0} and side length 1/q, and q is large enough so that Q′(x_{0}; 1/q) ⊂ ω. In view of (3.2), (3.4) also reads as

W_{\{ε^{R}\}}(x_{0}; \overline{F}) = lim_{q→∞} \frac{q^{2}}{2} J_{\{ε^{R}\}}(\overline{F} x; Q′(x_{0}; 1/q)). (3.5)

For q large enough, let {v_{ε^{R}}^{q}} ⊂ W^{1,p}(Q′(x_{0}; 1/q) × (−1, 1); ℝ^{3}) be such that

\begin{cases}v_{ε^{R}}^{q} → 0 in L^{p}(Q′(x_{0}; 1/q) × (−1, 1); ℝ^{3}), \\ J_{\{ε^{R}\}}(\overline{F} x; Q′(x_{0}; \frac{1}{q})) = \\ lim_{ε^{R}→0^{+}} ∫_{Q′(x_{0}; \frac{1}{q})×(−1,1)} W (x_{3}; \overline{F} + D_{α}v_{ε^{R}}^{q} |\frac{1}{ε^{R}} D_{3}v^{q}) dx_{α}dx_{3}. \end{cases} (3.6)

Such a sequence exists according to Theorem 2.5. Set

v_{q,ε^{R}}(x_{α}, x_{3}) := qv_{ε^{R}}^{q} (x_{0} + \frac{x_{α}}{q} , x_{3}) , x_{α} ∈ Q′.

Thus, by virtue of (3.6), (3.5) reads as

W_{\{ε^{R}\}}(x_{0}; \overline{F}) = (3.7)

\frac{1}{2} lim_{q→∞} lim_{ε^{R}→0^{+}} ∫_{Q′×(−1,1)} W (x_{3}; \overline{F} + D_{α}v_{q,ε^{R}} |\frac{1}{qε^{R}} D_{3}v_{q,ε^{R}}) dx_{α}dx_{3}.

The inequality

W_{\{ε^{R}\}}(x_{0}; \overline{F}) ≥ \underline{W} (\overline{F}) (3.8)

would then be immediate if v_{q,ε^{R}} = 0 on ∂Q′ × (−1, 1), because in such a case Fubini’s theorem would imply that, for a.e. x_{3} ∈ (−1, 1), v_{q,ε^{R}} ∈ W_{0}^{1,p}(Q′; ℝ^{3}) and (3.7) would become

W_{\{ε^{R}\}}(x_{0}; \overline{F}) ≥ \frac{1}{2} ∫_{−1}^{1} \underline{W} (\overline{F}) dx_{3} = \underline{W} (\overline{F}).

Unfortunately, such may not be the case and we have to modify v_{q,ε^{R}} accordingly. To this effect we firstly note that, at the expense of extracting a subsequence of {q, ε^{R}}, still labeled {q, ε^{R}}, we are always at liberty, in view of the coercive character of W (cf. (3.1)), to assume that the sequence {λ_{q,ε^{R}}} of nonnegative Radon measures

λ_{q,ε^{R}} := (1 + |D_{α}v_{q,ε^{R}}|^{p} + |\frac{1}{qε^{R}} D_{3}v_{q,ε^{R}}|^{p}) χ_{Q′×(−1,1)}L^{3}

converges weak-* in the sense of measures to a nonnegative finite Radon measure λ as {q, ε^{R}} → (∞, 0). We then define, for all Borel sets B of ℝ^{2}, λ̂(B) := λ(B × [−1, 1]).

We now introduce, for k ≥ 2,

w_{k,q,ε^{R}} := ϕ_{k}v_{q,ε^{R}},

where ϕ_{k} ∈ C_{0}^{∞}(Q′) is such that

\begin{cases}0 ≤ ϕ_{k} ≤ 1, ||D_{α}ϕ_{k}||_{L^{∞}} ≤ Ck^{2}, \\ ϕ_{k} = \begin{cases}1 & if x_{α} ∈ Q′(0, 1 − 1/k), \\ 0 & if x_{α} ∉ Q′(0, 1 − 1/(k + 1)). \end{cases} \end{cases}

Note that w_{k,q,ε^{R}}(·, x_{3}) ∈ W_{0}^{1,p}(Q′; ℝ^{3}) for a.e. x_{3} ∈ (−1, 1). Thus, recalling (3.7),

W_{\{ε^{R}\}}(x_{0}; \overline{F}) ≥ (3.9)

\frac{1}{2} lim inf_{q→∞} lim inf_{ε^{R}→0^{+}} ∫_{Q′(0,1− \frac{1}{k})×(−1,1)} W (x_{3}; \overline{F} + D_{α}w_{k,q,ε^{R}} |\frac{1}{qε^{R}} D_{3}w_{k,q,ε^{R}}) dx_{α}dx_{3}

≥ \frac{1}{2} lim inf_{q→∞} lim inf_{ε^{R}→0^{+}} \{ ∫_{Q′×(−1,1)} W (x_{3}; \overline{F} + D_{α}w_{k,q,ε^{R}} |\frac{1}{qε^{R}} D_{3}w_{k,q,ε^{R}}) dx_{α}dx_{3}

−β ∫_{(Q′\Q′(0,1− \frac{1}{k}))×(−1,1)}(1 + |\overline{F} |^{p}) dx_{α}dx_{3}

−Ck^{2p} ∫_{(Q′(0,1− \frac{1}{k+1} )\Q′(0,1− \frac{1}{k}))×(−1,1)} |v_{q,ε^{R}}|^{p} dx_{α}dx_{3}

−C∫_{(Q′(0,1− \frac{1}{k+1} )\Q′(0,1− \frac{1}{k}))×(−1,1)}(1 + |D_{α}v_{q,ε^{R}}|^{p} + |\frac{1}{qε^{R}} D_{3}v_{q,ε^{R}} |^{p})dx_{α}dx_{3}\}

≥ \underline{W} (\overline{F}) − \frac{C}{k^{2}} − lim sup_{q→∞} lim sup_{ε^{R}→0^{+}} Ck^{2p} ∫_{Q′×(−1,1)} |v_{q,ε^{R}}|^{p} dx_{α}dx_{3}

−C lim sup_{q→∞} lim sup_{ε^{R}→0^{+}} λ_{q,ε^{R}} ((Q′ (0, 1 − \frac{1}{k + 1}) \ Q′ (0, 1 − \frac{1}{k} )) × (−1, 1)) .

Now, in view of (3.6),

lim sup_{q→∞} lim sup_{ε^{R}→0^{+}} ∫_{Q′×(−1,1)} |v_{q,ε^{R}}|^{p} dx_{α}dx_{3} =

lim sup_{q→∞} q^{p+2} lim_{ε^{R}→0^{+}} ∫_{Q′(x_{0}, \frac{1}{q})×(−1,1)} |v_{ε^{R}}^{q}|^{p} dx_{α}dx_{3} = 0,

while

lim sup_{q→∞} lim sup_{ε^{R}→0^{+}} λ_{q,ε^{R}} ((Q′(0, 1 − \frac{1}{k + 1}) \ Q′ (0, 1 − \frac{1}{k} ) × (−1, 1))

≤ λ (\overline{(Q′ (0, 1 − \frac{1}{k + 1}) \ Q′ (0, 1 − \frac{1}{k} )) × (−1, 1)})

≤ λ̂ (Q′ \ Q′ (0, 1 − \frac{1}{k − 1})) .

Thus, (3.9) becomes

W_{\{ε^{R}\}}(x_{0}; \overline{F}) ≥ \underline{W} (\overline{F}) − \frac{C}{k^{2}} − Cλ̂ (Q′ \ Q (0, 1 − \frac{1}{k − 1})) ,

and (3.8) is obtained by letting k tend to ∞ since Q′(0, 1−1/(k−1)) is an increasing sequence of open sets with set limits Q′.

Conversely, for any given η > 0, let λ > 0, ϕ ∈ W^{1,∞}(Q′ × (−1, 1); ℝ^{3}) with ϕ = 0 on ∂Q′ × (−1, 1), be such that

\underline{W} (\overline{F}) + η ≥ \frac{1}{2} ∫_{Q′×(−1,1)} W (x_{3}; \overline{F} + D_{α}ϕ|λD_{3}ϕ) dx_{α}dx_{3}. (3.10)

This is legitimate because of the density of W^{1,∞}(Q′ × (−1, 1); ℝ^{3}) into W^{1,p}(Q′ × (−1, 1); ℝ^{3}) – both with zero trace on the boundary ∂Q′ × (−1, 1) – and of the bound from above in (3.1). Set

v_{ε^{R}}(x_{α}, x_{3}) := \overline{F} x_{α} + λε^{R}ϕ (\frac{x_{α}}{λε^{R}} , x_{3}) ,

where ϕ has been laterally extended by Q′-periodicity. Then,

v_{ε^{R}} → \overline{F} x_{α} in L^{p}(Ω; ℝ^{3}).

Furthermore, for any open set A ⊂ ω,

J_{\{ε^{R}\}}(\overline{F} x_{α}; A) ≤ lim inf_{ε^{R}→0^{+}} ∫_{A×(−1,1)} W (x_{3}; D_{α}v_{ε^{R}} |\frac{1}{ε^{R}} D_{3}v_{ε^{R}}) dx_{α}dx_{3} (3.11)

= lim inf_{ε^{R}→0^{+}} ∫_{A×(−1,1)} W (x_{3}; \overline{F} + D_{α}ϕ (\frac{x_{α}}{λε^{R}} , x_{3}) |λD_{3}ϕ (\frac{x_{α}}{λε^{R}} , x_{3})) dx_{α}dx_{3}.

Since ∫_{1}^{1} W (x_{3}; \overline{F} + D_{α}ϕ(·, x_{3})|λD_{3}ϕ(·, x_{3})) dx_{3} is a periodic function in L^{∞}(ℝ^{2}), it converges weak-* to its average and (3.11) becomes, in view of (3.10),

J_{\{ε^{R}\}}(\overline{F} x_{α}; A) ≤ L^{2}(A) ∫_{−1}^{1} ∫_{Q′} W (x_{3}; \overline{F} + D_{α}ϕ|λD_{3}ϕ) dx_{α}dx_{3}
≤ 2L^{2}(A) \underline{W} (\overline{F}) + 2ηL^{2}(A).

Letting η tend to 0^{+} yields

J_{\{ε^{R}\}}(\overline{F} x_{α}; A) ≤ 2L^{2}(A) \underline{W} (\overline{F}). (3.12)

But, according to Theorem 2.5, (3.12) also reads as

∫_{A} W_{\{ε^{R}\}}(x_{0}; \overline{F})dx_{α} ≤ L^{2}(A) \underline{W} (\overline{F}),

so that, upon choosing x_{0} ∈ Ω to be a Lebesgue point for W_{\{ε^{R}\}}(·; \overline{F}) and A to be a small ball centered at x_{0} and of vanishing radius, we obtain

W_{\{ε^{R}\}}(x_{0}; \overline{F}) ≤ \underline{W} (\overline{F}).

Since finally W_{\{ε^{R}\}}(x_{0}; \overline{F}) does not depend upon the choice of sequence \{ε^{R}\}, we conclude that there is no need to extract a subsequence from {ε}.

In light of Proposition 7.11 in [9], the proof of Theorem 3.1 is now complete.

Remark 3.2. Since W_{{ε^{R}}}(x_{α}; ·) is the integrand of a lower semicontinuous functional on W^{1,p}(ω; ℝ^{3}) – namely the Γ(L^{p})–limit of E_{ε^{R}}(·, ω) –, it is quasiconvex (see Statement II.5 in [2]). Thus \underline{W} (\overline{F}) defined in (3.3) is actually quasiconvex.

Remark 3.3. Note that, if W does not depend upon x_{3}, then

\underline{W} (\overline{F}) = Q\overline{W} (\overline{F}), \overline{F} ∈ ℝ^{3×2}.

In other words, the result of [24] is recovered in the homogeneous case. Indeed, clearly,

\underline{W} (\overline{F}) ≥ inf_{ϕ} \{\frac{1}{2} ∫_{−1}^{1} ∫_{Q′} \overline{W} (\overline{F} + D_{α}ϕ) dx_{α}dx_{3} : ϕ ∈ W^{1,p}(Q′ × (−1, 1); ℝ^{3}),

ϕ = 0 on ∂Q′ × (−1, 1)}

≥ \frac{1}{2} ∫_{−1}^{1} Q\overline{W} (\overline{F}) dx_{3} = Q\overline{W} (\overline{F}),

so that

\underline{W} (\overline{F}) ≥ Q\overline{W} (\overline{F}).

Conversely, for any η > 0, there exist, by virtue of the measurability selection criterion together with the density of W_{0}^{1,∞}(Q′; ℝ^{3}) into W_{0}^{1,p}(Q′; ℝ^{3}) and into L^{p}(Q′; ℝ^{3}), and the upper bound on W, functions ϕ^{η}, ξ^{η} ∈ W_{0}^{1,∞}(Q′; ℝ^{3}) such that

Q\overline{W} (\overline{F}) + η ≥ ∫_{Q′} W (\overline{F} + D_{α}ϕ^{η}|ξ^{η}) dx_{α}.

Extend ϕ^{η}, ξ^{η} Q′-periodically to ℝ^{2} and set

ϕ_{n}^{η}(x_{α}, x_{3}) := \frac{1}{n}ϕ^{η}(nx_{α}) + \frac{1}{n^{2}} x_{3}ξ^{η}(nx_{α}).

Then ϕ_{n}^{η} ∈ W^{1,p}(Q′ × (−1, 1); ℝ^{3}) with ϕ_{n}^{η} = 0 on ∂Q′ × (−1, 1), and so

\underline{W} (\overline{F}) ≤ \frac{1}{2} lim inf_{n→+∞} ∫_{Q′×(−1,1)} W (\overline{F} + D_{α}ϕ_{n}^{η}|n^{2}D_{3}ϕ_{n}^{η}) dx_{α}dx_{3}

= \frac{1}{2} lim inf_{n→+∞} ∫_{Q′×(−1,1)} W (\overline{F} + D_{α}ϕ^{η}(nx_{α}) + \frac{1}{n}x_{3}D_{α}ξ(nx_{α})|ξ(nx_{α})) dx_{α}dx_{3}
≤ \frac{1}{2} lim inf_{n→+∞} ∫_{Q′×(−1,1)} W (\overline{F} + D_{α}ϕ^{η}(nx_{α})|ξ(nx_{α})) dx_{α}dx_{3},

where the uniformly continuous character of W on compact sets has been used in the last inequality (see e.g. the proof of Lemma 4.1 in [17] for more details). But W (\overline{F} + D_{α}ϕ^{η}(·)|ξ(·)) is a periodic function in L^{∞}(ℝ^{2}), thus weak-* converges to its average and we obtain

\underline{W} (\overline{F}) ≤ ∫_{Q′} W (\overline{F} + D_{α}ϕ^{η}|ξ^{η}) dx_{α} ≤ Q\overline{W} (\overline{F}) + η.

The result is obtained upon letting η tend to 0.

Remark 3.4. We believe that the result of Theorem 3.1, appropriately extended, still holds true in the case of an energy density that also depends upon x_{α}, although we are not at present in a position to offer a full proof in such a setting.

4. Second application –The periodic case

In this section it is assumed that W (x_{α}, x_{3}; F) is a Carathéodory function from Q′ × (−1, 1) × ℝ^{9} into ℝ satisfying

β′|F |^{p} ≤ W (x_{α}, x_{3}; F) ≤ β(1 + |F |^{p}),

with 1 ≤ p < ∞ and β′, β > 0. The function W is extended by Q′-periodicity to ℝ^{2} × (−1, 1) × ℝ^{9} and we set

W_{ε}(x_{α}, x_{3}; F) := W (\frac{x_{α}}{ε} , x_{3}; F ) .

Also, we assume that f is a continuous function from Q′ into [0, 1] with 0 < γ ≤ min f and we set

f_{ε}(x_{α}) := f (\frac{x_{α}}{ε} ) .

We define, for any \overline{F} ∈ ℝ^{3×2},

W_{hom}(\overline{F}) := lim inf_{t↗∞} g(t),

where, for any t > 0 ,

g(t) := \frac{1}{t^{2}} inf_{ϕ} \{ ∫_{(tQ′)^{f}}W (x_{α}, x_{3}; \overline{F} + D_{α}ϕ|D_{3}ϕ) dx_{α}dx_{3} : ϕ ∈ W^{1,p}((tQ′)^{f}; ℝ^{3}),

ϕ(x_{α}, x_{3}) = 0 if x_{α} ∈ ∂(tQ′), |x_{3}| < f(x_{α})},

and where, for A ⊂ ℝ^{2}, A^{f} := {(x_{α}, x_{3}) : x_{α} ∈ A, |x_{3}| < f(x_{α})}.

Remark 4.1. It is easily shown that

W_{hom}(\overline{F}) = inf_{t>0} g(t),

and also that, in the definition of g(t), periodic boundary conditions on the test functions can be imposed in lieu of Dirichlet boundary conditions.

The following theorem holds true:

Theorem 4.2. If u ∈ W^{1,p}(ω; ℝ^{3}) and if A is an open subset of ω, then

Γ(L^{p})– lim E_{ε}(u; A) = ∫_{A} W_{hom}(D_{α}u) dx_{α}.

Proof. Consider a sequence {ε}, with ε ↘ 0^{+}. Application of Theorem 2.5 permits to assert the existence of a subsequence {ε^{R}} of {ε} and of a Carathéodory function W_{{ε^{R}}} such that

J_{{ε^{R}}}(u; A) = ∫_{A} W_{{ε^{R}}}(x_{α}; D_{α}u) dx_{α}.

We firstly show that W_{{ε^{R}}} is independent of x_{α}.

Fix \overline{F} ∈ ℝ^{3×2}, and let x_{0}, y_{0} ∈ ω be Lebesgue points for W_{\{ε^{R}\}}(·; \overline{F}), so that

(4.1)  W_{\{ε^{R}\}}(x_{0}; \overline{F}) = lim_{δ→0^{+}} \frac{1}{δ^{2}} ∫_{Q′(x_{0},δ)} W_{\{ε^{R}\}}(x_{α}; \overline{F}) dx_{α}
= lim_{δ→0^{+}} \frac{1}{δ^{2}} J_{\{ε^{R}\}}(\overline{F} x_{α}; Q′(x_{0}, δ)),

– idem for y_{0}. Assume that δ is small enough. According to Lemma 2.6, there exists a sequence {ψ_{ε^{R}}^{δ}} with ψ_{ε^{R}}^{δ} = 0 on {(x_{α}, x_{3}) : |x_{3}| < f(x_{α}/ε^{R}), x_{α} ∈ ∂Q′(x_{0}, δ)} and ψ_{ε^{R}}^{δ}χ_{Q′(x_{0},δ)_{ε^{R}}} → 0 in L^{p}(Ω; ℝ^{3}) (with, as usual, Q′(x_{0}, δ)_{ε^{R}} := {(x_{α}, x_{3}) : |x_{3}| < f(x_{α}/ε^{R}), x_{α} ∈ Q′(x_{0}, δ)}), such that

J_{\{ε^{R}\}}(\overline{F} x_{α}; Q′(x_{0}, δ)) = lim_{ε^{R}→0^{+}} E_{ε^{R}}(\overline{F} x_{α} + ψ_{ε^{R}}^{δ}; Q′(x_{0}, δ)). (4.2)

Define the vector τ_{ε^{R}} ∈ ε^{R}ℤ^{N} as

(τ_{ε^{R}})_{i} := ε^{R}|[\frac{(y_{0} − x_{0})_{i}}{ε^{R}} ]|, for i = 1, …, N.

Cleary τ_{ε^{R}} → y_{0} − x_{0} as ε^{R} → 0^{+}. Let

ϕ_{ε^{R}}^{δ}(x_{α}, x_{3}) := ψ_{ε^{R}}^{δ}(x_{α} − τ_{ε^{R}}, x_{3}),

where we have extended ψ_{ε^{R}}^{δ} by 0 to [ℝ^{2} \ Q′(x_{0}, δ)]_{ε^{R}}.

Fix r > 1 and consider ε^{R} small enough so that

Q′(y_{0} − τ_{ε^{R}}, δ) ⊂ Q′(x_{0}, rδ). (4.3)

Since ϕδ ; \overline{ε}χ_{Q′(y_{0},δ)_{ε^{R}}} → 0 in L^{p}(Ω; ℝ^{3}), we have

J_{\{ε^{R}\}}(\overline{F} x_{α}; Q′(y_{0}, δ)) ≤

lim inf_{ε^{R}→0^{+}} ∫_{Q′(y_{0},δ)_{ε^{R}}} W (\frac{x_{α}}{ε^{R}} , x_{3}; \overline{F} + D_{α}ϕ_{ε^{R}}^{δ} |\frac{1}{ε^{R}} D_{3}ϕ_{ε^{R}}^{δ}) dx_{α}dx_{3}

≤ lim inf_{ε^{R}→0^{+}} ∫_{Q′(y_{0}−τ_{ε^{R}} ,δ)_{ε^{R}}} W (\frac{x_{α} + τ_{ε^{R}}}{ε^{R}} , x_{3}; \overline{F} + D_{α}ϕδ ; \overline{ε}(x_{α} + τ_{ε^{R}}, x_{3})|

\frac{1}{ε^{R}} D_{3}ϕ_{ε^{R}}^{δ}(x_{α} − τ_{ε^{R}}, x_{3})) dx_{α}dx_{3}

≤ lim inf_{ε^{R}→0^{+}} ∫_{Q′(x_{0},rδ)_{ε^{R}}} W (\frac{x_{α}}{ε^{R}} , x_{3}; \overline{F} + D_{α}ψδ ; \overline{ε}(x_{α}, x_{3})|

\frac{1}{ε^{R}} D_{3}ψ_{ε^{R}}^{δ}(x_{α}, x_{3})) dx_{α}dx_{3}

≤ J_{\{ε^{R}\}}(\overline{F} x_{α}; Q′(x_{0}, δ)) + 2β(1 + |\overline{F} |^{p})L^{2}(Q′(x_{0}, rδ) \ Q′(x_{0}, δ))

where we have used (4.2), (4.3) and the periodicity of W (·, x_{3}). Letting r → 1, we finally obtain

J_{\{ε^{R}\}}(\overline{F} x_{α}; Q′(y_{0}, δ)) ≤ J_{\{ε^{R}\}}(\overline{F} x_{α}; Q′(x_{0}, δ)),

hence, in view of (4.1),

W_{\{ε^{R}\}}(y_{0}; \overline{F}) = lim_{δ→0^{+}} \frac{1}{δ^{2}} J_{\{ε^{R}\}}(\overline{F} x_{α}; Q′(y_{0}, δ)) ≤ W_{\{ε^{R}\}}(x_{0}; \overline{F}).

Given the arbitrariness of x_{0} and y_{0} we conclude that

W_{\{ε^{R}\}}(y_{0}; \overline{F}) = W_{\{ε^{R}\}}(x_{0}; \overline{F}) =: W_{\{ε^{R}\}}(\overline{F}).

We now identify W_{\{ε^{R}\}}(\overline{F}). Assuming, without loss of generality, that 0 ∈ ω and Q′ ⊂ ω, by virtue of Lemma 2.6 there exists a sequence {ψ_{ε^{R}}} with ψ_{ε^{R}} = 0 on {(x_{α}, x_{3}) : |x_{3}| < f(x_{α}/ε^{R}), x_{α} ∈ ∂Q′} and ψ_{ε^{R}}χ_{Q′_{ε^{R}}} → 0 in L^{p}(Ω; ℝ^{3}), such that

W_{\{ε^{R}\}}(\overline{F}) = J_{\{ε^{R}\}}(\overline{F} x_{α}; Q′) = lim_{ε^{R}→0^{+}} E_{ε^{R}}(\overline{F} x_{α} + ψ_{ε^{R}}; Q′).

Define

ϕ_{ε^{R}}(x_{α}, x_{3}) := \frac{1}{ε^{R}} ψ_{ε^{R}}(ε^{R}x_{α}, x_{3}).

Then, ϕ_{ε^{R}} ∈ W^{1,p} ([(0, 1/ε^{R})^{2}]^{f} ; ℝ^{3}), and it is equal to 0 as soon as x_{α} ∈

∂(0, 1/ε^{R})^{2}; thus it is an admissible test function in the definition of g(1/ε^{R}) and

lim sup_{ε^{R}→0^{+}} g (\frac{1}{ε^{R}} ) ≤ (4.4)

lim sup_{ε^{R}→0^{+}} ε^{R}^{2} ∫_{_{[(0, \frac{1}{ε^{R}})_{2}]}_{f}} W (x_{α}, x_{3}; \overline{F} + D_{α}ϕ_{ε^{R}}|D_{3}ϕ_{ε^{R}}) dx_{α}dx_{3}

= lim sup_{ε^{R}→0^{+}} ∫_{Q′_{ε^{R}}} W (\frac{x_{α}}{ε^{R}} , x_{3}; \overline{F} + D_{α}ψ_{ε^{R}} |\frac{1}{ε^{R}} D_{3}ψ_{ε^{R}}) dx_{α}dx_{3}

= W_{\{ε^{R}\}}(\overline{F}).

Conversely, consider λ_{n} ↗ ∞ such that g(λ_{n}) → lim inf_{t↗∞} g(t). For each n, take ϕ_{n} ∈ W^{1,p}({(0, λ_{n})^{2} × (−1, 1) : |x_{3}| < f(x_{α})}; ℝ^{3}) with ϕ_{n} = 0 if x_{α} ∈ ∂(0, λ_{n})^{2}, and such that

g(λ_{n}) + \frac{1}{λ_{n}^{3}} ≥ \frac{1}{λ_{n}^{2}} ∫_{[(0,λ_{n})^{2}]_{f}} W (x_{α}, x_{3}; \overline{F} + D_{α}ϕ_{n}|D_{3}ϕ_{n}) dx_{α}dx_{3}. (4.5)

Set ψ_{ε^{R}}^{n} := ε^{R}ϕ_{n} (\frac{x_{α}}{ε^{R}} , x_{3}) , where ϕ_{n} has been extended by zero to (|[λ_{n}]| + 1)^{2}, and then to the whole of ℝ^{2} by (|[λ_{n}]| + 1)^{2}-periodicity. Then,

J_{\{ε^{R}\}}(\overline{F} x_{α}; Q′) ≤ lim inf_{ε^{R}→0^{+}} E_{ε^{R}}(\overline{F} x_{α} + ψ_{ε^{R}}^{n}; Q′)

= lim inf_{ε^{R}→0^{+}} ∫_{Q′_{ε^{R}}} W (\frac{x_{α}}{ε^{R}} , x_{3}; \overline{F} + D_{α}ψ_{ε^{R}}^{n} |\frac{1}{ε^{R}} D_{3}ψ_{ε^{R}}^{n}) dx_{α}dx_{3}

= lim inf_{ε^{R}→0^{+}} ∫_{Q′} [ ∫_{−f(\frac{xα}{ε^{R}})}^{f(\frac{xα}{ε^{R}})} W (\frac{x_{α}}{ε^{R}} , x_{3}; \overline{F} + D_{α}ϕ_{n} (\frac{x_{α}}{ε^{R}} , x_{3}) |

D_{3}ϕ_{n} (\frac{x_{α}}{ε^{R}} , x_{3}) ) dx_{3}]dx_{α}

= \frac{1}{(|[λ_{n}]| + 1)^{2}} ∫_{(0,λ_{n})^{2}} [ ∫_{−f(x_{α})}^{f(x_{α})} W (x_{α}, x_{3}; \overline{F} + D_{α}ϕ_{n}(x_{α}, x_{3})|

D_{3}ϕ_{n}(x_{α}, x_{3})) dx_{3}]dx_{α}

+\frac{1}{(|[λ_{n}]| + 1)^{2}} ∫_{[(0,|[λ_{n}]|+1)^{2}\(0,λ_{n})^{2}]_{f}} W (x_{α}, x_{3}; \overline{F} |0) dx_{α}dx_{3}

≤ \frac{λ_{n}^{2}}{(|[λ_{n}]| + 1)^{2}} (g(λ_{n}) + \frac{1}{λ_{n}^{3}} ) + O (\frac{1}{n}) ,

where we have used (4.5) as well as the (|[λ_{n}]| + 1)^{2}-periodic character of

∫_{f(·)}^{f(·)} W (·, x_{3}; \overline{F} + D_{α}ϕ^{n}(·, x_{3})|D_{3}ϕ_{n}(·, x_{3})) dx_{3}.

Thus, letting n tend to ∞,

J_{\{ε^{R}\}}(\overline{F} x_{α}; Q′) ≤ lim inf_{t↗∞} g(t),

or still

W_{\{ε^{R}\}}(\overline{F}) ≤ lim inf_{t↗∞} g(t). (4.6)

Recalling (4.4), (4.6), we obtain

lim inf_{t↗∞} g(t) ≤ lim sup_{ε^{R}→0^{+}} g (\frac{1}{ε^{R}} ) ≤ W_{\{ε^{R}\}}(\overline{F}) ≤ lim inf_{t↗∞} g(t),

which proves the desired result. Since the Γ(L^{p})–limit of E_{ε^{R}}(u; A) is independent of the specific sequence {ε^{R}}, in light of Proposition 7.11 in [9] we conclude that E_{ε}(u; A) Γ(L^{p})-converges to ∫_{A} W_{hom}(D_{α}u) dx_{α}.

Remark 4.3. Theorem 4.2 still holds if we only assume that 0 ≤ f ≤ 1. In general, the description of the Γ–limit is not complete, as there may exist a u ∉ W^{1,p}(ω; ℝ^{3}) such that J_{{ε^{R}}}(u; Ω) < +∞. Nevertheless, some degenerate cases can be dealt with in the spirit of the homogenization of domain with soft inclusions. This can be done, for example, if we suppose that for some γ > 0 the set B_{γ} := {x_{α} ∈ ℝ^{2} : γ < f(x_{α})} contains a periodic connected Lipschitz set (see related work in [4] and [9] Chapter 19).

5. Third application – Optimal design of a thin film

The kind of dimensional reduction performed in this paper has proved relevant in the analysis and design of thin films. We refer the interested reader to [17] and references therein for a detailed motivation of the problem considered below and for relevant results in the so-called cylindrical case (see Remark 5.3 below).

It is thus assumed in this section that W_{i}(F), i = 1, 2, is a continuous real-valued function on ℝ^{3×3} such that

β′|F |^{p} ≤ W_{i}(F) ≤ β(1 + |F |^{p}), 0 < β′ ≤ β < ∞, 1 ≤ p < ∞, (5.1)

Our goal is to compute , for any v ∈ W^{1,p}(ω; ℝ^{3}), any θ ∈ L^{∞}(ω × (−1, 1); [0, 1]), any open subdomain A ⊂ ℝ^{2},

J(v; θ; A) := inf_{{ε}↘0^{+}} J_{{ε}}(v; θ; A),

where

J_{\{ε\}}(v; θ; A):= inf_{\{χ_{ε}\},\{v_{ε}\}} \{ lim inf_{ε→0^{+}} ∫_{A×(−1,1)} (χ_{ε}W_{1} + (1 − χ_{ε})W_{2}) (D_{α}v_{ε}|\frac{1}{ε}D_{3}v_{ε})dx_{α}dx_{3} :

v_{ε} ∈ W^{1,p}(A × (−1, 1); ℝ^{3}), χ_{ε} ∈ L^{∞}(A × (−1, 1); {0, 1}),

v_{ε} → v in L^{p}(A × (−1, 1); ℝ^{3}), χ_{ε} ⇀^{∗} θin L^{∞}(A × (−1, 1); [0, 1])}.

Let us define, for any \overline{F} ∈ ℝ^{3×2}, θ ∈ [0, 1],

Ŵ(θ, \overline{F}) := (5.2)

inf_{λ>0} inf_{ϕ,χ} \{\frac{1}{2} ∫_{Q′×(−1,1)}(χW_{1} + (1 − χ)W_{2})(\overline{F} + D_{α}ϕ|λD_{3}ϕ) dx_{α}dx_{3} :

ϕ ∈ W^{1,p}(Q′ × (−1, 1); ℝ^{3}), ϕ = 0 on ∂Q′ × (−1, 1),

χ ∈ L^{∞}(Q′ × (−1, 1); \{0, 1\}), \frac{1}{2} ∫_{Q′×(−1,1)} χ dx_{α}dx_{3} = θ\}

= inf_{k>0} inf_{ϕ,χ} \{\frac{1}{2k} ∫_{Q′×(−k,k)}(χW_{1} + (1 − χ)W_{2})(\overline{F} + D_{α}ϕ|D_{3}ϕ) dx_{α}dx_{3} :

ϕ ∈ W^{1,p}(Q′ × (−k, k); ℝ^{3}), ϕ = 0 on ∂Q′ × (−k, k),

χ ∈ L^{∞}(Q′ × (−k, k); \{0, 1\}), \frac{1}{2k} ∫_{Q′×(−k,k)} χ dx_{α}dx_{3} = θ\}.

Remark 5.1. It is easily proved that Ŵ is an upper-semicontinuous function of (θ, \overline{F}) ∈ [0, 1] × ℝ^{3×2}. The proof is a strict analogue to that of Proposition 2.9 in [17].

The following theorem holds true:

Theorem 5.2.

J(v; θ; A) ≥ 2 ∫_{A} Ŵ (\frac{1}{2} ∫_{−1}^{1} θ(x_{α}, s) ds, Dv(x_{α})) dx_{α}.

Further, equality holds if θ ∈ L^{∞}(ω; [0, 1]) and if W satisfies the following symmetry property:

W (\overline{F} |F_{3}) = W (\overline{F} | − F_{3}), \overline{F} ∈ ℝ^{3×2}, F_{3} ∈ ℝ^{3}. (5.3)

Remark 5.3. In contrast with the setting investigated in [17] the material distribution – the characteristic functions χ_{ε} –is not restricted to cylindrical geometries; in other words χ_{ε} may also depend on the transverse variable x_{3}. This is the only difference between Theorem 5.2 and Theorem 2.3 in [17]. Note that the equality in Theorem 5.2 holds if θ is independent of x_{3}, that is if we are truly in a plate-like setting.

Remark 5.4. The section heading is somewhat misleading since, in truth, Theorem 5.2 is not a mere application of the results of Section 2; specifically, Theorem 2.5 cannot be invoked in the current setting because of the presence of the additional field χ_{ε}. The proof of Theorem 2.3 in [17] can however be revisited in the light of the method used to prove Theorem 2.5 so as to prove Theorem 5.2. This is the object of the proof of Theorem 5.2 below.

Remark 5.5. We conjecture that the symmetry condition (5.3), although satisfied in many applications, is not necessary, but confess to our inability at doing away with it at present. We remark that this hypothesis is not required in the case of cylindrical inclusions (see [17]).

Proof. The first part of Theorem 5.2 is contained in the more precise Lemma 5.6 stated below which is in turn the strict analogue of (part of) Theorem 2.5 in the current setting.

Lemma 5.6. For any sequence {ε} ↘ 0^{+}, there exists a subsequence {ε^{R}} such that

J_{\{ε^{R}\}}(v; θ; A) ≥ 2 ∫_{A} Ŵ (\frac{1}{2} ∫_{−1}^{1} θ(x_{α}, s) ds, Dv(x_{α})) dx_{α},

thus, in particular,

J(v; θ; A) ≥ 2 ∫_{A} Ŵ (\frac{1}{2} ∫_{−1}^{1} θ(x_{α}, s) ds, Dv(x_{α})) dx_{α}.

Proof. The subsequence {ε^{R}} is defined exactly as in Section 2 (see argument leading to (2.7)). The proof is then divided into two steps which will be sketched below. Only those parts of the argument that differ from analogous parts in the proofs of Theorem 2.5 or of Theorem 2.3 in [17] will be detailed. The first step is devoted to a proof that J_{{ε^{R}}}(v; θ; ·) is a finite nonnegative Radon measure which is absolutely continuous with respect to L^{2}⌊ω. A second step establishes that the Radon-Nykodim derivative \frac{dJ_{\{ε^{R}\}}(v;θ;·)}{dL^{2}}(x_{0}) is, for suitable x_{0}’s in ω, greater than

or equal to 2Ŵ (1/2 ∫_{−1}^{1} θ(x_{0}, s) ds, D_{α}v(x_{0})) .

Step 1. Step 1 is a near verbatim reproduction of Steps 1–3 in the proof of Theorem 2.5. Firstly, it is observed, exactly as in Lemma 2.6, that approximating sequences for v may as well take the value v on the lateral boundary of A × (−1, 1). The proof is identical to that of Lemma 2.6. Then the inner regularity of J_{{ε^{R}}}(v; θ; ·) is established exactly as for (2.16). We now address the subadditive character of J_{{ε^{R}}}(v; θ; ·). Once again the proof is nearly identical to that of (2.18). Note

however that the recovery sequences for B^{δ}, D^{δ} are pairs (χ_{ε^{R}}^{B^{δ}}, v_{ε^{R}}^{B^{δ}}) ∈ L^{∞}(B^{δ} ×

(−1, 1); {0, 1}) × W^{1,p}(B^{δ} × (−1, 1); ℝ^{3}) – idem for D^{δ} –, with

\begin{cases}χ_{ε^{R}}^{B^{δ}} ⇀^{∗} θ & in L^{∞}(B^{δ} × (−1, 1); [0, 1]), \\ v_{ε^{R}}^{B^{δ}} → v & in L^{p}(B^{δ} × (−1, 1); ℝ^{3}); \end{cases}

idem for D^{δ}. A new pair-sequence is defined as

\begin{cases}χ\overline{ε} & := χ_{η_{0}}^{δ}χD^{δ} ; \overline{ε} + (1 − χ_{η_{0}}^{δ})χB^{δ} ; \overline{ε} + χ_{(A\(B^{δ}∪D^{δ}))\overline{ε}}h\overline{ε} \\ v_{ζ,\overline{ε}} & := ϕ_{ζ}^{δ}vD^{δ} ; \overline{ε} + (1 − ϕ_{ζ}^{δ})vB^{δ} ; \overline{ε} + χ_{(A\(B^{δ}∪D^{δ}))\overline{ε}}u, \end{cases}

where ϕ_{ζ} was defined in (2.22), \{h\overline{ε}\} is any sequence in L^{∞}(Ω; \{0, 1\}) converging to θ weakly-* in L^{∞}(Ω; [0, 1]), and χ_{η_{0}}^{δ} is a characteristic function such that

χ_{η_{0}}^{δ} = \begin{cases}1 & if x_{α} ∉ B^{δ} or dist(x_{α}, ∂B^{δ}) ≤ η_{0}, \\ 0 & otherwise. \end{cases}

Then, clearly, as \overline{ε} ↘ 0^{+},

\begin{cases}χ\overline{ε} ⇀^{∗} θ & in L^{∞}(A × (−1, 1); [0, 1]), \\ v_{ζ,\overline{ε}} → v & in L^{p}(A × (−1, 1); ℝ^{3}), \end{cases}

and the remainder of the proof of (2.18) proceeds as before. Finally, (2.26), (2.27) are unchanged and Lemma 7.3 permits to conclude that J_{{ε^{R}}}(v; θ; ·) is a finite nonnegative Radon measure. The bound from above in (5.1) immediately implies that it is absolutely continuous with respect to L^{2}⌊ω.

Step 2. From the definition of Radon-Nykodim derivative, for almost every x_{0} ∈ ω,

\frac{dJ_{\{ε^{R}\}}(v; θ; ·)}{dL^{2}}(x_{0}) = lim_{δ→0^{+}} \frac{1}{δ^{2}} J_{\{ε^{R}\}}(v; θ; Q′(x_{0}, δ)) (5.4)

= lim_{δ→0^{+}} \frac{1}{δ^{2}} lim_{\overline{ε}→0^{+}} ∫_{Q′(x_{0},δ)×(−1,1)} (χ\overline{ε}W_{1} + (1 − χ\overline{ε})W_{2}) (D_{α}v\overline{ε}|\frac{1}{\overline{ε}}D_{3}v\overline{ε}) dx_{α}dx_{3},

where \{\overline{ε}\} is a subsequence of \{ε^{R}\} and

\begin{cases}χ\overline{ε} ⇀^{∗} θ & in L^{∞}(Q′(0, δ) × (−1, 1); [0, 1]), \\ v\overline{ε} → v & in L^{p}(Q′(0, δ) × (−1, 1); ℝ^{3}). \end{cases}

Take x_{0} ∈ ω to be a Lebesgue point for ∫_{−1}^{1} θ(·, s) ds and a point of approximate differentiability for v. Setting

\begin{cases}χ_{δ,\overline{ε}} := χ\overline{ε}(x_{0} + δx_{α}, x_{3}), \\ v_{δ,\overline{ε}} := \frac{v\overline{ε}(x_{0} + δx_{α}, x_{3}) − v(x_{0})}{δ} \end{cases}

(5.4) now reads as

\frac{dJ_{\{ε^{R}\}}(v; θ; ·)}{dL^{2}}(x_{0}) = (5.5)

lim_{δ→0^{+}} lim_{\overline{ε}→0^{+}} ∫_{Q′×(−1,1)}(χ_{δ,\overline{ε}}W_{1} + (1 − χ_{δ,\overline{ε}})W_{2}) (D_{α}v_{δ,\overline{ε}}|\frac{δ}{\overline{ε}}D_{3}v_{δ,\overline{ε}}) dx_{α}dx_{3}.

Note that

(5.6)

lim_{δ→0^{+}} lim_{\overline{ε}→0^{+}} \frac{1}{2} ∫_{Q′×(−1,1)} χ_{δ,\overline{ε}} dx_{α}dx_{3} = \frac{1}{2} lim_{δ} lim_{\overline{ε}} \frac{1}{δ^{2}} ∫_{Q′(x_{0},δ)×(−1,1)} χ\overline{ε} dx_{α}dx_{3}
= \frac{1}{2} lim_{δ→0^{+}} \frac{1}{δ^{2}} ∫_{Q′(x_{0},δ)×(−1,1)} θ dx_{α}dx_{3}

= \frac{1}{2} lim_{δ→0^{+}} \frac{1}{δ^{2}} ∫_{Q′(x_{0},δ)} (∫_{−1}^{1} θ(x_{α}, s) ds) dx_{α}

= \frac{1}{2} ∫_{−1}^{1} θ(x_{0}, s) ds,

because x_{0} is a Lebesgue point for ∫_{−1}^{1} θ(x_{α}, s) ds.

Set A_{δ,\overline{ε}} := \{(x_{α}, x_{3}) ∈ Q′ × (−1, 1) : χ_{δ,\overline{ε}}(x_{α}, x_{3}) = 1\}. Then (5.6) implies that

lim_{δ→0^{+}} lim_{\overline{ε}→0^{+}} \frac{1}{2}L^{3}(A_{δ,\overline{ε}}) = \frac{1}{2} ∫_{−1}^{1} θ(x_{0}, s) ds =: Θ.

The remaining of the proof would then be obvious from (5.5) and the definition (5.2) of Ŵ(Θ, Dv(x_{0})) if

\frac{1}{2}L^{3}(A_{δ,\overline{ε}}) = Θ

and if v_{δ,\overline{ε}} = D_{α}(x_{0})x_{α} on ∂Q′ × (−1, 1). Unfortunately, there is no guarantee that the above holds true and the sequence \{χ_{δ,\overline{ε}}, v_{δ,\overline{ε}}\} must be modified accordingly. This procedure is identical to that described in the proof of Lemma 3.1 in [17] – up to changing the names of the indices q, n to δ, \overline{ε}, and also up to replacing θ(x_{0}) by Θ – and the interested reader is invited to consult pages 185 to 189 of that paper. Note that in the case where Θ ≡ 1 one should remark, in translating the proof into our context, that, by virtue of Remark 3.3 and also of (3.3), (5.2),

Q\overline{W}_{1}(\overline{F}) = \underline{W}_{1}(\overline{F}) = Ŵ(1, \overline{F}).

The proof of Lemma 5.6 is complete.

We now address the second part of Theorem 5.2 and thus assume, from now onward, that θ is independent of x_{3}.

The proof is divided into two steps. In a first step, it is assumed that θ ≡ θ_{∞} and v ≡ v_{∞} are, respectively, constant and affine functions and we get the following result:

Lemma 5.7. Let T be a triangle on the plane. Then there exists a sequence {v_{n}, χ_{n}} ∈ W^{1,p}(T × (−1, 1); ℝ^{3}) × L^{∞}(T × (−1, 1); {0, 1}) with v_{n} = v_{∞} on ∂T × (−1, 1) such that, as n ↗ ∞,

\begin{cases}χ_{n} ⇀^{∗} θ_{∞} & in L^{∞}(T × (−1, 1); [0, 1]), \\ v_{n} → v_{∞} & in L^{p}(T × (−1, 1); ℝ^{3}), \end{cases}

and

J(v_{∞}; θ_{∞}; T) = lim_{n→+∞} ∫_{T ×(−1,1)}(χ_{n}W_{1} + (1 − χ_{n})W_{2})(D_{α}v_{n}|nD_{3}v_{n}) dx_{α}dx_{3}
= 2L^{2}(T) Ŵ(θ_{∞}, D_{α}v_{∞}).

A second step addresses the case of general domains A ⊂ ω, general θ’s and general v’s, and yields Theorem 5.2.

Proof of Lemma 5.7. The proof is a blow-up argument in the spirit of [19]. From the definition (5.2) of Ŵ, together with the density of W^{1,∞}(Q′ × (−1, 1); ℝ^{3}) into W^{1,p}(Q′ × (−1, 1); ℝ^{3}), we deduce that, for any η > 0, there exist λ^{η} > 0, ϕ^{η} ∈ W^{1,∞}(Q′ × (−1, 1); ℝ^{3}), with ϕ^{η} = 0 on ∂Q′ × (−1, 1), and χ^{η} ∈ L^{∞}(Q′ × (−1, 1); {0, 1}) with 1/2 ∫_{Q′×(−1,1)} χ^{η} dx_{α}dx_{3} = θ_{∞} such that

2Ŵ(θ_{∞}, D_{α}v_{∞}) ≥ ∫_{Q′×(−1,1)} (χ^{η}W_{1} + (1 − χ^{η})W_{2})(D_{α}v_{∞} + D_{α}ϕ^{η}|λ^{η}D_{3}ϕ^{η}) dx_{α}dx_{3} − η. (5.7)

Extend ϕ^{η}, χ^{η} to be Q′ × (−2, 2)-periodic by setting

\overline{ϕ}^{η}, \overline{χ}^{η}(x_{α}, x_{3}) := \begin{cases}ϕ^{η}, χ^{η}(x_{α}, x_{3}), & −1 ≤ x_{3} ≤ 1, \\ ϕ^{η}, χ^{η}(x_{α}, −2 − x_{3}), & −2 ≤ x_{3} ≤ −1, \\ ϕ^{η}, χ^{η}(x_{α}, 2 − x_{3}), & 1 ≤ x_{3} ≤ 2, \end{cases}

then extend \overline{ϕ}^{η}, \overline{χ}^{η} by Q′ × (−2, 2)-periodicity to ℝ^{3}. Note that, in view of (5.3), (5.7) also reads as

4Ŵ(θ_{∞}, D_{α}v_{∞})≥∫_{Q′×(−2,2)} !(\overline{χ}^{η}W_{1} + (1 − \overline{χ}^{η})W_{2})(D_{α}v_{∞} + D_{α}\overline{ϕ}^{η}|λ^{η}D_{3}\overline{ϕ}^{η}) dx_{α}dx_{3} − 2η. (5.8)

Remark 5.8. This is the only instance where assumption (5.3) is used in this study.

Set

\begin{cases}χ_{n}^{η}(x_{α}, x_{3}) & := \overline{χ}^{η}(n^{2}x_{α}, nx_{3}), \\ v_{n}^{η}(x_{α}, x_{3}) & := v_{∞}(x_{α}) + \frac{1}{n^{2}} \overline{ϕ}^{η}(n^{2}x_{α}, nλ^{η}x_{3}). \end{cases}

If Q′(a, r) is a square on ℝ^{2}, then it is easily checked that, as n ↗ ∞,

\begin{cases}χ_{n}^{η} ⇀^{∗} θ_{∞} & in L^{∞}(Q′(a, r) × (−1, 1); [0, 1]), \\ v_{n}^{η} → v_{∞} & in L^{p}(Q′(a, r) × (−1, 1); ℝ^{3}). \end{cases} (5.9)

Further, in view of (5.8), together with the periodic character of v_{n}^{η}, χ_{n}^{η},

J(v_{∞}; θ_{∞}; Q′(a, r)) ≤

lim inf_{η→0^{+}} lim_{n→+∞} ∫_{Q′(a,r)×(−1,1)}(χ_{n}^{η}W_{1} + (1 − χ_{n}^{η})W_{2})(D_{α}v_{n}^{η}|nD_{3}v_{n}^{η}) dx_{α}dx_{3}

≤ lim sup_{η→0^{+}} lim_{n→+∞} ∫_{Q′(a,r)×(−1,1)}(χ_{n}^{η}W_{1} + (1 − χ_{n}^{η})W_{2})(D_{α}v_{n}^{η}|nD_{3}v_{n}^{η}) dx_{α}dx_{3}

≤ 2L^{2}(Q′(a, r)) Ŵ(θ_{∞}, D_{α}v_{∞}).

But, in view of Lemma 5.6,

J(v_{∞}; θ_{∞}; Q′(a, r)) ≥ 2L^{2}(Q′(a, r)) Ŵ(θ_{∞}, D_{α}v_{∞}),

so that

J(v_{∞}; θ_{∞}; Q′(a, r)) =

lim_{η→0^{+}} lim_{n→+∞} ∫_{Q′(a,r)×(−1,1)}(χ_{n}^{η}W_{1} + (1 − χ_{n}^{η})W_{2})(D_{α}v_{n}^{η}|nD_{3}v_{n}^{η}) dx_{α}dx_{3}

= 2L^{2}(Q′(a, r)) Ŵ(θ_{∞}, D_{α}v_{∞}),

hence, by virtue of Lemma 7.1 in the Appendix, there exists a sequence {η(n)}_{n} such that v_{n} := v_{n}^{η(n)}, χ_{n} := χ_{n}^{η(n)} satisfy (5.9) as well as

(5.10)

J(v_{∞}; θ_{∞}; Q′(a, r)) = lim_{n→+∞} ∫_{Q′(a,r)×(−1,1)} (χ_{n}W_{1} + (1 − χ_{n})W_{2})(D_{α}v_{n}|nD_{3}v_{n}) dx_{α}dx_{3}
= 2L^{2}(Q′(a, r)) Ŵ(θ_{∞}, D_{α}v_{∞}).

Consider now a triangle T covered with squares of the type Q′(a, r), a ∈ ℝ^{2}, r > 0, up to a set of measure 1/m, i.e.,

T_{m} := ∪_{i=1}^{N(m)}Q′(a_{i}^{m}, r_{i}^{m}) ⊂⊂ T,

with L^{2}(T \T_{m}) ≤ 1/m. An easy construction, identical to that of Step 2 in Lemma 4.2 in [17] (see Remark 4.4 in [17]) yields (5.10) for T in lieu of Q′(a, r). The proof of Lemma 5.7 is complete.

In view of Lemma 5.7 – and in particular of the boundary condition v_{n} = v_{∞} for the corresponding sequence –, Lemma 5.7 also holds true if T is replaced by a triangulation of the plane on which v_{∞} is continuous and piecewise affine and θ_{∞} piecewise constant. Then, let v, θ be arbitrary elements of W^{1,p}(ω; ℝ^{3}) and L^{∞}(ω; [0, 1]), respectively, and consider {w_{k}, θ_{k}} a piecewise affine/piecewise constant pair defined on triangulations of the plane such that

\begin{cases}θ_{k} → θ & a.e. in ω, \\ w_{k} → v & in W^{1,p}(ω; ℝ^{3}). \end{cases}

For each pair (w_{k}, θ_{k}) there exists a pair-sequence {v_{k}^{n}, θ_{k}^{n}} defined on the same triangulation satisfying the properties of Lemma 5.7 for that triangulation and for v_{∞} := w_{k}, θ_{∞} := θ_{k}. A diagonalization process (with n replaced by n(k)) immediately yields a sequence {v_{k}, χ_{k}} ∈ W^{1,p}(A × (−1, 1); ℝ^{3}) × L^{∞}(A × (−1, 1); {0, 1}) such that

\begin{cases}χ_{k} ⇀^{∗} θ & in L^{∞}(A × (−1, 1); [0, 1]), \\ v_{k} → v & in L^{p}(A × (−1, 1); ℝ^{3}), \end{cases} (5.11)

and

(5.12)

J(v; θ; A) ≤ lim inf_{k→+∞} ∫_{A×(−1,1)}(χ_{k}W_{1} + (1 − χ_{k})W_{2})(D_{α}v_{k}|n(k)D_{3}v_{k}) dx_{α}dx_{3}
= lim inf_{k→+∞} J(v_{k}; θ_{k}; A)
= lim inf_{k} 2 ∫_{A} Ŵ(θ_{k}, D_{α}v_{k}) dx_{α}.

The bound from above for W_{i}, the first convergence in (5.11), Fatou’s lemma, and the upper semicontinuity property of Ŵ (see Remark 5.1) imply that

lim inf_{k→+∞} ∫_{A}{β(1+|D_{α}v_{k}|^{p})−Ŵ(θ_{k}, D_{α}v_{k})}dx_{α} ≥ ∫_{A}{β(1+|D_{α}v|^{p})−Ŵ(θ, D_{α}v)}dx_{α},

thus

∫_{A} Ŵ(θ, D_{α}v) dx_{α} ≥ lim sup_{k→+∞} ∫_{A} Ŵ(θ_{k}, D_{α}v_{k}) dx_{α},

which, together with (5.12), yields

J(v; θ; A) ≤ 2 ∫_{A} Ŵ(θ, D_{α}v) dx_{α}.

Lemma 5.6 provides the other inequality and the proof of Theorem 5.2 is complete.

6. Final Remarks

This paper provides some insight into the characterization of effective energies for thin structures with varying profiles within a nonlinear setting, and some of our results have already been used and referred to in the literature on equilibria of thin structures, such as the papers by Ansini and Braides [3], Braides and Fonseca [10] and Shu [26]. It is, by no means, a completed subject, as we have pointed out throughout the text. From the technical point of view, we believe that Theorem 3.1 may be extended to the case where the energy density also depends upon x_{α} (see Remark 3.4), and condition (5.3) should not be requested for proving Theorem 5.2 (see Remarks 5.5 and 5.8).

Finally, although Theorem 2.5 holds for arbitrary sets Ω_{ε} (see Remark 2.9), in order to have a complete description of the limit problem it is now known that some geometrical and structural conditions need to be imposed on Ω_{ε}, as illustrated by the example of Braides and Batthacharya [6] where the limit problem is 3D and there is no dimensional reduction in the resulting effective energy.

7. Appendix

Lemma 7.1 and Lemma 7.2 are trivial diagonalization lemmata.

Lemma 7.1. Let a_{k,j} be a doubly indexed sequence of real numbers (k, j ↗ ∞). If

lim_{k} lim_{j} a_{k,j} = L,

then there exists a subsequence {k(j)}_{j} ↗ ∞ such that

lim_{j} a_{k(j),j} = L.

Lemma 7.2. Let a_{k,j} be a doubly indexed sequence of real numbers (k, j ↗ ∞). If

sup_{k} lim_{j} a_{k,j} = L,

then there exists a subsequence {k(j)}_{j} ↗ ∞ such that

lim_{j} a_{k(j),j} = L.

Let a_{k,j} be a doubly indexed sequence of real numbers (k, j ↗ ∞). If

lim_{k} lim sup_{j} a_{k,j} = L = lim_{k} lim inf_{j} a_{k,j},

then there exists a subsequence {k(j)}_{j} ↗ ∞ such that

lim_{j} a_{k(j),j} = L.

The third lemma in this appendix provides sufficient conditions for a mapping π : O → [0, +∞) to be the trace of a Radon measure, where O is the set of open subsets of ω and ω is an open subset of ℝ^{N}. It is close in spirit to De Giorgi-Letta’s criterion [16].

Lemma 7.3. Let π be a mapping from O into ℝ^{+} and μ be a finite nonnegative Radon measure on ℝ^{N}. If, for any A, B, C ∈ O,

(i) π(A) ≤ π(A \ \overline{C}) + π(B), \overline{C} ⊂ B ⊂ A, (ii) for any ε > 0, there exists C_{ε} ∈ O with \overline{C}_{ε} ⊂ A and π(A \ \overline{C}_{ε}) ≤ ε, (iii) π(ω) ≥ μ(ℝ^{N}), (iv) π(A) ≤ μ(\overline{A}), then, π is the restriction of μ to O.

Proof. Recalling (ii), consider d_{ε} := dist(\overline{C}_{ε}, ∂A) > 0. Then, \overline{C}_{ε} ⊂ B_{ε} := \{x ∈ A; dist(x, ∂A) > d_{ε}/2\}, while \overline{B}_{ε} ⊂ A. Thus, by (i, iv), and since \overline{C}_{ε} ⊂ B_{ε} ⊂ A,

π(A) ≤ ε + π(B_{ε}) ≤ ε + μ(\overline{B}_{ε}) ≤ ε + μ(A).

Hence, letting ε tend to 0,

π(A) ≤ μ(A).

Conversely, since μ is a Radon measure, it is inner regular, so that for any ε > 0, there exists C_{ε} ∈ O with \overline{C}_{ε} ⊂ A and μ(A) ≤ ε + μ(C_{ε}). Hence, with the help of (iii) and of the previously derived inequality,

μ(A) ≤ ε+μ(\overline{C}_{ε}) = ε+μ(ω)−μ(ω\\overline{C}_{ε}) ≤ ε+μ(ℝ^{N})−π(ω\\overline{C}_{ε}) ≤ ε+π(ω)−π(ω\\overline{C}_{ε}),

and, since \overline{C}_{ε} ⊂ A ⊂ ω, (i) implies that μ(A) ≤ ε + π(A), so that the result is obtained upon letting ε tend to 0.

Acknowledgments. The authors would like to thank Emilio Acerbi for several interesting and stimulating discussions on the subject of this paper. The research of A. Braides was partially supported by Marie-Curie fellowship ERBFMBICT972023 of the European Union program “Training and Mobility of Researchers”. The research of I. Fonseca was partially supported by the National Science Foundation under Grants No. DMS-9500531 and DMS-9731957 and through the Center for Nonlinear Analysis (CNA). The research of Gilles Francfort was partially supported through the TMR-EC contract CT98/0229 “Phase Transitions and Crystalline Solids”. The research of the three authors was partially supported by the Max-Planck Institute for Mathematics in the Sciences, Leipzig (Germany).

References

[1] E. Acerbi, G. Buttazzo, D. Percivale. A variational definition for the strain energy of an elastic string. J. Elasticity, 25, 1991, 137–148.

[2] E. Acerbi, N. Fusco. Semicontinuity problems in the calculus of variations. Arch. Rat. Mech. Anal., 86, 1984, 125–145.

[3] Nadia Ansini, Andrea Braides. Homogenization of oscillating boundaries and applications to thin films. Preprint SISSA, Trieste 1999.

[4] E. Acerbi, V. Chiad`0 Piat, G. Dal Maso, D. Percivale An extension theorem from connected sets, and homogenization in general periodic domains. Nonlinear Analysis, 18, 1992, 481-496.

[5] E. Anzelotti, S. Baldo, D. Percivale. Dimensional reduction in variational problems, asymptotic developments in Γ–convergence, and thin structures in elasticity. Asymptotic Anal., 9, 1994, 61–100.

[6] K. Batthacharya, A. Braides Thin films with many small cracks. Preprint SISSA 1999.

[7] H. Ben–Belgacem. Modélisation de structures minces en élasticité non–linéaire. Thèse de l’Université de Pierre et Marie Curie, Paris, 1996.

[8] D. Blanchard, G.A. Francfort. Asymptotic thermoelastic behaviour of flat plates. Quart. Appl. Math., 45(4), 1987, 645–667.

[9] A. Braides, A. Defranceschi. Homogenization of multiple integrals. Oxford Lecture Series in Mathematics and its Applications, 12, Oxford, 1998.

[10] A. Braides, I. Fonseca Brittle thin films. Applied Math. and Optimization., to appear.

[11] G. Buttazzo. Semicontinuity, relaxation and integral representation in the calculus of variations. Pitman, London, 1989.

[12] D. Caillerie. Thin elastic and periodic plates. Math. Methods in Appl. Sci., 6, 1984 , 159–191.

[13] P.G. Ciarlet. A justification of the von Kàrmàn equations. Arch. Rat. Mech. Anal., 73, 1980 , 349–389.

[14] P.G. Ciarlet. Introduction to Linear Shell Theory. North Holland, Amsterdam, 1998.

[15] P.G. Ciarlet, Ph. Destuynder. A justification of the two–dimensional linear plate model. J. Mécanique, 18, 1979, 315–344.

[16] E. De Giorgi, G. Letta. Une notion générale de convergence faible pour des fonctions croissantes d’ensemble. Ann. Sc. Norm. Sup. Pisa Cl. Sci., 4, 1977, 61–99.

[17] I. Fonseca, G.A. Francfort. 3D–2D asymptotic analysis of an optimal design problem for thin films. J. reine angew. Math., 505, 1998, 173–202.

[18] I. Fonseca, G.A. Francfort. In preparation.

[19] I. Fonseca, S. Muller. Quasiconvex integrands and lower semicontinuity in L^{1}. SIAM J. Math. Anal., 23, 1992, 1081–1098.

[20] D. Fox, A. Raoult, J.C. Simo. A justification of nonlinear properly invariant plate theories. Arch. Rat. Mech. Anal., 124, 1993, 157–199.

[21] G. Geymonat, F. Krasucki, J.J. Marigo. Stress distribution in anisotropic elastic composite beams. In P.G. Ciarlet, E. Sanchez–Palencia, editors, Applications of multiple scalings in mechanics, Masson, Paris, 1987, 118–133.

[22] G. Geymonat, F. Krasucki, J.J. Marigo. Sur la commutativité des passages à la limite en théorie asymptotique des poutres composites. C.R. Acad. Sci. Paris, 305(II), 1987 , 225–228.

[23] R.V. Kohn, M. Vogelius. A new model for thin plates with rapidly varying thickness II, III. Quart. Appl. Math., 43, 44, 1985, 1986, 1–22, 35–48.

[24] H. Le Dret, A. Raoult. The nonlinear membrane model as variational limit of nonlinear three–dimensional elasticity. J. Math. Pures Appl.,74, 1995, 549–578.

[25] F. Murat, A. Sili. Problèmes monotones dans des cylindres de faible diamètre. C. R. Acad. Sci. Paris, 319(I), 1995, 567–572.

[26] Y. Shu Heterogeneous thin films of martensitic materials. Preprint Caltech, Pasadena 1999.

(Andrea Braides) S.I.S.S.A, 34014 Trieste, Italy

(Irene Fonseca) Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, PA 15213, USA

E-mail address, I. Fonseca: fonseca@andrew.cmu.edu

(Gilles Francfort) L.P.M.T.M., Université Paris-Nord, 93430 Villetaneuse, France