Journal of Convex Analysis Volume 3 (1996), No. 2, 221–243

A Variational Model for Non Linear Elastic Plates

Valentina Casarino

Dipartimento di Matematica dell’universitá di Genova Via Dodecaneso 37, 16100 Genova, Italy. e-mail: casarino@dima.unige.it

Danilo Percivale

Istituto Matematico di Ingegneria, Fiera del mare Pad. D, Piazzale J.F.Kennedy, 16100 Genova, Italy. e-mail: percival@dima.unige.it

Received September 11, 1995 Revised manuscript received January 2, 1996

By adopting the variational point of view, the constitutive equations of a non linear elastic plate are deduced under kinematical constraints on the admissible deformations.

1. Introduction

A classical approach to the study of thin structures in elasticity consists in starting from three-dimensional models and deducing the behaviour of two-dimensional or onedimensional thin elastic bodies by passing to the limit when one or two dimensions go to zero. Though the two-dimensional linear model of an elastic plate has exhaustively been studied from different points of view (see [1], [3], [8], [9]), to our knowledge it doesn’t exist a rigorous theory which permits to deduce a non linear model of elastic plate as a limit (for instance, in the sense of [3]) of non linear three-dimensional thin elastic bodies. The most difficulty which arises in treating these problems is that, without some kinematical constraints on the deformations, no information about the compactness of minimizers of the energy functional of the three-dimensional elastic bodies can be expected and therefore it is very difficult to formulate any reasonable conjecture about the “limit” functional, that is the energy functional of the limit plate. The simplest method to avoid this difficulty is that of Kirchhoff, which consists in the linearization of the Green-St.Venant energy functional

∫_{Ω} (μ|E|^{2} + \frac{λ}{2}(tr E)^{2})dx  (1.1)

(where E = \frac{1}{2}(^{T} DφDφ − I), φ being the deformation), thus obtaining the classical functional

∫_{Ω} (μ|e(u)|^{2} + \frac{λ}{2}(tr e(u))^{2})dx  (1.2)

ISSN 0944-6532 / $ 2.50 ◯^{c} Heldermann Verlag

222 V.Casarino, D. Percivale / A variational model for non linear elastic plates

with u(x) = φ(x) − x. When one of the dimensions of Ω goes to zero and μ, λ become greater and greater in a suitable way (see [1]) the limit functional can be calculated explicitly. In this paper we adopt another point of view: we start again, roughly speaking, from the Green-St.Venant energy functional, but we impose a different kinematical constraint on the deformation φ, which will be assumed sufficiently regular. More precisely, if Σ_{ε} is the reference configuration of the three-dimensional elastic body having Σ as middle surface, we assume, following Podio-Guidugli [14] that “material fibers orthogonal to the middle surface before loading remain approximately orthogonal to it after loading”; this restriction takes the form E(φ)(\underline{n}) = 0 in Σ_{ε}, where \underline{n} is the normal unit vector to Σ. In the two dimensional case this is enough to obtain the required compactness properties of the minimizing sequences of the energy functional, while in the case n = 3 (the elastic plate) another internal constraint which takes into account some symmetry properties of the deformation must be assumed. The method we used here is closely related to Γ-convergence, already used in similar situations; nevertheless, for the reader’s convenience, all results are stated and proven without using any specific knowledge about Γ-limits. For a more precise setting of the problem in this framework we refer to [1].

2. The bidimensional case: notations and statements

Let us set Σ = [0, 1] and Σ_{ε} := {(x_{1}, x_{2}) ∈ IR^{2} : x_{1} ∈ Σ, |x_{2}| ≤ ε}, which will be viewed as the reference configuration of a two dimensional elastic body. For every φ ∈ L^{2}(Σ_{ε}; IR^{2}) we set

φ̃(x_{1}) = \frac{1}{2ε} ∫_{−ε}^{ε} φ(x_{1}, x_{2})dx_{2}

and for every φ ∈ C^{2}(Σ_{ε}; IR^{2}) we denote by C the right Cauchy-Green strain tensor:

C = ^{T} DφDφ

and by E the Green-St.Venant strain tensor:

E = \frac{1}{2}(C − I).

We denote by e_{i}, i = 1, 2 the canonical basis in IR^{2} and we define:

A_{ε}(Σ_{ε}) := {φ ∈ C^{2}(Σ_{ε}; IR^{2}) : E(φ)(e_{2}) = 0, det Dφ > 0, φ(0, x_{2}) = (0, x_{2}) }.

For every ε > 0 and φ ∈ C^{2}(Σ_{ε}; IR^{2}) we set:

F_{ε}(φ) = \{ \begin{matrix}∫_{Σ_{ε}} W (E(φ))dx & if φ ∈ A_{ε}(Σ_{ε}) \\ +∞ & otherwise, \end{matrix}

where the deformation energy density is supposed of the form:

W (E) = μ|E|^{2} + \frac{λ}{2}(tr E)^{2};  (2.1)

λ, μ ∈ L^{∞}(Σ, [0, +∞)), μ(x_{1}) ≥ μ̄ > 0 a.e. on [0, 1] are called the Lamè moduli of the material; we set also Λ(x_{1}) = (μ + \frac{λ}{2})(x_{1}). Let f ∈ C^{0}(Σ_{1}; IR^{2}) and g ∈ C^{0}(Σ_{1}; IR^{2}) be given functions, corresponding to the applied load system; we set from now on f_{0} = f_{|Σ} and g_{0} = g_{|Σ}.

For every ε > 0 we define the functionals G_{ε} : L^{2}(Σ_{ε}; IR^{2}) → IR ∪ {+∞}:

G_{ε}(φ) = \frac{1}{ε^{3}} [F_{ε}(φ) − ε^{2}∫_{Σ_{ε}}⟨f, φ⟩dx − ε^{3}∫_{Σ_{ε}∩\{x_{2}=ε\}}⟨g, φ⟩dx_{1}] .

The presence of the rescaling factor \frac{1}{ε^{3}} is essential in proving some compactness properties on the admissible deformations, as we shall see in proposition 3.2. Since F_{ε} is not polyconvex neither quasiconvex (we refer to [9] and [11] for more details) we cannot expect that G_{ε} has an absolute minimum since it is not l.s.c. in the weak topology of H^{1}(Σ_{ε}, IR^{2}). A sequence φ^{ε} ∈ L^{2}(Σ_{ε}; IR^{2}) will be called a minimizing sequence for G_{ε} if lim_{ε→0} {G_{ε}φ^{ε} − inf G_{ε}} = 0; our goal is to study the asymptotic behaviour of such sequences and to our aim we are led to introduce the set of limiting admissible deformations

A(Σ) := \{v ∈ H^{2}(Σ; IR^{2}) : |v̇(x_{1})|^{2} = 1 for every x_{1} ∈ [0, 1], v(0) = \underline{0}, v̇(0) = \underline{e}_{1}\}.

For every v ∈ L^{2}(Σ; IR^{2}) let us define:

F_{0}(v) = \{ \begin{matrix}\frac{2}{3}∫_{Σ} Λ(x_{1}) \left|{ \atop v̈}\right|^{2} dx_{1} & if v ∈ A(Σ) \\ +∞ & otherwise, \end{matrix}

and also the functional G_{0} : L^{2}(Σ; IR^{2}) → IR ∪ {+∞}:

G_{0}(v) = F_{0}(v) − 2 ∫_{Σ} ⟨f_{0}, v⟩dx_{1} − ∫_{Σ} ⟨g_{0}, v⟩dx_{1},

which will be the limit functional. Indeed the main result of this section is the following:

Theorem 2.1. Let (φ^{ε})_{ε>0} ⊂ L^{2}(Σ_{ε}; IR^{2}) be a minimizing sequence for G_{ε}. Then the sequence (φ̃^{ε})_{ε>0} is compact in L^{2}(Σ; IR^{2}) and, if v_{0} is one of its limit points, then v_{0} ∈ A(Σ), φ̃^{ε} → v_{0} in H^{1}(Σ; IR^{2}) and

G_{0}(v_{0}) = min {G_{0}(v) : v ∈ A(Σ)} = lim_{ε→0} G_{ε}(φ^{ε}).

3. Some preliminary lemmas

We shall prove in the following lemma that the deformations φ that belong to the set A_{ε}(Σ_{ε}) admit a particular representation.

Lemma 3.1. If φ ∈ A_{ε}(Σ_{ε}), then there exist v ∈ C^{2}(Σ; IR^{2}), w ∈ C^{1}(Σ; IR^{2}) such that:

φ(x_{1}, x_{2}) = v(x_{1}) + x_{2}w(x_{1}), |v̇(x_{1})| ≠ 0 for every x_{1} ∈ [0, 1],

w(x_{1}) = \frac{1}{|v̇(x_{1})|}(−v̇_{2}(x_{1}), v̇_{1}(x_{1})) for every x_{1} ∈ [0, 1], v(0) = \underline{0}, w(0) = \underline{e}_{2}.

Moreover, if v ∈ C^{2}(Σ; IR^{2})∩A(Σ), the deformation field φ defined by φ(x_{1}, x_{2}) = v(x_{1})+ x_{2}w(x_{1}), where w = (−v̇_{2}, v̇_{1}), belongs to A_{ε}(Σ_{ε}).

Proof. Let φ be in C^{2}(Σ_{ε}; IR^{2}). The condition (^{T} DφDφ − I) · e_{2} = 0 yields:

⟨φ_{x_{1}}, φ_{x_{2}}⟩ = 0  (3.1)
|φ_{x_{2}}|^{2} = 1  (3.2)

Deriving (3.1) with respect to x_{2} and (3.2) with respect to x_{1}, we obtain: ⟨φ_{x_{1}x_{2}}, φ_{x_{2}}⟩ + ⟨φ_{x_{1}}, φ_{x_{2}x_{2}}⟩ = 0 and ⟨φ_{x_{2}x_{1}}, φ_{x_{2}}⟩ = 0. It results therefore ⟨φ_{x_{1}}, φ_{x_{2}x_{2}}⟩ = 0 and also ⟨φ_{x_{2}}, φ_{x_{2}x_{2}}⟩ = 0, because |φ_{x_{2}}|^{2} = 1. Having in mind the condition ⟨φ_{x_{1}}, φ_{x_{2}}⟩ = 0, we conclude that φ_{x_{2}x_{2}} = 0, that is:

φ(x_{1}, x_{2}) = v(x_{1}) + x_{2}w(x_{1}),

for some v, w : Σ → IR^{2}. We remark that v ∈ C^{2}(Σ; IR^{2}), because φ ∈ C^{2}(Σ_{ε}; IR^{2}) and that from (3.1) and (3.2) it follows:

|w(x_{1})|^{2} = 1 for every x_{1} ∈ [0, 1]  (3.3)

⟨v̇(x_{1}), w(x_{1})⟩ = 0 for every x_{1} ∈ [0, 1]  (3.4)

We define now: w^{⊥} = (w_{2}, −w_{1}). The constraint det Dφ > 0, that is: ⟨v̇ + x_{2}ẇ, w^{⊥}⟩ > 0, implies in particular that ⟨v̇, w^{⊥}⟩ > 0 and |v̇(x_{1})| ≠ 0 for every x_{1} ∈ [0, 1]. We can therefore set by (3.3) and (3.4):

w(x_{1}) = \frac{(−v̇_{2}(x_{1}), v̇_{1}(x_{1}))}{|v̇(x_{1})|}

for every x_{1} ∈ [0, 1].

In the following for every φ which belongs to A_{ε}(Σ_{ε}) we shall write:

φ(x_{1}, x_{2}) = v(x_{1}) + x_{2}w(x_{1}),

with v, w such as in lemma 3.1 and we observe also that if φ ∈ A_{ε}(Σ_{ε}) we have φ̃ = v.

In the following proposition some compactness properties on the sequences (v^{ε}) and (w^{ε}) are proven, which shall permit us to demonstrate the main theorem and also provide a “a posteriori” justification of the choice of the means φ̃^{ε}.

Proposition 3.2. Assume that (φ^{ε})_{ε>0} ⊂ A_{ε}(Σ_{ε}) and F_{ε}(φ^{ε}) ≤ Cε^{3} for some C > 0. Then:

i) (v^{ε})_{ε>0} is compact in H^{1}(Σ; IR^{2}) and (w^{ε})_{ε>0} is relatively weakly compact in H^{1}(Σ; IR^{2}); ii) if v^{ε} → v in L^{2}(Σ, IR^{2}), then v ∈ A(Σ) and w^{ε} ⇀ (−v̇_{2}, v̇_{1}) in H^{1}(Σ; IR^{2}); iii) the sequence (⟨v̇^{ε}, ẇ^{ε}⟩)_{ε>0} is bounded in L^{2}(Σ; IR^{2}) and, if v^{ε} → v in L^{2}(Σ, IR^{2}), then

⟨v̇^{ε}, ẇ^{ε}⟩ ⇀ ⟨v̇, ẇ⟩ in L^{2}(Σ).

Proof. First of all, we observe that if φ ∈ A_{ε}(Σ_{ε}), then:

(tr E(φ))^{2} = |E(φ)|^{2} .

We get therefore:

F_{ε}(φ^{ε}) = ∫_{Σ_{ε}} [μ |E(φ^{ε})|^{2} + \frac{λ}{2}(tr E(φ^{ε}))^{2}]dx

= ∫_{Σ_{ε}} Λ(x_{1}) |E(φ^{ε})|^{2}dx

= ∫_{Σ_{ε}} \frac{1}{4}Λ(x_{1}) |^{T} Dφ^{ε}Dφ^{ε} − I|^{2}dx

= \frac{1}{4} ∫_{Σ_{ε}} Λ(x_{1}) |U^{ε} + I|^{2} · |U^{ε} − I|^{2}dx,

(3.5)

where U^{ε} = \sqrt{^{T} Dφ^{ε}Dφ^{ε}} is defined as the unique symmetric and positive definite matrix such that (U^{ε})^{2} =^{T} Dφ^{ε}Dφ^{ε}. By the polar factorization of an invertible matrix, for every ε > 0 we may find a positive rotation R^{ε} such that Dφ^{ε} = R^{ε}U^{ε}, from which it follows:

F_{ε}(φ^{ε}) = \frac{1}{4} ∫_{Σ_{ε}} Λ(x_{1})|^{T} R_{ε}Dφ^{ε} − I|^{2}|^{T} R_{ε}Dφ^{ε} + I|^{2}dx.

It is easy to prove that there exists C > 0 such that |^{T} R_{ε}Dφ^{ε} + I|^{2} ≥ C for every ε > 0, therefore we get:

\begin{aligned}F_{ε}(φ^{ε}) & ≥ C · \frac{1}{4} ∫_{Σ_{ε}} Λ(x_{1}) |^{T} R_{ε}Dφ^{ε} − I|^{2}dx_{1}dx_{2} \\ & = C · \frac{1}{4} ∫_{Σ_{ε}} Λ(x_{1}) |Dφ^{ε} − R^{ε}|^{2}dx_{1}dx_{2}. \end{aligned}

By applying the Cayley-Hamilton theorem to the matrix U^{ε} = \sqrt{^{T} Dφ^{ε}Dφ^{ε}} we can determine the matrix R^{ε}. We obtain:

R^{ε} = ( \begin{matrix}\frac{(φ_{1}^{ε})_{x_{1}}}{|φ_{x_{1}}^{ε}|} & w_{1}^{ε} \\ \frac{(φ_{2}^{ε})_{x_{1}}}{|φ_{x_{1}}^{ε}|} & w_{2}^{ε} \end{matrix} )

226 V.Casarino, D. Percivale / A variational model for non linear elastic plates

and by the uniqueness of the rotation R^{ε} we conclude that:

R^{ε} = ( \begin{matrix}w_{2}^{ε} & w_{1}^{ε} \\ −w_{1}^{ε} & w_{2}^{ε} \end{matrix} ) .

We get now:

Cε^{3} ≥ F_{ε}(φ^{ε})

≥ \frac{c}{4} ∫_{Σ_{ε}} Λ(x_{1})( |v̇_{1}^{ε} + x_{2}ẇ_{1}^{ε} − w_{2}^{ε}|^{2} + |v̇_{2}^{ε} + x_{2}ẇ_{2}^{ε} + w_{1}^{ε}|^{2} )dx

≥ \frac{c}{4}μ̄ ∫_{Σ_{ε}} (|v̇_{1}^{ε} − w_{2}^{ε}|^{2} + x_{2}^{2} |ẇ_{1}^{ε}|^{2} + |v̇_{2}^{ε} + w_{1}^{ε}|^{2} + x_{2}^{2} |ẇ_{2}^{ε}|^{2} )dx.  (3.6)

By integrating with respect to x_{2}, we obtain : ∫_{Σ} |ẇ^{ε}|^{2} dx_{1} ≤ C, for some C > 0; since w^{ε}(0) = \underline{e}_{2} for every ε > 0, the sequence (w^{ε})_{ε>0} is relatively weakly compact in H^{1}(Σ; IR^{2}) and then there exists a subsequence, which we always denote with w^{ε}, such that w^{ε} ⇀ w = (w_{1}, w_{2}) in H^{1}(Σ; IR^{2}). By using inequality (3.6) we get v̇_{1}^{ε} → w_{2} and v̇_{2}^{ε} → −w_{1} in L^{2}(Σ; IR^{2}), therefore (v^{ε}) is relatively compact in the strong topology of H^{1}(Σ; IR^{2}) and if v^{ε} → v in L^{2}(Σ, IR^{2}) we have v̇_{1} = w_{2} and v̇_{2} = −w_{1}, then v̈ =: (v̈_{1}, v̈_{2}) = (ẇ_{2}, −ẇ_{1}) ∈ L^{2}(Σ, IR^{2}) and v ∈ H^{2}(Σ; IR^{2}). We observe now that |v̇(x_{1})|^{2} = 1 for every x_{1} ∈ [0, 1], since w^{ε} converges, up to a subsequence, to (−v̇_{2}, v̇_{1}) uniformly on Σ and |w^{ε}(x_{1})|^{2} = 1 for every x_{1} ∈ [0, 1]. To prove iii) we observe that:

F_{ε}(φ^{ε}) = ∫_{Σ_{ε}} \frac{1}{4}Λ(x_{1}) |^{T} Dφ^{ε}Dφ^{ε} − I|^{2}dx

= \frac{1}{4} ∫_{Σ_{ε}} Λ(x_{1})( |v̇^{ε} + x_{2}ẇ^{ε}|^{2} − 1)^{2}dx

= \frac{1}{4} ∫_{Σ_{ε}} Λ(x_{1})( |v̇^{ε}|^{2} − 1 + 2x_{2}⟨v̇^{ε}, ẇ^{ε}⟩ + x_{2}^{2} |ẇ^{ε}|^{2} )^{2}dx

= \frac{1}{4} ∫_{Σ_{ε}} Λ(x_{1})[( |v̇^{ε}|^{2} − 1 + x_{2}^{2} |ẇ^{ε}|^{2})^{2} + 4x_{2}^{2} |⟨v̇^{ε}, ẇ^{ε}⟩|^{2} ]dx

≥ ∫_{Σ_{ε}} Λ(x_{1}) · x_{2}^{2}|⟨v̇^{ε}, ẇ^{ε}⟩|^{2}dx

≥ μ̄ ∫_{Σ_{ε}} x_{2}^{2}|⟨v̇^{ε}, ẇ^{ε}⟩|^{2}dx.

By the hypotheses F_{ε}(φ^{ε}) ≤ Cε^{3} we obtain now:

\frac{2}{3}μ̄ ∫_{Σ} |⟨v̇^{ε}, ẇ^{ε}⟩|^{2}dx_{1} ≤ C.  (3.7)

Recalling that v̇^{ε} → v̇ in L^{2}(Σ; IR^{2}) and ẇ^{ε} ⇀ ẇ in L^{2}(Σ; IR^{2}), it is easy to verify that ⟨v̇^{ε}, ẇ^{ε}⟩ ⇀ ⟨v̇, ẇ⟩ in the sense of distributions and therefore by using (3.7) we get iii).

4. Proof of the main result

Proposition 4.1. For every v ∈ L^{2}(Σ; IR^{2}), for every sequence (φ^{ε})_{ε>0} such that φ̃^{ε} = v^{ε} → v in L^{2}(Σ; IR^{2}) we have:

lim inf_{ε→0} G_{ε}(φ^{ε}) ≥ G_{0}(v).  (4.1)

Proof. We may suppose G_{ε}(φ^{ε}) ≤ C, since otherwise (4.1) is obvious. Then we have φ^{ε} ∈ A_{ε}(Σ_{ε}) and:

F_{ε}(φ^{ε}) ≤ Cε^{3} + ε^{2} ∫_{Σ_{ε}} ⟨f, φ^{ε}⟩dx + ε^{3} ∫_{Σ_{ε}∩{x_{2}=ε}} ⟨g, φ^{ε}⟩dx_{1}

≤ C(ε^{3} + ε^{2} ∫_{Σ_{ε}} |φ^{ε}|^{2} dx + ε^{2} ∫_{Σ_{ε}} |f|^{2} dx
+ ε^{3}∫_{Σ_{ε}∩{x_{2}=ε}} |φ^{ε}(x_{1}, ε)|^{2} + ε^{3} ∫_{Σ_{ε}∩{x_{2}=ε}} |g|^{2})

≤ C(ε^{3} + ε^{2} ∫_{Σ_{ε}} |v^{ε} + x_{2}w^{ε}|^{2}dx + ε^{3} ∫_{Σ} |v^{ε} + εw^{ε}|^{2} dx_{1})
≤ C(ε^{3} + ε^{3} ∫_{Σ} |v^{ε}|^{2}dx_{1}dx_{2} + ε^{5} ∫_{Σ} |w^{ε}|^{2}dx_{1}dx_{2})
≤ Cε^{3},

because v^{ε} → v in L^{2}(Σ; IR^{2}), |w^{ε}|^{2} = 1 on Σ and f and g are continuous; hence by proposition 3.2 it follows that v ∈ A(Σ). As in proposition 3.2 we get now:

\frac{1}{ε^{3}} F_{ε}(φ^{ε}) = \frac{1}{ε^{3}} ∫_{Σ_{ε}} \frac{1}{4}(μ + \frac{λ}{2})(x_{1}) |^{T} Dφ^{ε}Dφ^{ε} − I|^{2}dx

= \frac{1}{ε^{3}} ∫_{Σ_{ε}} Λ(x_{1}) · x_{2}^{2} |⟨v̇^{ε}, ẇ^{ε}⟩|^{2}dx

= \frac{2}{3} ∫_{Σ} Λ(x_{1}) |⟨v̇^{ε}, ẇ^{ε}⟩|^{2} dx_{1}

≥ \frac{2}{3}∫_{Σ} Λ(x_{1})[ |⟨v̇, ẇ⟩|^{2} + ⟨v̇, ẇ⟩ · (⟨v̇^{ε}, ẇ^{ε}⟩ − ⟨v̇, ẇ⟩)]dx_{1}.

Recalling that (⟨v̇^{ε}, ẇ^{ε}⟩)_{ε>0} converges weakly in L^{2}(Σ; IR^{2}) to ⟨v̇, ẇ⟩, we get finally:

lim inf_{ε→0} \frac{1}{ε^{3}} F_{ε}(φ^{ε}) ≥ \frac{2}{3} ∫_{Σ} Λ(x_{1}) |⟨v̇, ẇ⟩|^{2} dx_{1}

= \frac{2}{3} ∫_{Σ} Λ(x_{1}) \left|{ \atop v̈}\right|^{2} dx_{1},

where the last equality follows from the fact that ⟨ẇ, v̈⟩ = 0 by proposition 3.2 and ⟨v̇, v̈⟩ = 0 because \left|{ \atop v̇}\right|^{2} = 1 on Σ.

We can now conclude:

lim inf_{ε→0} G_{ε}(φ^{ε}) = lim inf_{ε→0} \frac{1}{ε^{3}} [F_{ε}(φ^{ε}) − ε^{2} ∫_{Σ_{ε}} ⟨f, v^{ε} + x_{3}w^{ε}⟩dx
− ε^{3} ∫_{Σ_{ε}∩{x_{2}=ε}} ⟨g, v^{ε} + εw^{ε}⟩dx_{1}]

≥ \frac{2}{3} ∫_{Σ} Λ(x_{1}) \left|{ \atop v̈}\right|^{2} dx_{1} − 2 ∫_{Σ} ⟨f_{0}, v⟩dx_{1} − ∫_{Σ} ⟨g_{0}, v⟩dx_{1}.

We are going now to prove that the limit functional G_{0} is as good as possible, with respect to the convergence of the means introduced above.

Proposition 4.2. For every v ∈ L^{2}(Σ; IR^{2}) there exists a sequence (φ^{ε})_{ε>0} ⊂ A_{ε}(Σ_{ε}) such that

v^{ε} = φ̃^{ε} → v in L^{2}(Σ; IR^{2}) and lim_{ε→0} G_{ε}(φ^{ε}) = G_{0}(v).

Proof. For every v ∈ L^{2}(Σ; IR^{2}) we set:

G^{+}(v) = inf{lim sup_{ε→0} G_{ε}(φ^{ε}) : φ^{ε} ∈ L^{2}(Σ_{ε}, IR^{2}), v^{ε} = φ̃^{ε} → v in L^{2}(Σ; IR^{2})}.

It can be proven that the infimum is actually a minimum by the growth condition on the functional, therefore by proposition 2.1 we need only prove:

G^{+}(v) ≤ G_{0}(v)  (4.2)

for every v ∈ L^{2}(Σ; IR^{2}). We observe first that inequality 4.2 is obvious when v doesn’t belong to A(Σ) since for such a v we have G_{0}(v) = +∞. When v ∈ C^{2} ∩ A(Σ) we define for every ε > 0

φ^{ε}(x_{1}, x_{2}) = v(x_{1}) + x_{2}w(x_{1}),

where w = (−v̇_{2}, v̇_{1}). We get now:

F_{ε}(φ^{ε}) = ∫_{Σ_{ε}} \frac{1}{4}Λ(x_{1}) |^{T} Dφ^{ε}Dφ^{ε} − I|^{2}dx

= \frac{1}{4} ∫_{Σ_{ε}} Λ(x_{1})( |v̇ + x_{2}ẇ|^{2} − 1)^{2}dx

= \frac{1}{4} ∫_{Σ_{ε}} Λ(x_{1})(2x_{2}⟨v̇, ẇ⟩ + x_{2}^{2} \left|{ \atop ẇ}\right|^{2} )^{2}dx

= \frac{1}{4} ∫_{Σ} Λ(x_{1})[\frac{8}{3}ε^{3} |⟨v̇, ẇ⟩|^{2} + \frac{2}{5}ε^{5}|ẇ|^{4}]dx_{1},

and finally, since v ∈ C^{2}(Σ; IR^{2}):

lim sup_{ε→0} \frac{1}{ε^{3}} F_{ε}(φ^{ε}) = \frac{2}{3} ∫_{Σ} Λ(x_{1}) \left|{ \atop v̈}\right|^{2}dx_{1},

which yields G^{+}(v) ≤ G_{0}(v) for every v ∈ C^{2}(Σ) and then recalling that G^{+} and G_{0} are weakly l.s.c. in L^{2}(Σ, IR^{2}) and that C^{2} is dense in L^{2}(Σ, IR^{2}) inequality 4.2 is proven.

Proof of Theorem 2.1 If m^{ε} = inf{G_{ε}(φ) : φ ∈ A_{ε}(Σ_{ε})} and u^{∗} denotes the identity function in IR^{2} we get:

m^{ε} ≤ G_{ε}(u^{∗}) = −\frac{1}{ε} ∫_{Σ_{ε}} ⟨f, u^{∗}⟩dx − ∫_{Σ_{ε}∩\{x_{2}=ε\}} ⟨g, u^{∗}⟩dx_{1} ≤ C(f, g)

and therefore:

\begin{aligned}F_{ε}(φ^{ε}) & ≤ ε^{3}m^{ε} + ε^{2} ∫_{Σ_{ε}} ⟨f, φ^{ε}⟩dx + ε^{3} ∫_{Σ_{ε}∩\{x_{2}=ε\}} ⟨g, φ^{ε}⟩dx_{1} + ε^{4} \\ & ≤ C̃(f, g)ε^{3}. \end{aligned}

The same argument described in proposition 3.2 shows that the sequence (φ̃^{ε}) is compact in H^{1}(Σ; IR^{2}) and the limit point v_{0} belongs to H^{2}(Σ; IR^{2}). We have therefore only to prove:

G_{0}(v) ≥ G_{0}(v_{0}) for every v ∈ A(Σ).  (4.3)

By prop.4.2 there exists a sequence (ψ_{ε})_{ε>0} such that ψ_{ε} ∈ A_{ε}(Σ_{ε}), ψ̃_{ε} → v in H^{1}(Σ; IR^{2}) and

lim_{ε→0} G_{ε}(ψ_{ε}) = G_{0}(v)  (4.4).

Let us now observe that G_{ε}(ψ_{ε}) ≥ G_{ε}(φ^{ε}) − ε, from the hypotheses on (φ^{ε}); hence by prop.4.1 we have:

lim inf_{ε→0} G_{ε}(ψ_{ε}) ≥ lim inf_{ε→0} G_{ε}(φ^{ε}) ≥ G_{0}(v_{0}).

Finally by (4.4) we get (4.3).

5. Comparison with the linear approach

We approach in this section the same constrained problem by using the linear strain measure instead of the non linear one (for the unconstrained case we refer to [3]). We recall that, if u denotes the displacement, that is φ(x) = x+u(x), the linearized strain tensor is defined as:

e(u) = \frac{1}{2}(^{T} Du + Du),  (5.1)

and it is easy to verify that E(u) = e(u) + \frac{1}{2}^{T} DuDu.

Neglecting the second order term we shall determine a limit functional F_{0}^{lin}, defined on a suitable class of functions X^{lin}(Σ), which is different from F_{0}. To deal with this case, we introduce the set of admissible displacements:

A^{lin}(Σ_{ε}) := {u ∈ C^{2}(Σ_{ε}; IR^{2}) : e(u) · (e_{2}) = 0, u(0, x_{2}) = (0, 0)},

and we define then W, F_{ε} and G_{ε} as in section 2, with e = e(u) at the place of E = E(φ). Finally we introduce the limit functionals. We set

A^{lin}(Σ) := {z ∈ H^{2}(Σ; IR^{2}) : z_{1}(x_{1}) ≡ 0, z_{2}(0) = 0, ż_{2}(0) = 0 }.

For every z ∈ H^{2}(Σ; IR) we define:

F_{0}^{lin}(z) = \{ \begin{matrix}\frac{2}{3} ∫_{Σ} Λ(x_{1}) \left|{ \atop z̈_{2}}\right|^{2} dx_{1} & if z ∈ A^{lin}(Σ) \\ +∞ & otherwise, \end{matrix}

and

G_{0}^{lin}(z) = \{ \begin{matrix}F_{0}^{lin}(z_{2}) − 2 ∫_{Σ} f_{2}z_{2}dx_{1} − ∫_{Σ} g_{2}z_{2}dx_{1} & if z ∈ A^{lin}(Σ) \\ +∞ & otherwise. \end{matrix}

We observe that the condition e(u)(e_{2}) = 0 is equivalent to:

\{ \begin{matrix}\frac{∂u_{2}}{∂x_{1}} + \frac{∂u_{1}}{∂x_{2}} = 0 \\ \frac{∂u_{2}}{∂x_{2}} = 0, \end{matrix}

thus obtaining the following representation for the displacements u in A^{lin}(Σ_{ε}):

Lemma 5.1. If u ∈ A^{lin}(Σ_{ε}), then there exist h, u_{2} ∈ C^{2}(Σ; IR^{2}), such that:

u(x_{1}, x_{2}) = (h(x_{1}) − x_{2}u̇_{2}(x_{1}), u_{2}(x_{1})),

h(0) = 0, u_{2}(0) = 0, u̇_{2}(0) = 0.

Let us now define the means as in (2.2); if u ∈ A^{lin}(Σ_{ε}) we get : ũ = (h, u_{2}). We give now a compactness result analogous to proposition 3.2:

Proposition 5.2. Assume that (u^{ε})_{ε>0} ⊂ A^{lin}(Σ_{ε}) and F_{ε}(u^{ε}) ≤ Cε^{3} for some C > 0. Then, up to a subsequence, ũ^{ε} → z in L^{2}(Σ, IR^{2}) and z ∈ A^{lin}(Σ). Moreover h^{ε} → 0 in H^{1}(Σ, IR^{2}) and u_{2}^{ε} ⇀ z_{2} in H^{2}(Σ, IR^{2}).

Proof. If u^{ε} ∈ A^{lin}(Σ_{ε}), it results:

e(u^{ε}) = ( \begin{matrix}ḣ^{ε} − x_{2}ü_{2}^{ε} & 0 \\ 0 & 0 \end{matrix} ) ,

hence (tr e(u^{ε}))^{2} = |e(u^{ε})|^{2} and

Cε^{3} ≥ F_{ε}(φ^{ε}) = ∫_{Σ_{ε}} [μ |e(u^{ε})|^{2} + \frac{λ}{2}(tr e(u^{ε}))^{2}]dx

= ∫_{Σ_{ε}} Λ(x_{1}) |ḣ^{ε} − x_{2}ü_{2}^{ε}|^{2}dx

= ∫_{Σ_{ε}} Λ(x_{1})( |ḣ^{ε}|^{2} + x_{2}^{2} |ü_{2}^{ε}|^{2}dx)

≥ μ̄ ∫_{Σ} (2ε |ḣ^{ε}|^{2} + \frac{2}{3} |ü_{2}^{ε}|^{2} )dx_{1}.

In particular ḣ^{ε} converges to 0 in L^{2}(Σ; IR^{2}). Since h^{ε}(0) = 0 we deduce that h^{ε} → 0 in H^{1}(Σ; IR^{2}) and finally, recalling that u_{2}^{ε}(0) = u̇_{2}^{ε}(0) = 0 we get that (u_{2}^{ε}) is weakly compact in H^{2}(Σ, IR^{2}).

Proposition 5.3. For every z ∈ L^{2}(Σ; IR^{2}), for every sequence (u^{ε})_{ε>0} such that ũ_{2}^{ε} → z in L^{2}(Σ, IR^{2}) we have:

lim inf_{ε→0} G_{ε}(u^{ε}) ≥ G_{0}^{lin}(z).

Proof. We may suppose that G_{ε}(u^{ε}) ≤ C for every ε > 0, otherwise the thesis is obvious and as in proposition 4.1 we have: F_{ε}(u^{ε}) ≤ C̃ε^{3} for every ε > 0 and for a suitable C̃ > 0, hence, by proposition 5.2, h^{ε} → 0 in H^{1}(Σ; IR^{2}) and u_{2}^{ε} ⇀ z_{2}. We have now:

lim inf_{ε→0} \frac{1}{ε^{3}} F_{ε}(u^{ε}) ≥ lim inf_{ε→0} \frac{1}{ε^{3}} ∫_{Σ_{ε}} Λ(x_{1})x_{2}^{2} |ü_{2}^{ε}|^{2}dx

≥ \frac{2}{3} ∫_{Σ} Λ(x_{1}) \left|{ \atop z̈_{2}}\right|^{2} dx_{1},

and finally:

lim inf_{ε→0} G_{ε}(u^{ε}) ≥ F_{0}^{lin}(z_{2}) − 2 ∫_{Σ} f_{2}z_{2}dx_{1} − ∫_{Σ} g_{2}z_{2}dx_{1}.

Proposition 5.4. For every z ∈ L^{2}(Σ; IR^{2}) there exists a sequence

(u^{ε})_{ε>0} ⊂ A^{lin}(Σ_{ε})

such that

ũ_{2}^{ε} = u_{2}^{ε} → z in L^{2}(Σ; IR^{2}) and lim_{ε→0} G_{ε}(u^{ε}) = G_{0}^{lin}(z).

Proof. If z ∉ A^{lin}(Σ) it is easy to find a sequence u^{ε} ∈ L^{2}(Σ_{ε}, IR^{2}) such that ũ^{ε} → z in L^{2}(Σ; IR^{2}) and G_{ε}(u^{ε}) = +∞, thus proving the thesis. When z ∈ A^{lin}(Σ) the proof is analogous to that of proposition 4.2, choosing:

u^{ε}(x_{1}, x_{2}) = (−x_{2}ż(x_{1}), z(x_{1}))

for every (x_{1}, x_{2}) ∈ Σ_{ε} and for every ε > 0.

Remark 5.5. By arguing as in the previous section it is now easy to prove that minimizing sequences of the functionals G_{ε} are compact in the sense of theorem 2.1, that every limit point is a minimizer of the functional G_{0}^{lin}, while a direct computation shows that G_{0}^{lin} < G_{0}. We observe that in [1] it has been proven that the “limit” of the sequence of functionals

G_{ε}^{#}(u) = \{ \begin{matrix}G_{ε}^{lin}(u) & if u ∈ H^{1}(Σ_{ε}, IR^{2}) \\ +∞ & elsewhere in L^{2}(Σ_{ε}, IR^{2}) \end{matrix}

is G_{0}^{lin} that is, roughly speaking, the constraints don’t modify the limit.

6. The elastic plate

In this section Σ will be an open bounded subset of IR^{2} having Lipschitz boundary, Γ a subset of ∂Σ having non zero one-dimensional Hausdorff measure; we set for every ε > 0

Σ_{ε} := {x = (x_{1}, x_{2}, x_{3}) ∈ IR^{3} : (x_{1}, x_{2}) ∈ Σ, |x_{3}| < ε},

which will be considered as the reference configuration of a three-dimensional elastic body. As in section 2 if φ ∈ L^{2}(Σ_{ε}; IR^{3}) we set:

φ̃(x_{1}, x_{2}) = \frac{1}{2ε} ∫_{−ε}^{ε} φ(x_{1}, x_{2}, x_{3})dx_{3}

and if φ ∈ C^{2}(\overline{Σ_{ε}}; IR^{3}) is a deformation we may define the right Cauchy-Green strain tensor C = ^{T} DφDφ and the Green-St.Venant strain tensor: E = \frac{1}{2}(C − I). For every a, b, c ∈ IR^{3} we will denote by (a|b|c) the matrix whose columns are a, b, c; by 𝕊_{+}^{3} we mean the set of all symmetric, positive definite 3 × 3 matrices. We denote by e_{i}, i = 1, 2, 3 the canonical basis in IR^{3} and we define the class of admissible deformations of \overline{Σ_{ε}}:

A_{ε}(Σ_{ε}) := \{φ ∈ C^{2}(\overline{Σ_{ε}}, IR^{3}) : E(φ)(\underline{e}_{3}) = 0, det Dφ > 0,
φ_{x_{1}} ∧ φ_{x_{1}x_{3}} = φ_{x_{2}x_{3}} ∧ φ_{x_{2}} in \overline{Σ_{ε}}, φ(x) = x on Γ × [−ε, ε]\}.

It is worth noticing that this class contains an additional constraint with respect to the two dimensional case. As we shall prove in proposition 7.2, however, this additional constraint is automatically satisfied by the admissible limit deformations if the metric associated to the coordinate system is euclidean. Moreover, in remark 7.3 we exhibit a class of deformations that satisfy this constraint and also allow us to consider the bidimensional case just studied as a particular case of this, more general, problem. For every ε > 0 and φ ∈ L^{2}(Σ; IR^{3}) we define:

F_{ε}(φ) = \{ \begin{matrix}∫_{Σ_{ε}} W (E(φ))dx & if φ ∈ A_{ε}(Σ_{ε}) \\ +∞ & otherwise, \end{matrix}

where W (E) = μ|E|^{2} + \frac{λ}{2}(tr E)^{2}, λ, μ ∈ L^{∞}(Σ, [0, +∞)) being the Lamè moduli of the material and μ = μ(x_{1}, x_{2}) ≥ μ̄ > 0 a.e. on Σ.

As in the previous section let f, g ∈ C^{0}(\overline{Σ}_{1}; IR^{3}) be given functions, corresponding to the applied loads and let f_{0}, g_{0} be their restrictions to the “middle plane” Σ. For every ε > 0 and for every φ ∈ L^{2}(Σ_{ε}, IR^{3}) we set now

G_{ε}(φ) = \frac{1}{ε^{3}} [F_{ε}(φ) − ε^{2}∫_{Σ_{ε}}⟨f, φ⟩dx − ε^{3}∫_{Σ_{ε}∩\{x_{3}=ε\}}⟨g, φ⟩dx_{1}dx_{2}]

and as in section 2 a sequence φ^{ε} ∈ L^{2}(Σ_{ε}, IR^{3}) will be called a minimizing sequence for G_{ε} if

lim_{ε→0}[G_{ε}(φ^{ε}) − inf G_{ε}(φ)] = 0.

In order to study the asymptotic behaviour of such a sequence we introduce now the set of limiting admissible deformations:

A(Σ) := {v ∈ H^{1}(Σ; IR^{3}) : |v_{x_{1}}|^{2} = 1, |v_{x_{2}}|^{2} = 1, ⟨v_{x_{1}}, v_{x_{2}}⟩ = 0 on Σ,
w = v_{x_{1}} ∧ v_{x_{2}} ∈ H^{1}(Σ; IR^{3}), v(x_{1}, x_{2}) = (x_{1}, x_{2}, 0), w(x_{1}, x_{2}) = \underline{e}_{3} on Γ\}.

For every v ∈ L^{2}(Σ; IR^{3}) let us define:

F_{0}(v) = \{ \begin{matrix}\frac{2}{3}∫_{Σ} [(μ + \frac{λ}{2})H^{2} − 2μK]dx_{1}dx_{2} & if v ∈ A(Σ) \\ +∞ & otherwise, \end{matrix}

where K = ⟨v_{x_{1}}, w_{x_{1}}⟩⟨v_{x_{2}}, w_{x_{2}}⟩−|⟨v_{x_{2}}, w_{x_{1}}⟩|^{2} and H = −⟨Dv, Dw⟩ represent respectively the gaussian and the mean curvature. We define also the functional:

G_{0}(v) = F_{0}(v) − 2 ∫_{Σ} ⟨f_{0}, v⟩dx_{1} − ∫_{Σ} ⟨g_{0}, v⟩dx_{1}.

We shall prove the following:

Theorem 6.1. Let (φ^{ε})_{ε>0} ⊂ L^{2}(Σ_{ε}, IR^{3}) be a minimizing sequence for G_{ε}. Then the sequence (φ̃^{ε})_{ε>0} is compact in L^{2}(Σ; IR^{3}) and, if v_{0} is one of its limit points, then v_{0} ∈ A(Σ) (in particular v_{0}_{x_{1}} ∧ v_{0}_{x_{2}} ∈ H^{1}(Σ; IR^{3})), φ̃^{ε} → v_{0} in H^{1}(Σ; IR^{3}) and

G_{0}(v_{0}) = min {G_{0}(v) : v ∈ A(Σ)} = lim_{ε→0} G_{ε}(φ^{ε}).

7. Some preliminary lemmas

We prove now the analogous of lemma 3.1.

Lemma 7.1. If φ ∈ A_{ε}(Σ_{ε}) then there exist v ∈ C^{2}(Σ; IR^{3}), w ∈ C^{1}(Σ; IR^{3}) such that

φ(x_{1}, x_{2}, x_{3}) = v(x_{1}, x_{2}) + x_{3}w(x_{1}, x_{2}), |w|^{2} = 1, w = \frac{v_{x_{1}} ∧ v_{x_{2}}}{|v_{x_{1}} ∧ v_{x_{2}}|},

v_{x_{1}}∧w_{x_{1}} = w_{x_{2}}∧v_{x_{2}} for every (x_{1}, x_{2}) ∈ Σ, v(x_{1}, x_{2}) = (x_{1}, x_{2}, 0) and w(x_{1}, x_{2}) = (0, 0, 1) for every (x_{1}, x_{2}) ∈ Γ. Moreover, if v ∈ C^{2}(Σ; IR^{3}) ∩ A(Σ), the deformation field φ defined by φ(x_{1}, x_{2}, x_{3}) = v(x_{1}, x_{2}) + x_{3}w(x_{1}, x_{2}), where w = v_{x_{1}} ∧ v_{x_{2}}, belongs to A_{ε}(Σ_{ε}).

Proof. Let φ be in C^{2}(Σ_{ε}; IR^{3}). The condition (^{T} DφDφ) · (e_{3}) = 0 implies:

⟨φ_{x_{1}}, φ_{x_{3}}⟩ = 0,  (7.1)
⟨φ_{x_{2}}, φ_{x_{3}}⟩ = 0,  (7.2)
|φ_{x_{3}}|^{2} = 1.  (7.3)

Deriving again (7.1) and (7.2) with respect to x_{3} we obtain now:

⟨φ_{x_{1}x_{3}}, φ_{x_{3}}⟩ + ⟨φ_{x_{1}}, φ_{x_{3}x_{3}}⟩ = 0  (7.4)
⟨φ_{x_{2}x_{3}}, φ_{x_{3}}⟩ + ⟨φ_{x_{2}}, φ_{x_{3}x_{3}}⟩ = 0.  (7.5)

Deriving equation (7.3) with respect to x_{1} and x_{2} we have that:

\begin{aligned}⟨φ_{x_{3}x_{1}}, φ_{x_{3}}⟩ & = 0 \\ ⟨φ_{x_{2}x_{3}}, φ_{x_{3}}⟩ & = 0, \end{aligned}

therefore it follows from (7.4) and (7.5) that ⟨φ_{x_{1}}, φ_{x_{3}x_{3}}⟩ = 0 and ⟨φ_{x_{2}}, φ_{x_{3}x_{3}}⟩ = 0. From (7.3) it results also ⟨φ_{x_{3}}, φ_{x_{3}x_{3}}⟩ = 0; because of (7.1), (7.2) and the orientationpreserving condition detDφ > 0 it must be: φ_{x_{3}x_{3}} ≡ 0, that is φ(x_{1}, x_{2}, x_{3}) = v(x_{1}, x_{2}) + x_{3}w(x_{1}, x_{2}), for some v, w; Σ → IR^{3}, with:

\begin{aligned}⟨v_{x_{1}}, w⟩ & = 0 \\ ⟨v_{x_{2}}, w⟩ & = 0 \\ |w(x_{1}, x_{2})|^{2} & = 1 ∀ (x_{1}, x_{2}) ∈ Σ \end{aligned}  (7.6)

We remark that v ∈ C^{2}(Σ; IR^{3}), because φ ∈ C^{2}(Σ_{ε}; IR^{3}). We have now: det Dφ = ⟨(v_{x_{1}} + x_{3}w_{x_{1}}), (v_{x_{2}} + x_{3}w_{x_{2}}) ∧ w⟩. If det Dφ > 0, for x_{3} = 0 we obtain in particular ⟨v_{x_{1}} ∧ v_{x_{2}}, w⟩ > 0. By (7.6) we can therefore define:

w = \frac{v_{x_{1}} ∧ v_{x_{2}}}{|v_{x_{1}} ∧ v_{x_{2}}|}.

We have now that: φ_{x_{1}} ∧ φ_{x_{1}x_{3}} = (v_{x_{1}} + x_{3}w_{x_{1}}) ∧ w_{x_{1}} = v_{x_{1}} ∧ w_{x_{1}} and w_{x_{2}} ∧ (v_{x_{2}} + x_{3}w_{x_{2}}) = w_{x_{2}} ∧ v_{x_{2}}. Finally, the boundary conditions on v and w are easily deduced from the boundary conditions on φ.

We observe that if φ ∈ A_{ε}(Σ_{ε}) and φ(x_{1}, x_{2}, x_{3}) = v(x_{1}, x_{2}) + x_{3}w(x_{1}, x_{2}), then φ̃ = v. In the following if φ ∈ A_{ε}(Σ_{ε}) we shall write φ(x_{1}, x_{2}, x_{3}) = v(x_{1}, x_{2}) + x_{3}w(x_{1}, x_{2}), with v, w such as in lemma 7.1.

We will now prove that the internal constraint v_{x_{1}} ∧ w_{x_{1}} = w_{x_{2}} ∧ v_{x_{2}} is satisfied by a wide class of functions. Indeed we have the following:

Proposition 7.2. Let v : Σ → IR^{3} a C^{2} function such that:

|v_{x_{1}}|^{2} = 1, |v_{x_{2}}|^{2} = 1, ⟨v_{x_{1}}, v_{x_{2}}⟩ = 0.  (7.7)

Then, if w = v_{x_{1}} ∧ v_{x_{2}}, we have:

v_{x_{1}} ∧ w_{x_{1}} = w_{x_{2}} ∧ v_{x_{2}}.

Proof. By the conditions (7.7) we get:

0 = ⟨v_{x_{1}}, v_{x_{1}x_{1}}⟩ = ⟨v_{x_{2}}, v_{x_{2}x_{2}}⟩ = ⟨v_{x_{1}}, v_{x_{1}x_{2}}⟩ = ⟨v_{x_{2}}, v_{x_{1}x_{1}}⟩,

hence there exist α, β, γ, δ, θ : Σ → IR such that:

\begin{aligned}v_{x_{1}x_{1}} & = αv_{x_{2}} + βv_{x_{1}} ∧ v_{x_{2}} \\ v_{x_{2}x_{2}} & = γv_{x_{1}} + δv_{x_{1}} ∧ v_{x_{2}} \\ v_{x_{1}x_{2}} & = θv_{x_{1}} ∧ v_{x_{2}}. \end{aligned}

We get now:

w_{x_{1}} = (v_{x_{1}} ∧ v_{x_{2}})_{x_{1}} = βv_{x_{1}} − θv_{x_{2}},  (7.8)
w_{x_{2}} = (v_{x_{1}} ∧ v_{x_{2}})_{x_{2}} = −θv_{x_{1}} − δv_{x_{2}}.  (7.9)

We get now from (7.8) and (7.9):

v_{x_{1}} ∧ w_{x_{1}} = θv_{x_{1}} ∧ v_{x_{2}} = w_{x_{2}} ∧ v_{x_{2}}.

Broadly speaking, the previous result shows that, if the metric associated to the coordinate system is euclidean, then the internal constraint is always satisfied.

Remark 7.3. It is interesting to observe that also the functions of the type:

v(x_{1}, x_{2}, x_{3}) = (v_{1}(x_{1}), x_{2}, v_{2}(x_{1}))

verify v_{x_{1}} ∧ w_{x_{1}} = w_{x_{2}} ∧ v_{x_{2}}, because this allows us to consider the case studied in the previous section as a particular case of the threedimensional problem. In particular if we take Σ = (0, 1) × (α, β) simple computations show that:

F_{0}(v) = \frac{2}{3} ∫_{Σ} Λ(x_{1}, x_{2}) |⟨v_{x_{1}}, w_{x_{1}}⟩|^{2}dx_{1}dx_{2}

and, since in this case ⟨v_{x_{1}}, w_{x_{1}}⟩ = \left|{ \atop v̈}\right|^{2} we have:

F_{0}(v) = \frac{2}{3} ∫_{0}^{1} Λ̃(x_{1}) \left|{ \atop v̈}\right|^{2}dx_{1}, where Λ̃(x_{1}) = ∫_{α}^{β} Λ(x_{1}, x_{2})dx_{2}.

Lemma 7.4. Let a^{ε}, b^{ε}, c^{ε}: Σ → IR^{3} be such that: |a^{ε}|^{2} = |b^{ε}|^{2} = 1, ⟨a^{ε}, b^{ε}⟩ = 0, c^{ε} = a^{ε} ∧ b^{ε}. If c^{ε} → c in L^{2}(Σ; IR^{2}), then a^{ε_{j}} → a and b^{ε_{j}} → b in L^{2}(Σ; IR^{2}) for some subsequences (a^{ε_{j}})_{j} and (b^{ε_{j}})_{j} of (a^{ε})_{ε} and (b^{ε})_{ε}.

Proof. It holds for every ε > 0:

a^{ε} = b^{ε} ∧ (a^{ε} ∧ b^{ε}).

There exists a subsequence (b^{ε_{j}}) of (b^{ε}) such that b^{ε_{j}} ⇀ b and a^{ε} ∧ b^{ε} → c, therefore a^{ε_{j}} converges weakly to b ∧ c in L^{2}(Σ; IR^{2}). Moreover it results a = b ∧ c and also: a ∧ b = (b ∧ c) ∧ b = c. We have now that for every measurable subset A ⊂ Σ a^{ε_{j}} ⇀ a in L^{2}(A) so that by the lower semicontinuity of the norm we deduce:

∫_{A} |a|^{2}dx ≤ lim inf ∫_{A} |a^{ε_{j}}|^{2}dx = |A|.

It results therefore:

\frac{1}{|A|}∫_{A} |a|^{2}dx ≤ 1

and also

\frac{1}{|A|}∫_{A} |b|^{2}dx ≤ 1

for every A ∈ Σ, A measurable. This implies |a|^{2} ≤ 1 and |b|^{2} ≤ 1 a.e. on Σ. We have now that:

1 = |a ∧ b| ≤ |a| · |b| ≤ 1,

that is |a| = |b| = 1 and ⟨a, b⟩ = 0. We get finally

∫_{Σ} |a^{ε_{j}} − a|^{2} dx_{1}dx_{2} = ∫_{Σ} (|a^{ε_{j}}|^{2} + |a|^{2} − 2⟨a^{ε_{j}}, a⟩)dx_{1}dx_{2} = 2 ∫_{Σ} (1 − ⟨a^{ε_{j}}, a⟩),

hence a^{ε_{j}} → a in L^{2}(Σ; IR^{2}). Analogously we obtain b^{ε_{j}} → b.

Proposition 7.5. Assume that φ^{ε} ∈ A_{ε}(Σ_{ε}) and F_{ε}(φ^{ε}) ≤ Cε^{3}. Then:

i) (v^{ε})_{ε} is relatively compact in H^{1}(Σ; IR^{3}) and (w^{ε}) is relatively weakly compact in H^{1}(Σ; IR^{3});

ii) if v^{ε} → v in L^{2}(Σ, IR^{3}) then v ∈ A(Σ); iii)

⟨v_{x_{i}}^{ε}, w_{x_{i}}^{ε}⟩ ⇀ ⟨v_{x_{i}}, w_{x_{i}}⟩, i = 1, 2
⟨v_{x_{1}}^{ε}, w_{x_{2}}^{ε}⟩ + ⟨w_{x_{1}}^{ε}, v_{x_{2}}^{ε}⟩ ⇀ ⟨v_{x_{1}}, w_{x_{2}}⟩ + ⟨w_{x_{1}}, v_{x_{2}}⟩
in L^{2}(Σ).

Proof. Exactly as in proposition 3.2 by applying the polar factorization of the invertible matrix Dφ^{ε} we get:

F_{ε}(φ^{ε}) ≥ C ∫_{Σ_{ε}} |Dφ^{ε} − R^{ε}|^{2}dx,

where R^{ε} is the rotation such that:

Dφ^{ε} = R^{ε}\sqrt{^{T} Dφ^{ε}Dφ^{ε}}.

We represent now R^{ε} as: R^{ε} = (a^{ε}|b^{ε}|c^{ε}), with |a^{ε}|^{2} = |b^{ε}|^{2} = |c^{ε}|^{2} = 1, c^{ε} = a^{ε} ∧ b^{ε} and we observe that the matrix :

T R^{ε}Dφ^{ε} = ( \begin{matrix}⟨a^{ε}, φ_{x_{1}}^{ε}⟩ & ⟨a^{ε}, φ_{x_{2}}^{ε}⟩ & ⟨a^{ε}, w^{ε}⟩ \\ ⟨b^{ε}, φ_{x_{1}}^{ε}⟩ & ⟨b^{ε}, φ_{x_{2}}^{ε}⟩ & ⟨b^{ε}, w^{ε}⟩ \\ ⟨a^{ε}, w^{ε}⟩ & ⟨b^{ε}, w^{ε}⟩ & ⟨c^{ε}, w^{ε}⟩ \end{matrix} )

belongs to 𝕊_{+}^{3} as square root of a symmetric and positive definite matrix.

Since^{T} R^{ε}Dφ^{ε} = \sqrt{^{T} Dφ^{ε}Dφ^{ε}} we get ⟨a^{ε}, w^{ε}⟩ = 0, ⟨b^{ε}, w^{ε}⟩ = 0 and ⟨c^{ε}, w^{ε}⟩ = 1, therefore c^{ε} = w^{ε} for every ε > 0. The symmetry condition yields:

⟨a^{ε}, φ_{x_{2}}^{ε}⟩ = ⟨b^{ε}, φ_{x_{1}}^{ε}⟩ for every ε > 0.

Recalling that b^{ε} = w^{ε} ∧ a^{ε} we get:

⟨φ_{x_{2}}^{ε} − φ_{x_{1}}^{ε} ∧ w^{ε}, a^{ε}⟩ = 0.

We observe first of all that the vector:

φ_{x_{2}}^{ε} − φ_{x_{1}}^{ε} ∧ w^{ε} = (v_{x_{2}}^{ε} − v_{x_{1}}^{ε} ∧ w^{ε}) + x_{3}(w_{x_{2}}^{ε} − w_{x_{1}}^{ε} ∧ w^{ε})

is orthogonal to a^{ε} and to w^{ε} and also that there exists σ = σ(x_{1}, x_{2}) such that:

φ_{x_{2}}^{ε} − φ_{x_{1}}^{ε} ∧ w^{ε} = [1 + x_{3}σ(x_{1}, x_{2})] · (v_{x_{2}}^{ε} − v_{x_{1}}^{ε} ∧ w^{ε}).  (7.10)

In fact the vectors v_{x_{2}}^{ε} − v_{x_{1}}^{ε} ∧ w^{ε} and w_{x_{2}}^{ε} − w_{x_{1}}^{ε} ∧ w^{ε} are both orthogonal to w^{ε} and also satisfy:

(v_{x_{2}}^{ε} − v_{x_{1}}^{ε} ∧ w^{ε}) ∧ (w_{x_{2}}^{ε} − w_{x_{1}}^{ε} ∧ w^{ε}) = 0.  (7.11)

To prove (7.11) we observe first of all that:

0 = ⟨w^{ε}, v_{x_{2}}^{ε}⟩_{x_{1}} = ⟨w_{x_{1}}^{ε}, v_{x_{2}}^{ε}⟩ + ⟨w^{ε}, v_{x_{2}x_{1}}^{ε}⟩
0 = ⟨w^{ε}, v_{x_{1}}^{ε}⟩_{x_{2}} = ⟨w_{x_{2}}^{ε}, v_{x_{1}}^{ε}⟩ + ⟨w^{ε}, v_{x_{1}x_{2}}^{ε}⟩.

Since v^{ε} belong to C^{2}(Σ_{ε}; IR^{3}) we have:

⟨w_{x_{1}}^{ε}, v_{x_{2}}^{ε}⟩ = ⟨w_{x_{2}}^{ε}, v_{x_{1}}^{ε}⟩.

We get now:

(v_{x_{2}}^{ε} − v_{x_{1}}^{ε} ∧ w^{ε}) ∧ (w_{x_{2}}^{ε} − w_{x_{1}}^{ε} ∧ w^{ε}) = v_{x_{2}}^{ε} ∧ w_{x_{2}}^{ε} − w_{x_{1}}^{ε}⟨v_{x_{2}}^{ε}, w^{ε}⟩ + w^{ε}⟨v_{x_{2}}^{ε}, w_{x_{1}}^{ε}⟩+
v_{x_{1}}^{ε}⟨w^{ε}, w_{x_{2}}^{ε}⟩ − w^{ε}⟨v_{x_{1}}^{ε}, w_{x_{2}}^{ε}⟩ + w_{x_{1}}^{ε}⟨v_{x_{1}}^{ε} ∧ w^{ε}, w^{ε}⟩ − w^{ε}⟨v_{x_{1}}^{ε} ∧ w^{ε}, w_{x_{1}}^{ε}⟩ =
v_{x_{2}}^{ε} ∧ w_{x_{2}}^{ε} − w^{ε}⟨v_{x_{1}}^{ε} ∧ w^{ε}, w_{x_{1}}^{ε}⟩ = v_{x_{2}}^{ε} ∧ w_{x_{2}}^{ε} − w^{ε}⟨w_{x_{1}}^{ε} ∧ v_{x_{1}}^{ε}, w^{ε}⟩ =
v_{x_{2}}^{ε} ∧ w_{x_{2}}^{ε} − w_{x_{1}}^{ε} ∧ v_{x_{1}}^{ε} = 0

and (7.11) is proven. Because of the orthogonality between φ_{x_{2}}^{ε} − φ_{x_{1}}^{ε} ∧ w^{ε} and a^{ε} and w^{ε}, we get that v_{x_{2}}^{ε} − v_{x_{1}}^{ε} ∧ w^{ε} is orthogonal to a^{ε} and w^{ε}. Moreover it holds: v_{x_{2}}^{ε} − v_{x_{1}}^{ε} ∧ w^{ε} ≠ 0, because, if v_{x_{2}}^{ε} − v_{x_{1}}^{ε} ∧ w^{ε} = 0, it would be:

det Dφ^{ε}_{|x_{3}=0} = ⟨v_{x_{1}}^{ε}, v_{x_{2}}^{ε} ∧ w^{ε}⟩ = −⟨v_{x_{1}}^{ε}, w^{ε} ∧ v_{x_{2}}^{ε}⟩ = −|v_{x_{1}}^{ε}|^{2} < 0,

a contradiction. We have therefore:

b^{ε} = \frac{v_{x_{2}}^{ε} − v_{x_{1}}^{ε} ∧ w^{ε}}{|v_{x_{2}}^{ε} − v_{x_{1}}^{ε} ∧ w^{ε}|}

and also:

a^{ε} = b^{ε} ∧ w^{ε} = \frac{v_{x_{1}}^{ε} + v_{x_{2}}^{ε} ∧ w^{ε}}{|v_{x_{1}}^{ε} + v_{x_{2}}^{ε} ∧ w^{ε}|}.

We get now as in proposition 3.2:

Cε^{3} ≥ ∫_{Σ_{ε}} ( |v_{x_{1}}^{ε} + x_{3}w_{x_{1}}^{ε} − a^{ε}|^{2} + |v_{x_{2}}^{ε} + x_{3}w_{x_{2}}^{ε} − b^{ε}|^{2}
+ |w^{ε} − c^{ε}|^{2} )dx

= ∫_{Σ_{ε}} ( |v_{x_{1}}^{ε} − a^{ε}|^{2} + x_{3}^{2} |w_{x_{1}}^{ε}|^{2} + |v_{x_{2}}^{ε} − b^{ε}|^{2} + x_{3}^{2} |w_{x_{2}}^{ε}|^{2} )dx

= 2ε ∫_{Σ} ( |v_{x_{1}}^{ε} − a^{ε}|^{2} + |v_{x_{2}}^{ε} − b^{ε}|^{2} )dx_{1}dx_{2} + \frac{2}{3}ε^{3} ∫_{Σ} |Dw|^{2}dx_{1}dx_{2}.

We have ∫_{Σ} |w^{ε}|^{2}dx_{1}dx_{2} + ∫_{Σ} |Dw^{ε}|^{2}dx_{1}dx_{2} ≤ C, for some C > 0; there exist therefore w ∈ H^{1}(Σ; IR^{3}) and a subsequence of (w^{ε}), always indicated by (w^{ε}), such that w^{ε} converges weakly in H^{1}(Σ; IR^{3}) to w. In particular, w^{ε} converges strongly to w in L^{2}(Σ, IR^{3}); therefore c^{ε} → w in L^{2}(Σ, IR^{3}). By using Lemma 7.4 we obtain v_{x_{1}}^{ε} → a and v_{x_{2}}^{ε} → b in L^{2}(Σ, IR^{3}), possibly passing to a subsequence, with |a|^{2} = |b|^{2} = 1 and ⟨a, b⟩ = 0, therefore (v^{ε}) is relatively compact in H^{1}(Σ; IR^{3}), with |v_{x_{1}}|^{2} = 1, |v_{x_{2}}|^{2} = 1 and ⟨v_{x_{1}}, v_{x_{2}}⟩ = 0 a.e. on Σ.

V.Casarino, D. Percivale / A variational model for non linear elastic plates 239

To prove iii) we observe that:

F_{ε}(φ^{ε}) = ∫_{Σ_{ε}} [μ(x_{1}, x_{2}) |E(φ^{ε})|^{2} + \frac{λ}{2}(x_{1}, x_{2})(tr E(φ^{ε}))^{2}]dx

≥ μ̄ ∫_{Σ_{ε}} |E(φ^{ε})|^{2}dx

= \frac{μ̄}{4} ∫_{Σ_{ε}} [( |φ_{x_{1}}^{ε}|^{2} − 1)^{2} + ( |φ_{x_{2}}^{ε}|^{2} − 1)^{2} + 2 |⟨φ_{x_{1}}^{ε}, φ_{x_{2}}^{ε}⟩|^{2} ]dx

= \frac{μ̄}{4} ∫_{Σ_{ε}} \{( |v_{x_{1}}^{ε} + x_{3}w_{x_{1}}^{ε}|^{2} − 1)^{2} + ( |v_{x_{2}}^{ε} + x_{3}w_{x_{2}}^{ε}|^{2} − 1)^{2}

+ 2[⟨v_{x_{1}}^{ε}, v_{x_{2}}^{ε}⟩ + x_{3}(⟨v_{x_{1}}^{ε}, w_{x_{2}}^{ε}⟩ + ⟨v_{x_{2}}^{ε}, w_{x_{1}}^{ε}⟩)
+ x_{3}^{2}⟨w_{x_{1}}^{ε}, w_{x_{2}}^{ε}⟩]^{2}}dx

= \frac{μ̄}{4} ∫_{Σ_{ε}} \{[ |v_{x_{1}}^{ε}|^{2} − 1 + x_{3}^{2} |w_{x_{1}}^{ε}|^{2} ]^{2} + [ |v_{x_{2}}^{ε}|^{2} − 1 + x_{3}^{2} |w_{x_{2}}^{ε}|^{2} ]^{2}

+ 4x_{3}^{2}|⟨v_{x_{1}}^{ε}, w_{x_{1}}^{ε}⟩|^{2} + 4x_{3}^{2}|⟨v_{x_{2}}^{ε}, w_{x_{2}}^{ε}⟩|^{2} + 2[⟨v_{x_{1}}^{ε}, v_{x_{2}}^{ε}⟩
+ x_{3}^{2}|⟨w_{x_{1}}^{ε}, w_{x_{2}}^{ε}⟩|^{2}]^{2} + 2x_{3}^{2}|⟨w_{x_{1}}^{ε}, v_{x_{2}}^{ε}⟩ + ⟨v_{x_{1}}^{ε}, w_{x_{2}}^{ε}⟩|^{2}}

≥ \frac{μ̄}{4} ∫_{Σ_{ε}} 4x_{3}^{2}[ |⟨v_{x_{1}}^{ε}, w_{x_{1}}^{ε}⟩|^{2} + |⟨v_{x_{2}}^{ε}, w_{x_{2}}^{ε}⟩|^{2} ]

+ 2x_{3}^{2} |⟨v_{x_{1}}^{ε}, w_{x_{2}}^{ε}⟩ + ⟨w_{x_{1}}^{ε}, v_{x_{2}}^{ε}⟩|^{2} dx.

By the hypothesis F_{ε}(φ^{ε}) ≤ Cε^{3} it results therefore that (∥⟨v_{x_{1}}^{ε}, w_{x_{1}}^{ε}⟩∥_{L^{2}}) (∥⟨v_{x_{2}}^{ε}, w_{x_{2}}^{ε}⟩∥_{L^{2}}) and (∥⟨v_{x_{1}}^{ε}, w_{x_{2}}^{ε}⟩ + ⟨w_{x_{1}}^{ε}, v_{x_{2}}^{ε}⟩∥_{L^{2}}) are bounded. Since ⟨v_{x_{1}}^{ε}, w_{x_{1}}^{ε}⟩, ⟨v_{x_{2}}^{ε}, w_{x_{2}}^{ε}⟩ and ⟨v_{x_{1}}^{ε}, w_{x_{2}}^{ε}⟩+ ⟨w_{x_{1}}^{ε}, v_{x_{2}}^{ε}⟩ converge in the sense of the distribution respectively to ⟨v_{x_{1}}, w_{x_{1}}⟩, ⟨v_{x_{2}}, w_{x_{2}}⟩ and ⟨v_{x_{1}}, w_{x_{2}}⟩ + ⟨w_{x_{1}}, v_{x_{2}}⟩, and Dv^{ε} → Dv and Dw^{ε} ⇀ Dw in L^{2}(Σ; IR^{3}), we get the thesis.

8. Proof of the main result

Proposition 8.1. For every v ∈ H^{1}(Σ; IR^{3}), for every sequence (φ^{ε})_{ε>0} such that

φ^{ε} ∈ A_{ε}(Σ_{ε}) and φ̃^{ε} = v^{ε} → v in H^{1}(Σ; IR^{3})

we have:

lim inf_{ε→0} G_{ε}(φ^{ε}) ≥ G_{0}(v).  (8.1)

Proof. Exactly as in proposition 4.1 we may suppose G_{ε}(φ^{ε}) ≤ C and we get F_{ε}(φ^{ε}) ≤ Cε^{3}, hence v ∈ A(Σ) by proposition 7.5.

240 V.Casarino, D. Percivale / A variational model for non linear elastic plates

We have now:

lim inf_{ε→0} \frac{1}{ε^{3}} F_{ε}(φ^{ε}) = lim inf_{ε→0} \frac{1}{ε^{3}} ∫_{Σ_{ε}} [μ(x_{1}, x_{2}) |E(φ^{ε})|^{2} + \frac{λ}{2}(x_{1}, x_{2})(tr E(φ^{ε}))^{2}]dx

= lim inf_{ε→0} (I_{ε}^{1} + I_{ε}^{2}).

As in proposition 7.5 we get:

lim inf_{ε→0} I_{ε}^{1} ≥ lim inf_{ε→0} \frac{1}{ε^{3}} ∫_{Σ_{ε}} \frac{μ}{4}\{4x_{3}^{2}[ |⟨v_{x_{1}}^{ε}, w_{x_{1}}^{ε}⟩|^{2} + |⟨v_{x_{2}}^{ε}, w_{x_{2}}^{ε}⟩|^{2} ]
+ 2x_{3}^{2} |⟨v_{x_{1}}^{ε}, w_{x_{2}}^{ε}⟩ + ⟨w_{x_{1}}^{ε}, v_{x_{2}}^{ε}⟩|^{2} }dx

≥ \frac{1}{3} ∫_{Σ} μ[2( |⟨v_{x_{1}}, w_{x_{1}}⟩|^{2} + |⟨v_{x_{2}}, w_{x_{2}}⟩|^{2} )

+ |⟨v_{x_{1}}, w_{x_{2}}⟩ + ⟨w_{x_{1}}, v_{x_{2}}⟩|^{2} ]dx_{1}dx_{2}.

We have moreover:

I_{ε}^{2} = lim inf_{ε→0} \frac{1}{ε^{3}} ∫_{Σ_{ε}} \frac{λ}{2}(x_{1}, x_{2})(tr E(φ^{ε}))^{2}dx

= lim inf_{ε→0} \frac{1}{ε^{3}} ∫_{Σ_{ε}} \frac{λ}{8}[|φ_{x_{1}}^{ε}|^{2} + |φ_{x_{2}}^{ε}|^{2} − 2]^{2}dx

≥ lim inf_{ε→0} \frac{1}{ε^{3}} ∫_{Σ_{ε}} \{\frac{λ}{8}[|φ_{x_{1}}|^{2} + |φ_{x_{2}}|^{2} − 2]^{2}

+ 2(|φ_{x_{1}}|^{2} + |φ_{x_{2}}|^{2} − 2) · [|φ_{x_{1}}^{ε}|^{2} − |φ_{x_{1}}|^{2} + |φ_{x_{2}}^{ε}|^{2} − |φ_{x_{2}}|^{2}]}dx

= lim inf_{ε→0} (J_{ε}^{1} + J_{ε}^{2}).

We have:

lim inf_{ε→0} J_{ε}^{1} ≥ lim inf_{ε→0} \frac{1}{ε^{3}} ∫_{Σ_{ε}} \frac{λ}{8}[2x_{3}⟨Dv, Dw⟩ + x_{3}^{2} |Dw|^{2} ]^{2}dx

≥ lim inf_{ε→0} \frac{1}{ε^{3}} ∫_{Σ_{ε}} \frac{λ}{2}x_{3}^{2} |⟨Dv, Dw⟩|^{2}dx

= \frac{1}{3} ∫_{Σ} λ |⟨Dv, Dw⟩|^{2}dx_{1}dx_{2}.

V.Casarino, D. Percivale / A variational model for non linear elastic plates 241

We get finally:

lim inf_{ε→0} J_{ε}^{2} = lim inf_{ε→0} \frac{1}{ε^{3}} ∫_{Σ_{ε}} \frac{λ}{4}(|φ_{x_{1}}|^{2} + |φ_{x_{2}}|^{2} − 2) · (|Dφ^{ε}|^{2} − |Dφ|^{2})dx

= lim inf_{ε→0} \frac{1}{ε^{3}} ∫_{Σ_{ε}} \frac{λ}{4}[2x_{3}⟨Dv, Dw⟩ + x_{3}^{2} |Dw|^{2} ] · [ |Dv^{ε}|^{2} − |Dv|^{2} +

+ 2x_{3}(⟨Dv^{ε}, Dw^{ε}⟩ − ⟨Dv, Dw⟩) + x_{3}^{2}( |Dw^{ε}|^{2} − |Dw|^{2} )]dx

= lim inf_{ε→0} \frac{1}{ε^{3}} ∫_{Σ_{ε}} \frac{λ}{4}\{4x_{3}^{2}⟨Dv, Dw⟩ · [⟨Dv^{ε}, Dw^{ε}⟩ − ⟨Dv, Dw⟩]

+ x_{3}^{2} |Dw|^{2} ( |Dv^{ε}|^{2} − |Dv|^{2} ) + x_{3}^{2}|Dw|^{2}( |Dw^{ε}|^{2} − |Dw|^{2} )}dx.

The minimum limit of the first integral is 0 because ⟨Dv^{ε}, Dw^{ε}⟩ ⇀ ⟨Dv, Dw⟩ in L^{2}(Σ); for the second one we observe that

lim inf_{ε→0} ∫_{Σ} |Dw|^{2}( |Dv^{ε}|^{2} − |Dv|^{2})dx + lim inf_{ε→0} ∫_{Σ} |Dw|^{2}( |Dw^{ε}|^{2} − |Dw|^{2} )dx

≥ lim inf_{ε→0} ∫_{Σ} |Dw|^{2} |Dv^{ε}|^{2}dx_{1}dx_{2} − ∫_{Σ} |Dw|^{2} |Dv|^{2}dx_{1}dx_{2}

+ lim inf_{ε→0} ∫_{Σ} |Dw|^{2} |Dw^{ε}|^{2}dx_{1}dx_{2} − ∫_{Σ} |Dw|^{2} |Dw|^{2}dx_{1}dx_{2} ≥ 0

by the lower semicontinuity of the functional: η ↦ ∫_{Σ} |Dw|^{2} |η|^{2}dx_{1}dx_{2} with respect to the weak topology of H^{1}(Σ), hence

lim inf_{ε→0} I_{ε}^{2} ≥ \frac{1}{3} ∫_{Σ} λ(⟨v_{x_{1}}, w_{x_{1}}⟩ + ⟨v_{x_{2}}, w_{x_{2}}⟩)^{2}dx_{1}dx_{2}

and we can conclude as in proposition 4.1.

Proposition 8.2. For every v ∈ L^{2}(Σ; IR^{3}) there exists a sequence

(φ^{ε})_{ε>0} ⊂ A_{ε}(Σ_{ε})

such that

v^{ε} = φ̃^{ε} → v in L^{2}(Σ; IR^{3}) and lim_{ε→0} G_{ε}(φ^{ε}) = G_{0}(v).

Proof. Exactly as in proposition 4.2 we define for every v ∈ L^{2}(Σ; IR^{3}) the functional G^{+}(v). The inequality G^{+}(v) ≤ G_{0}(v) is obvious if v doesn’t belong to A(Σ). When v ∈ C^{2} ⋂ A(Σ) we define for every ε > 0

φ^{ε}(x_{1}, x_{2}, x_{3}) = v(x_{1}, x_{2}) + x_{3}w(x_{1}, x_{2}),

where w = v_{x_{1}} ∧ v_{x_{2}}.

We obtain now:

F_{ε}(φ^{ε}) = \frac{1}{4} ∫_{Σ_{ε}} \{μ[(2x_{3}⟨v_{x_{1}}, w_{x_{1}}⟩ + x_{3}^{2} |w_{x_{1}}|^{2} )^{2} + (2x_{3}⟨v_{x_{2}}, w_{x_{2}}⟩

+ x_{3}^{2} |w_{x_{2}}|^{2} )^{2} + |⟨v_{x_{1}} + x_{3}w_{x_{1}}, v_{x_{2}} + x_{3}w_{x_{2}}⟩|^{2} ]

+ \frac{λ}{2}[ |v_{x_{1}} + x_{3}w_{x_{1}}|^{2} − 1 + |v_{x_{2}} + x_{3}w_{x_{2}}|^{2} − 1]\}dx.

With the same computations of propositions 7.5 and 8.1 we get finally:

lim_{ε→0} G_{ε}(φ^{ε}) = G_{0}(v).

Proof of Theorem 6.1 The proof is exactly the same of Theorem 2.1.

Remark 8.3. It is possible generalize with slight modifications the previous result to the physically significant case, in which the energy becomes infinite when the volume locally vanishes. If we suppose that the deformation energy density is of the form:

W (E) = Θ(det(2E+I)) · [μ|E|^{2} + \frac{λ}{2}(tr E)^{2}],

where the function Θ : (0, +∞) → [0, +∞) satisfies the conditions:

i) Θ is continuous; ii) Θ(s) ≥ Θ(1) > 0 for every s ∈ (0, +∞); iii) lim_{s→0^{+}}Θ(s) = +∞, then all results continue to hold.

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