Philippe G Ciarlet

Giuseppe Geymonat
email: giuseppe.geymonat@lms.polytechnique.fr

Francoise Krasucki
email: krasucki@math.univ-montp2.fr

Mathematical Problems in Mechanics

teaching and research institutions in France or abroad, or from public or private research centers.

Nonlinear Donati compatibility conditions for the nonlinear Kirchhoff -von K arm an-Love plate theory

Résumé

Conditions de compatibilité non linéaires de Donati pour la théorie non linéaire des plaques de Kirchhoff-von Kármán-Love. Soit ω un domaine simplement connexe de R 2 et soit (E αβ ) et (F αβ ) deux champs de matrices 2 × 2 symétriques dont les composantes sont dans L 2 (ω). Dans cette Note, on identifie et justifie des conditions non linéaires de compatibilité "de type Donati" que doivent satisfaire les composantes E αβ et F αβ afin qu'il existe un champs de vecteurs (η1, η2, w) ∈ H 1 0 (ω) × H 1 0 (ω) × H 2 0 (ω) tel que 1 2 (∂αη β + ∂ β ηα + ∂αw∂ β w) = E αβ et ∂ αβ w = F αβ dans ω.

Les membres de gauche de ces relations sont les composantes de tenseurs trouvés dans la théorie de Kirchhoff-von Kármán-Love des plaques non linéairement élastiques.

Preliminaries

Greek indices vary in {1, 2} and the convention summation with respect to repeated indices is used. A domain in R 2 is a bounded, open, and connected subset ω of R 2 with a Lipschitz-continuous boundary ∂ω, the set ω being locally on the same side of ∂ω. Partial derivatives of the first, second, and third, order of functions of y = (y α ) ∈ ω are denoted ∂ α := ∂/∂y α , ∂ αβ := ∂ 2 /∂y α ∂y β , and ∂ αβσ := ∂ 3 /∂y α ∂y β ∂y σ ; the same notations are used for partial derivatives in the sense of distributions.

Vector fields and matrix fields, and spaces of vector fields, defined over ω are denoted by boldface letters. The usual Sobolev spaces over ω are denoted H m (ω), m ∈ Z, and H m 0 (ω), m ≥ 1; the norm in H m (ω), m ∈ Z, is denoted • m,ω ; in particular then, • 0,ω is the norm of H 0 (ω) = L 2 (ω). The notation L 2 (ω) designates the space of all 2 × 2 symmetric matrix fields with components in L 2 (ω). If S = (S αβ ) is a 2 × 2 matrix field with smooth enough components defined over ω, its divergence div S is the vector field defined by (div

S) α = ∂ α S αβ .
If X and Y are two (real) vector spaces and A is a linear operator from X to Y , Im A := {y ∈ Y ; y = Ax for at least one x ∈ X} and Ker A := {x ∈ X; Ax = 0}.

The notation X ′ •, • X designates the duality between a normed vector space X and its dual X ′ .

In the classical Kirchhoff-von Kármán-Love theory of nonlinearly elastic plates (see, e.g., Chapters 4 and 5 in [START_REF] Ciarlet | Theory of Plates[END_REF]), the unknown displacement field of the middle surfac ω of the plate minimizes an energy whose integrand contains a positive-definite quadratic function of the change of metric and change of curvature tensors, respectively defined by

E αβ := 1 2 (∂ α η β + ∂ β η α + ∂ α w∂ β w) and F αβ := ∂ αβ w, (1)
for an arbitrary displacement field (η

1 , η 2 , w) ∈ H 1 0 (ω) × H 1 0 (ω) × H 2 0
(ω) (we consider here plates that are clamped over their entire lateral face).

In the intrinsic approach to the same theory, the matrix fields (E αβ ) ∈ L 2 (ω) and (F αβ ) ∈ L 2 (ω) are considered as the sole unknowns. There thus arises the question as to whether there exist suitable compatibility conditions that the components E αβ and F αβ of these matrix fields should satisfy in order that there exists a vector field (η 1 , η 2 , w) ∈ H 1 0 (ω) × H 1 0 (ω) × H 2 0 (ω) satisfying [START_REF] Amrouche | On the characterizations of matrix fields as linearized strain tensor fields[END_REF]. As shown in [START_REF] Ciarlet | Nonlinear Saint-Venant compatibility conditions and the intrinsic approach for nonlinearly elastic plates[END_REF], if the domain ω is simply-connected, the nonlinear Saint-Venant compatibility conditions

∂ στ E αβ + ∂ αβ E στ -∂ ασ E βτ -∂ βτ E ασ + F αβ F στ -F ασ F βτ = 0 in H -2 (ω), ∂ σ F αβ -∂ β F ασ = 0 in H -1 (ω),
constitute one possible answer to this question. The objective of this Note is to give (cf. Theorem 4.2) a different answer to the same question, this time in the form of variational equations, which as such constitute examples of nonlinear Donati compatibility conditions (a general presentation of Saint-Venant and Donati compatibility conditions as they arise in three-dimensional linearized elasticity is found in Chapter 6 in [START_REF] Ciarlet | Linear and Nonlinear Functional Analysis with Applications[END_REF]).

Complete proofs and an application to intrinsic nonlinear plate theory will be found in [START_REF] Ciarlet | Nonlinear Donati compatibility conditions and the intrinsic approach for nonlinearly elastic plates[END_REF].

An existence theorem for an Airy-function

The following result is a "weak" version (already used in [START_REF] Ciarlet | Nonlinear Saint-Venant compatibility conditions and the intrinsic approach for nonlinearly elastic plates[END_REF]) of a classical result for smooth functions. Its proof is based on the two-dimensional version of the weak Poincaré lemma due to [START_REF] Ciarlet | Another approach to linearized elasticity and a new proof of Korn's inequality[END_REF] and then given a substantially simpler proof in [START_REF] Kesavan | On Poincaré's and J.L. Lions' lemmas[END_REF]; cf. also Thm. 6.17-4 in [START_REF] Ciarlet | Linear and Nonlinear Functional Analysis with Applications[END_REF].

Theorem 2.1 Let ω be a simply-connected domain in R 2 , and let there be given a matrix field

(F αβ ) ∈ L 2 (ω) that satisfies ∂ σ F αβ -∂ β F ασ = 0 in H -1 (ω).
Then there exists a function ϕ ∈ H 2 (ω), unique up to the addition of a polynomial of degree ≤ 1, such that

∂ αβ ϕ = F αβ in L 2 (ω).
Theorem 2.1 can be immediately recast as an existence result of an ad hoc Airy function (denoted ϕ in the next theorem) under low regularity assumptions. As such, it complements Thm. 2 of [START_REF] Geymonat | On the existence of the Airy function in Lipschitz domains. Application to the traces of H 2[END_REF], where the existence of an Airy function was established, again in the space H 2 (ω), but for non-simply-connected domains, under the assumption that the tensor field noted S in the next theorem satisfies in addition the usual global equilibrium equations. Theorem 2.2 Let ω be a simply-connected domain in R 2 , and let there be given a matrix field S = (S αβ ) ∈ L 2 (ω) that satisfies div S = 0 in H -1 (ω).

Then there exists a function ϕ ∈ H 2 (ω), unique up to the addition of a polynomial of degree ≤ 1, such that

∂ 11 ϕ = S 22 , ∂ 12 ϕ = -S 12 , ∂ 22 ϕ = S 11 in L 2 (ω). (2)
A function ϕ satisfying the relations ( 2) is called an Airy function associated with the matrix field S.

Linear Donati compatibility conditions for linearly elastic plates

For convenience, we consider separately the existence of the "horizontal" components η α , and that of the "vertical" component w of the unknown vector field. Theorem 3.1 Let ω be a domain in R 2 and let there be given a matrix field (e αβ ) ∈ L 2 (ω) that satisfies ω e αβ s αβ dy = 0 for all (s αβ ) ∈ L 2 (ω) such that ∂ α s αβ = 0 in H -1 (ω).

(

) 3
Then there exists a vector field

(η α ) ∈ H 1 0 (ω) × H 1 0 (ω) such that 1 2 (∂ α η β + ∂ β η α ) = e αβ in L 2 (ω),
and such a vector field (η α ) is uniquely determined.

Proof. See, e.g., [START_REF] Amrouche | On the characterizations of matrix fields as linearized strain tensor fields[END_REF], or [START_REF] Geymonat | Some remarks on the compatibility conditions in elasticity[END_REF] and [START_REF] Geymonat | Hodge decomposition for symmetric matrix fields and the elasticity complex in Lipschitz domains[END_REF].

Theorem 3.2 Let ω be a domain in R 2 and let there be given a matrix field

(F αβ ) ∈ L 2 (ω) that satisfies ω F αβ T αβ dy = 0 for all (T αβ ) ∈ L 2 such that ∂ αβ T αβ = 0 in H -2 (ω).
Then there exists a function w ∈ H 2 0 (ω) such that

∂ αβ w = F αβ in L 2 (ω), ( 4
)
and this function is uniquely determined.

Sketch of proof.

Let the continuous linear operator H : H 2 0 (ω) → L 2 (ω) be defined by

Hw =   ∂ 11 w ∂ 12 w ∂ 21 w ∂ 22 w   ∈ L 2 (ω) for each w ∈ H 2 0 (ω).
Then one shows that Im H is a closed subspace of L 2 (ω), and that the dual operator of H is div div : L 2 (ω) → H -2 (ω). The conclusion then follows from Banach closed range theorem.

Relation ( 3) and ( 4) constitute the linear Donati compatibility conditions corresponding to the Kirchhoff-Love theory of linearly elastic plates. Note that they hold regardless of whether the domain ω is simply-connected.

Nonlinear Donati compatibility conditions for nonlinearly elastic plates

The Green's formula (5) established in the next theorem is crucial to our subsequent analysis. Theorem 4.1 For all functions w ∈ H 2 0 (ω) and ϕ ∈ H 2 (ω),

ω (∂ 11 w∂ 22 w -∂ 12 w∂ 12 w)ϕ dy = ω - 1 2 (∂ 1 w) 2 ∂ 22 ϕ - 1 2 (∂ 2 w) 2 ∂ 11 ϕ + ∂ 1 w∂ 2 w∂ 12 ϕ dy. (5)
Sketch of proof. Both sides of ( 5) being continuous functions of (w, ϕ) ∈ H 2 0 (ω) × H 2 (ω), it is enough to establish (5) for all (w, ϕ) ∈ D(ω) × H 2 (ω). To this end, one uses the integration by parts formulas in Sobolev spaces.

The next theorem constitutes the main result of this Note. Theorem 4.2 Let ω be a simply-connected domain in R 2 . Given a matrix field S ∈ L 2 (ω) that satisfies div S = 0 in H -1 (ω), there exists a unique function ϕ ∈ H 2 (ω) such that (cf. Theorem 2.2) (ω) such that div S = 0 in H -1 (ω). ( 7)

∂
Then there exists a vector field

(η 1 , η 2 , w) ∈ H 1 0 (ω) × H 1 0 (ω) × H 2 0 (ω) such that 1 2 (∂ α η β + ∂ β η α + ∂ α w∂ β w) = E αβ in L 2 (ω), ∂ αβ w = F αβ in L 2 (ω),
and such a vector field (η 1 , η 2 , w) is uniquely determined.

Proof. Relation [START_REF] Ciarlet | Nonlinear Saint-Venant compatibility conditions and the intrinsic approach for nonlinearly elastic plates[END_REF] shows that there exists a uniquely determined function w ∈ H 2 0 (ω) such that (cf. Theorem 3.2)

F αβ = ∂ αβ w in L 2 (ω). (8)
Given any matrix field S = (S αβ ) ∈ L 2 (ω) such that div S = 0 in H -1 (ω), there exists a uniquely determined function ϕ ∈ H 2 (ω) such that (cf. Theorem 2.2)

S 11 = ∂ 22 ϕ, S 12 = -∂ 12 ϕ, S 22 = ∂ 11 ϕ in L 2 (ω).
Therefore, for any such matrix field S, The left-hand side of relation ( 6) can thus be rewritten as (cf. 5) Since this last relation holds for all S = (S αβ ) ∈ L 2 (ω) such that div S = 0 in H -1 (ω), there exists a uniquely determined vector field (η α ) ∈ H 1 0 (ω) × H 1 0 (ω) such that (cf. Theorem 3.1)

ω {E αβ S αβ + (det F )Φ(S)} dy = ω E 11 - 1 2 (∂ 1 w) 2 ∂ 22 ϕ -2 E 12 - 1 2 ∂ 1 w∂ 2 w ∂ 12 ϕ + E 22 - 1
E αβ - 1 2 ∂ α w∂ β w = 1 2 (∂ α η β + ∂ β η α ) in L 2 (ω).
This completes the proof.

Relations ( 6) and ( 7) constitute the nonlinear Donati compatibility conditions corresponding to the Kirchhoff-von Kármán-Love theory of nonlinearly elastic plates. Note that, when properly extended, they can also cover the case where the domain ω is not simply-connected; cf. [START_REF] Ciarlet | Nonlinear Donati compatibility conditions and the intrinsic approach for nonlinearly elastic plates[END_REF].

Finally, note that, as expected, the linearization of the nonlinear Donati compatibility conditions (7) reduce to the linear ones (cf. (3)).

  11 ϕ = S 22 , ∂ 12 ϕ = -S 12 , ∂ 22 ϕ = S 11 in L 2 (ω) and ω ϕ dy = ω ∂ α ϕ dy = 0. Let Φ : {S ∈ L 2 (ω); div S = 0 in H -1 (ω)} → {ψ ∈ H 2 (ω); ω ψ dy = ω ∂ α ψ dy = 0} denote the mapping defined in this fashion, i.e., by Φ(S) := ϕ. Let there be given two matrix fields (E αβ ) ∈ L 2 (ω) and F = (F αβ ) ∈ L 2 (ω) that satisfy ω F αβ T αβ dy = 0 for all T = (T αβ ) ∈ L 2 (ω) such that div div T = 0 in H -2 (ω), (6) ω {E αβ S αβ + (det F )Φ(S)} dy = 0 for all S = (S αβ ) ∈ L 2

ωE

  αβ S αβ dy = ω {E 11 ∂ 22 ϕ + E 22 ∂ 11 ϕ -2E 12 ∂ 12 ϕ} dy, and (cf. (7)) ω (det F )Φ(S) dy = ω (F 11 F 22 -(F 12 ) 2 )ϕ dy, = ω {(∂ 11 w∂ 22 w -∂ 12 w∂ 12 w)ϕ dy}.

  2 (∂ 2 w) 2 ∂ 11 ϕ dy 1 w) 2 S 11 + 2 E 22 -1 2 ∂ 1 w∂ 2 w S 12 + E 22 -1 2 (∂ 2 w) 2 S 22 dy.

	=	ω	E 11 -	1 2	(∂

Acknowledgment

This work was partially supported by a grant from the Research Grants Council of the Hong Kong Special Administrative Region, China [Project No. 9041738-CityU 100612].


Philippe G Ciarlet

Giuseppe Geymonat
email: giuseppe.geymonat@lms.polytechnique.fr

Franc ¸oise Krasucki
email: krasucki@math.univ-montp2.fr

Nonlinear Donati compatibility conditions and the intrinsic approach for nonlinearly elastic plates

Keywords: Nonlinear plate theory, Donati compatibility conditions, intrinsic approach Mathematics Subject Classification: 49J40, 74B20, 74K20

Linear Donati compatibility conditions guarantee that the components of symmetric tensor fields are those of linearized change of metric or linearized change of curvature tensor fields associated with the displacement vector field arising in a linearly elastic structure when it is subjected to applied forces. These compatibility conditions take the form of variational equations with divergence-free tensor fields as test-functions, by contrast with Saint-Venant compatibility conditions, which take the form of systems of partial differential equations.

In this paper, we identify and justify nonlinear Donati compatibility conditions that apply to a nonlinearly elastic plate modeled by the Kirchhoff-von Kármán-Love theory. These conditions, which to the authors' best knowledge constitute a first example of nonlinear Donati compatibility conditions, in turn allow to recast the classical approach to this nonlinear plate theory, where the unknown is the position of the deformed middle surface of the plate, into the intrinsic approach, where the change of metric and change of curvature tensor fields of the deformed middle surface of the plate are the only unknowns. The intrinsic approach thus provides a direct way to compute the stress resultants and the stress couples inside the deformed plate, often the unknowns of major interest in computational mechanics.

Résumé

Les conditions de compatibilité de Donati linéaires garantissent que les composantes de champs de tenseurs symétriques sont celles de tenseurs linéarisés de changement de métrique ou de changement de courbure, associés à un champ de déplacements apparaissant dans une structure élastique soumise à des forces appliquées. Ces conditions de compatibilité prennent la forme d'équations variationnelles avec des champs de tenseurs à divergence nulle comme fonctions-tests, par contraste avec les conditions de compatibilité de Saint-Venant, qui prennent la forme de systèmes d'équations aux dérivées partielles.

Dans cet article, nous identifions et justifions des conditions de compatibilité de Donati non linéaires, qui s'appliquent à une plaque non linéairement élastique modélisée selon la théorie de Kirchhoff-von Kármán-Love. Ces conditions, qui à la connaissance des auteurs constituent un premier exemple de conditions de compatibilité de Donati non linéaires, permettent ensuite de reformuler l'approche classique de cette théorie non linéaire de plaques sous la forme de l'approche intrinsèque, où les champs de tenseurs de changement de métrique et de changement de courbure de la surface moyenne déformée de la plaque sont les seules inconnues. L'approche intrinsèque fournit ainsi un moyen direct de calculer les efforts tranchants et les moments fléchissants à l'intérieur de la plaque déformée, souvent les inconnues les plus significatives en calcul des structures.

Introduction

Throughout this paper, Latin, resp. Greek, indices and exponents vary in the set {1, 2, 3}, resp. in the set {1, 2}, save when they are used for indexing sequences, resp. save ν in the notation ∂ ν . The summation convention with respect to repeated indices and exponents is used in conjunction with these rules.

A domain ω in R 2 is a bounded and connected open subset of R 2 with a Lipschitz-continuous boundary γ, the set ω being locally on the same side of γ. Given a domain ω ⊂ R 2 , the notations ∂ α := ∂/∂y α , ∂ αβ := ∂ 2 /∂y α ∂y β , etc., designate partial derivatives, possibly in the sense of distributions, of functions of (y α ) ∈ ω.

We now briefly describe the well-known, and often used, Kirchhoff-von Kármán-Love theory of a nonlinearly elastic plate, so named after Kirchhoff [START_REF] Kirchhoff | Über das Gleichgewicht und die Bewegung einer elastischen Scheibe[END_REF], von Kármán [START_REF] Von Kármán | Festigkeitsprobleme im Maschinenbau[END_REF], and Love [START_REF] Love | Treatise on the Mathematical Theory of Elasticity[END_REF].

Let ω be a domain in R 2 . Consider an elastic plate of thickness 2ε > 0 with ω as its middle surface, made up with a homogeneous and isotropic elastic material, and whose reference configuration ω × [-ε, ε] is a natural state. Let a αβστ := 4λµ λ + 2µ δ αβ δ στ + 2µ(δ ασ δ βτ + δ ατ δ βσ ) denote the components of the two-dimensional elasticity tensor of the plate, where λ ≥ 0 and µ > 0 denote the Lamé constants of the constituting material of the plate; let (p i ) ∈ L 2 (ω; R 3 ) and (q α ) ∈ L 2 (ω; R 2 ) respectively denote the resultants and couples of the given applied forces. Finally, assume that the plate is clamped on a dγ-measurable subset γ 0 of γ := ∂ω (note that γ 0 may be empty).

Let the functional J be defined for each vector field η := (η i ) ∈ H 1 (ω) × H 1 (ω) × H 2 (ω) by

J(η) := 1 2 ω ε 4 a αβστ (∂ σ η τ + ∂ τ η σ + ∂ σ η 3 ∂ τ η 3 )(∂ α η β + ∂ β η α + ∂ α η 3 ∂ β η 3 ) + ε 3 3 a αβστ ∂ στ η 3 ∂ αβ η 3 dy -L(η),
where L(η) := ω p i η i dyω q α ∂ α η 3 dy, and let the space V(ω) be defined by

V(ω) := {η = (η i ) ∈ H 1 (ω) × H 1 (ω) × H 2 (ω); η i = ∂ ν η 3 = 0 on γ 0 },
where ∂ ν designates the outer normal derivative operator along γ (the operator ∂ ν is well-defined dγ-almost everywhere along γ, since the unit outer normal vector is itself well-defined dγ-almost everywhere along the boundary of a domain). Then, according to the Kirchhoff-von Kármán-Love theory of a nonlinearly elastic plate (see, e.g., Ciarlet [START_REF] Ciarlet | Theory of Plates[END_REF]), the vector field η * = (η * i ), where η * i are the Cartesian components of the displacement vector field of the middle surface ω of the plate, should be the solution of the following minimization problem:

η * ∈ V(ω) and J(η * ) = inf η∈V(ω)

J(η).

If 0 < dγ-meas γ 0 ≤ dγ-meas γ, this minimization problem has at least one solution if the norms p α L 2 (ω) are small enough (see Ciarlet & Destuynder [START_REF] Ciarlet | A justification of a nonlinear model in plate theory[END_REF]; see also Nečas & Naumann [20] in the special case where p α = 0). If

γ 0 = ∅, in which case V(ω) = H 1 (ω) × H 1 (ω) × H 2 (
ω), this minimization problem has a solution if (and only if) the components p i and q α of the resultants and couples of the applied forces satisfy ad hoc compatibility conditions, and if the norms p α L 2 (ω) are again small enough (see Ciarlet & S. Mardare [START_REF] Ciarlet | Saint-Venant compatibility conditions and the intrinsic approach for nonlinearly elastic plates[END_REF]); note that the solution is never unique in this case, however (see the discussion given in ibid.).

Let L 2 (ω) denote the space of all 2 × 2 symmetric tensor fields with components in L 2 (ω). The nonlinear part of the integrand appearing in the functional J is a function of the change of metric tensor field (E αβ ) ∈ L 2 (ω) and of the change of curvature tensor field (F αβ ) ∈ L 2 (ω), the components of which are respectively defined for any vector field η ∈ V(ω) by

E αβ := 1 2 (∂ α η β + ∂ β η α + ∂ α η 3 ∂ β η 3 ) and F αβ := ∂ αβ η 3 2
(despite their names, which we adopt here because they are commonly used in nonlinear plate theory, these tensors are in effect only ad hoc approximations of the "true" change of metric and change of curvature tensors of the middle surface). By contrast with the classical approach described above, where the Cartesian components η * i of the displacement field are the unknowns, an intrinsic approach to the same problem consists instead in considering the components

E * αβ := 1 2 (∂ α η * β + ∂ β η * α + ∂ α η * 3 ∂ β η * 3 ) and F * αβ := ∂ αβ η * 3
of the corresponding change of metric and change of curvature tensor fields as the unknowns. The intrinsic approach thus provides a direct way to compute the stress resultants n αβ and the stress couples m αβ inside the plate since these are respectively defined as n αβ := a αβστ E * στ and m αβ := a αβστ F * στ . This feature constitutes an advantage of the intrinsic approach over the classical approach, inasmuch as the stress resultants n αβ and the stress couples m αβ are often considered to be the unknowns of interest (rather than the components η * i of the displacement vector field) in the computation of elastic structures; in this direction, see notably the pioneering contributions of W. Pietraszkiewicz and his school [START_REF] Opoka | Intrinsic equations for non-linear deformation and stability of thin elastic shells[END_REF].

In order to recast the minimization problem of the classical approach into one of the intrinsic approach, the first objective thus consists in finding necessary and sufficient conditions guaranteeing that, given two tensor fields (E αβ ) ∈ L 2 (ω) and (F αβ ) ∈ L 2 (ω), there exists a vector field η

= (η i ) ∈ V(ω) such that 1 2 (∂ α η β + ∂ β η α + ∂ α η 3 ∂ β η 3 ) = E αβ and ∂ αβ η 3 = F αβ .
A first answer to this question when

γ 0 = ∅, in which case V(ω) = H 1 (ω) × H 1 (ω) × H 2 (ω)
, was recently given in Ciarlet & S. Mardare [START_REF] Ciarlet | Saint-Venant compatibility conditions and the intrinsic approach for nonlinearly elastic plates[END_REF]. There, it was shown that, if the open set ω is simply-connected, a first class of such necessary and sufficient conditions can be expressed as a system of nonlinear partial differential equations in the sense of distributions, which take the form of the following nonlinear Saint-Venant compatibility conditions:

∂ στ E αβ + ∂ αβ E στ -∂ ασ E βτ -∂ βτ E ασ = F ασ F βτ -F αβ F στ in H -2 (ω), ∂ σ F αβ = ∂ β F ασ in H -1 (ω).
In this paper, we show that, when γ 0 = γ, in which case V(ω) = H 1 0 (ω) × H 1 0 (ω) × H 2 0 (ω), necessary and sufficient nonlinear compatibility conditions "of Donati type" can be found (Theorem 3.1), which this time take the form of nonlinear variational equations, with divergence-free tensor fields as test-functions; note that, at least to the authors' best knowledge, these constitute a first example of nonlinear compatibility conditions of Donati type.

Once such compatibility conditions are identified and justified, they in turn allow to recast the corresponding minimization problem into one of the intrinsic approach (Theorem 4.2) Some of the results of this paper were announced (mostly without proofs) in Ciarlet, Geymonat & Krasucki [START_REF] Ciarlet | Nonlinear Donati compatibility conditions for the nonlinear Kirchhoff-von Kármán-Love plate theory[END_REF].

Technical preliminaries

This section gathers various preliminary results that will all be used in Section 3 for identifying the nonlinear Donati compatibility conditions just alluded to.

Given a domain ω ⊂ R 2 , the notations H m (ω), H -m (ω), and H m 0 (ω), m ≥ 1, designate the usual Sobolev spaces and their dual spaces, the notations H 1/2 ( γ) and H -1/2 ( γ) designate the usual trace space and its dual space over any connected component γ of the boundary of ω, the notation D(ω) designates the space of infinitely differentiable functions with compact support in ω, and D ′ (ω) designates the space of distributions in ω.

Vector and matrix fields are designated by boldface letters. Spaces of vector, resp. symmetric matrix, fields are designated by boldface, resp. special Roman, letters.

Given a normed vector space V, the notation V ′ designates its dual space, and

V ′ •, • V designates the duality between V and V ′ . If W is a subspace of V, the notation W ֒→ V, resp. W ⋐ V, means that the canonical injection from W into V is continuous, resp. compact.
The following linear Saint-Venant compatibility conditions constitute an extension of a classical result for smooth functions to distributions. Theorem 2.1. Let there be given a simply-connected domain ω ⊂ R 2 and a symmetric matrix field (F αβ ) ∈ L 2 (ω).

Then there exists a function w ∈ H 2 (ω) such that

∂ αβ w = F αβ in L 2 (ω), if and only if ∂ σ F αβ -∂ β F ασ = 0 in H -1 (ω).
If this is the case, any other function

w ∈ H 2 (ω) such that ∂ αβ w = F αβ in L 2 (ω) is of the form w(y) = w(y) + α 0 + α 1 y 1 + α 2 y 2 for almost all y = (y α ) ∈ ω,
for some constants a 0 , α 1 , and α 2 .

Proof. It is clear that, given any domain ω ⊂ R 2 (i.e., simply-connected or not) and given any function w ∈ H 2 (ω), the matrix field (

F αβ ) := (∂ αβ w) ∈ L 2 (ω) satisfies ∂ σ F αβ = ∂ β F ασ in H -1 (ω).
To establish the converse property, we first observe that all the relations

∂ σ F αβ -∂ β F ασ = 0 in H -1 (ω) (i.e.
, for all α, β, σ ∈ {1, 2}) are satisfied if (and only if)

∂ 1 F 12 -∂ 2 F 11 = 0 and ∂ 1 F 22 -∂ 2 F 21 = 0.
The weak Poincaré lemma of the form established in Ciarlet & Ciarlet, Jr. [START_REF] Ciarlet | Another approach to linearized elasticity and a new proof of Korn's inequality[END_REF] (a simpler proof was subsequently given by Kesavan [START_REF] Kesavan | On Poincaré's and J.L. Lions' lemmas[END_REF]; see also Theorem 6.17-4 in Ciarlet [START_REF] Ciarlet | Linear and Nonlinear Functional Analysis with Applications[END_REF]) asserts that, given any vector field

(h α ) ∈ H -1 (ω) that satisfies ∂ α h β -∂ β h α = 0 in H -2 (ω),
there exists a function θ ∈ L 2 (ω), unique up to the addition of a constant function, such that

∂ α θ = h α in H -1 (ω).
Note that the assumption of simple-connectedness of the domain ω is an essential assumption in this lemma, as its proof relies on the "classical" Poincaré lemma (i.e., for smooth functions).

Since

∂ 1 F 12 -∂ 2 F 11 = 0, resp. ∂ 1 F 22 -∂ 2 F 21 = 0, in H -1 (ω),
there thus exists a function θ ∈ H 1 (ω), resp. χ ∈ H 1 (ω), such that

∂ 1 θ = F 11 and ∂ 2 θ = F 12 , resp. ∂ 1 χ = F 21 and ∂ 2 χ = F 22 , in L 2 (ω).
Since then

∂ 1 χ -∂ 2 θ = F 21 -F 12 = 0 in L 2 (ω),
another application of the same lemma shows that there exists a function w ∈ H 2 (ω) such that

∂ 1 w = θ and ∂ 2 w = χ in H 1 (ω),
hence such that

∂ 11 w = ∂ 1 θ = F 11 , ∂ 12 w = ∂ 1 χ = F 21 , ∂ 21 w = ∂ 2 θ = F 12 , ∂ 22 w = ∂ 2 χ = F 22 in L 2 (ω).
Such a function w is unique up to the addition of a polynomial of degree ≤ 1 in the variable y = (y α ) since any distribution T ∈ D ′ (ω) satisfying ∂ αβ T = 0 in ω is a polynomial of degree ≤ 1 in y (recall that a domain is connected by assumption).

A simple re-writing of the compatibility conditions of Theorem 2.1 shows that the function w found in this theorem can be also viewed as an Airy function. Recall that the divergence of a 2 × 2 tensor field S = (S αβ ) defined over a two-dimensional open set is the vector field div S := (∂ β S αβ ). Theorem 2.2. Let there be given a simply-connected domain ω ⊂ R 2 and a symmetric tensor field S = (S αβ ) ∈ L 2 (ω).

Then there exists a function w ∈ H 2 (ω) such that

∂ 11 w = S 22 , ∂ 12 w = -S 12 , ∂ 22 w = S 11 in L 2 (ω), if and only if div S = 0 in H -1 (ω).
If this is the case, the function w is uniquely determined up to the addition of a polynomial of degree ≤ 1 in the variable y = (y α ).

Proof. The relations

∂ 1 F 12 -∂ 2 F 11 = 0 and ∂ 1 F 22 -∂ 2 F 21 = 0 in H -1
(ω) used in the proof of Theorem 2.1 can be equivalently re-written as

div S = 0 in H -1 (ω), where S = (S αβ ) := F 22 -F 21 -F 12 F 11 ∈ L 2 (ω).
Hence the conclusions immediately follow from Theorem 2.1.

Given a symmetric matrix field (S αβ ) defined over ω, any function w such that ∂ 11 w = S 22 , ∂ 12 w = -S 12 , and ∂ 22 w = S 11 is called an Airy function for the field (S αβ ).

When the domain ω is not simply-connected, an Airy function still exists, provided the compatibility condition div S = 0 in H -1 (ω) found in Theorem 2.2 is complemented by other conditions, according to the following result due to Geymonat & Krasucki [START_REF] Geymonat | On the existence of the Airy function in Lipschitz domains. Application to the traces of H 2[END_REF], which constitutes a weak version of a classical result for smooth functions (see, e.g., Ciarlet & Rabier [START_REF] Ciarlet | Les Equations de von Kármán[END_REF]).

Theorem 2.3. Let ω ⊂ R 2 be a non-simply connected domain whose boundary γ consists of q ≥ 2 connected components γ i , 1 ≤ i ≤ q, let (ν α ) denote the unit outer normal vector along γ = q i=1 γ i , and let the functions p 0 , p 1 , p 2 : ω → R be respectively defined by p 0 (y) = 1, p 1 (y) = y 1 , p 2 (y) = y 2 for each y = (y α ) ∈ ω.

Let S = (S αβ ) ∈ L 2 (ω) be a symmetric tensor field. Then there exists a function w ∈ H 2 (ω) such that

∂ 11 w = S 22 , ∂ 12 w = -S 12 , ∂ 22 w = S 11 in L 2 (ω),
if and only if

div S = 0 in H -1 (ω), H -1/2 (γ i ) S αβ ν β , p 0 H 1/2 (γ i ) = 0, 1 ≤ i ≤ q, H -1/2 (γ i ) S 1β ν β , p 2 H 1/2 (γ i ) = H -1/2 (γ i ) S 2β ν β , p 1 H 1/2 (γ i ) , 1 ≤ i ≤ q.
If this is the case, the function w is uniquely determined up to the addition of a polynomial of degree ≤ 1 in the variable y.

Note that duality brackets such as H -1/2 (γ i ) S αβ ν β , p 0 H 1/2 (γ i ) are indeed well-defined, since any tensor field S ∈ L 2 (ω) satisfying div S = 0 in H -1 (ω) belongs to the space H(div; ω); consequently, each "restriction of S αβ ν β to γ i " is well defined as a distribution in the space H -1/2 (γ i ) (see, e.g., Girault & Raviart [START_REF] Girault | Finite Element Methods for Navier-Stokes Equations[END_REF] or Brezzi & Fortin [START_REF] Brezzi | Mixed and Hybrid Finite Element Methods[END_REF]).

The compatibility conditions guaranteeing that the components F αβ of a symmetric 2 × 2 matrix field in L 2 (ω) can be written as F αβ = ∂ αβ w for some function w ∈ H 2 (ω) were identified in Theorem 2.1 as a system of partial differential equations. We now show (Theorem 2.4(a)) that it is possible to find linear compatibility conditions, again bearing on the functions F αβ , of a completely different nature that achieve the same purpose. These different compatibility conditions are of Donati type, in the sense that they take the form of variational equations to be satisfied by divergence-free tensor fields. We also identify (Theorem 2.4(b)) another class of linear compatibility conditions of Donati type that will be used in the sequel.

Note that, by contrast with the compatibility conditions of Theorem 2.1, those of Theorem 2.4 below no longer require that the domain ω be simply-connected. Then there exists a function η ∈ H 2 0 (ω) such that

∂ αβ η = F αβ in L 2 (ω), if and only if ω F αβ T αβ dy = 0 for all (T αβ ) ∈ L 2 (ω) such that div div T := ∂ αβ T αβ = 0 in H -2 (ω).
If this is the case, the function η is uniquely determined. (b) Let there be given a domain ω ⊂ R 2 and a tensor field (e αβ ) ∈ L 2 (ω).

Then there exists a vector field

(η α ) ∈ H 1 0 (ω) × H 1 0 (ω) such that 1 2 (∂ α η β + ∂ β η α ) = e αβ in L 2 (ω),
if and only if

ω e αβ s αβ dy = 0 for all S = (s αβ ) ∈ L 2 (ω) such that div S = (∂ β s αβ ) = 0 in H -1 (ω).
If this is the case, the vector field (η α ) is uniquely determined.

Proof. Assume first that

F αβ = ∂ αβ η for some η ∈ H 2 0 (ω), so that ω F αβ T αβ dy = L 2 (ω) T αβ , ∂ αβ η L 2 (ω) = H -2 (ω) ∂ αβ T αβ , η H 2 0 (ω)
for each tensor field (T

αβ ) ∈ L 2 (ω). Hence ω F αβ T αβ dy = 0 if ∂ αβ T αβ = 0 in H -2 (ω)
. This proves the "only if" part of Theorem 2.4(a). Next, let

Aη := ∂ 11 η ∂ 12 η ∂ 21 η ∂ 22 η ∈ L 2 (ω) for each η ∈ H 2 0 (ω).
Then the linear operator A : H 2 0 (ω) → L 2 (ω) defined in this fashion is clearly continuous. Besides, since the semi-

norm η ∈→ α, β ∂ αβ η 2 L 2 (ω) 1/2
is a norm over the space H 2 0 (ω) which is equivalent to the norm • H 2 (ω) over this space, there exists a constant C such that

η H 2 (ω) ≤ C Aη L 2 (ω) for all η ∈ H 2 0 (ω).
Consequently, the image Im A of H 2 0 (ω) under A is closed in L 2 (ω). For any matrix field T = (T αβ ) ∈ L 2 (ω) and any function η ∈ H 2 0 (ω), we have

L 2 (ω) T, Aη L 2 (ω) := L 2 (ω) T αβ , ∂ αβ η L 2 (ω) = H -2 (ω) ∂ αβ T αβ , η H 2 0 (ω) = H -2 (ω) div div T, η H 2 0 (ω) ,
which shows that the dual operator of A :

H 2 0 (ω) → L 2 (ω) is div div : L 2 (ω) → H -2 (ω).

Banach closed range theorem therefore implies that

Im A = {F ∈ L 2 (ω); L 2 (ω) F, T L 2 (ω) = 0 for all T ∈ Ker (div div)}, which is exactly what the "if part" of Theorem 2.4(a) asserts.

That Ker A = {0} implies that the function denoted η is uniquely determined. This proves (a). The proof of (b) is well-known; see, e.g. Geymonat & Suquet [START_REF] Geymonat | Functional spaces for Norton-Hoff materials[END_REF], Geymonat & Krasucki [START_REF] Geymonat | Some remarks on the compatibility conditions in elasticity[END_REF][START_REF] Geymonat | Hodge decomposition for symmetric matrix fields and the elasticity complex in Lipschitz domains[END_REF], or Amrouche, Ciarlet, Gratie & Kesavan [START_REF] Amrouche | On the characterizations of matrix fields as linearized strain tensor fields[END_REF]. 6

Finally, we establish a specific Green's formula.

Theorem 2.5. Let ω be a domain in R 2 . Then, for any functions η ∈ H 2 0 (ω) and w ∈ H 2 (ω)

ω (∂ 11 η∂ 22 η -∂ 12 η∂ 12 η)w dy = ω - 1 2 (∂ 1 η) 2 ∂ 22 w - 1 2 (∂ 2 η) 2 ∂ 11 w + ∂ 1 η∂ 2 η∂ 12 w dy.
Proof. Considered as functions of (η, w) ∈ H 2 0 (ω) × H 2 (ω), both sides of the above relation are continuous (the space H 2 (ω) is continuously imbedded in the space W 1,4 (ω) since ω is a two-dimensional domain; this shows that the trilinear form in the right-hand side of this relation is continuous). Since D(ω) is dense in H 2 0 (ω) it thus suffices to establish the Green's formula for functions η ∈ D(ω) and w ∈ H 2 (ω). Using the usual formulas of Sobolev spaces (which is licit since ω is assumed to be a domain; cf., e.g., Nečas [START_REF] Nečas | Direct Methods in the Theory of Elliptic Equations[END_REF]) and noting that all the boundary integrals vanish in these formulas if η ∈ D(ω), we obtain, for any η ∈ D(ω) and w ∈ H 2 (ω),

1 2 ω (∂ 1 η) 2 ∂ 22 w dy = - 1 2 ω ∂ 2 (∂ 1 η) 2 ∂ 2 w dy = - ω ∂ 1 η∂ 12 η∂ 2 w dy = ω ∂ 2 (∂ 1 η∂ 12 η) w dy = ω (∂ 12 η∂ 12 η)w dy + ω (∂ 1 η∂ 122 η)w dy, 1 2 ω (∂ 2 η) 2 ∂ 11 w dy = ω (∂ 12 η∂ 12 η)w dy + ω (∂ 2 η∂ 112 η)w dy, ω ∂ 1 η∂ 2 η∂ 12 w dy = - ω ∂ 1 (∂ 1 η∂ 2 η)∂ 2 w dy = - ω (∂ 11 η∂ 2 η + ∂ 1 η∂ 12 η)∂ 2 w dy = ω ∂ 2 (∂ 11 η∂ 2 η) + ∂ 2 (∂ 1 η∂ 12 η) w dy = ω ∂ 112 η∂ 2 η + ∂ 11 η∂ 22 η + ∂ 12 η∂ 12 η + ∂ 1 η∂ 122 η w dy,
from which the announced Green's formula follows.

Nonlinear Donati compatibility conditions

The following theorem constitutes the first main result of this paper. For brevity, it is stated and established for a simply-connected domain ω, but it should be clear that a similar result (based on Theorem 2.3 instead of on Theorem 2.2) holds if ω is not simply-connected. Let there be given two matrix fields (E αβ ) ∈ L 2 (ω) and (F αβ ) ∈ L 2 (ω). Then there exists a vector field

(η 1 , η 2 , η 3 ) ∈ H 1 0 (ω) × H 1 0 (ω) × H 2 0 (ω) such that 1 2 (∂ α η β + ∂ β η α + ∂ α η 3 ∂ β η 3 ) = E αβ in L 2 (ω), ∂ αβ η 3 = F αβ in L 2 (ω),
if and only if the following nonlinear Donati compatibility conditions are satisfied:

ω F αβ T αβ dy = 0 for all T = (T αβ ) ∈ L 2 (ω) such that div div T = 0 in H -2 (ω), ω {E αβ S αβ + (det F)Φ(S)} dy = 0 for all S = (S αβ ) ∈ L 2 (ω) such that div S = 0 in H -1 (ω).
If this is the case, such a vector field (η 1 , η 2 , η 3 ) is uniquely determined.

Proof. The function w found in Theorem 2.2 is unique up to the addition of a polynomial of degree ≤ 1 in y; hence w becomes uniquely determined if it is subjected to satisfy in addition the relations ω w dy = ω ∂ α w dy = 0. This shows that the mapping Φ is well-defined. So, let two tensor fields (E αβ ) ∈ L 2 (ω) and (F αβ ) ∈ L 2 (ω) be given that satisfy the above nonlinear Donati compatibility conditions. We then first infer from Theorem 2.4(a) that there exists a uniquely determined function

η 3 ∈ H 2 0 (ω) such that F αβ = ∂ αβ η 3 in L 2 (ω).
Second, let S = (S αβ ) ∈ L 2 (ω) be a tensor field that satisfies div S = 0 in H -1 (ω). Hence there exists by Theorem 2.2 one and only one function w ∈ H 2 (ω) such that

S 11 = ∂ 22 w, S 12 = -∂ 12 w, S 22 = ∂ 11 w in L 2 (ω),
= ω E 11 - 1 2 (∂ 1 η 3 ) 2 ∂ 22 w -2 E 12 - 1 2 ∂ 1 η 3 ∂ 2 η 3 ∂ 12 w + E 22 - 1 2 (∂ 2 η 3 ) 2 ∂ 11 w dy = ω E 11 - 1 2 (∂ 1 η 3 ) 2 S 11 + 2 E 12 - 1 2 ∂ 1 η 3 ∂ 2 η 3 S 12 + E 22 - 1 2 (∂ 2 η 3 ) 2 S 22 dy.
Since this last relation holds for all S = (S αβ ) ∈ L 2 (ω) such that ∂ β S αβ = 0 in H -1 (ω), there exists a uniquely determined vector field (η α ) ∈ H 1 0 (ω) × H 1 0 (ω) such that (Theorem 2.4(b))

E αβ - 1 2 ∂ α η 3 ∂ β η 3 = 1 2 (∂ α η β + ∂ β η α ) in L 2 (ω).
This completes the proof of the "if part". The "only if part" follows from Theorem 2.4(a) for the first Donati compatibility conditions and by reversing the above computations for the second one.

Remark

As expected, the "linearization" of the nonlinear Donati compatibility conditions found in Theorem 3.1, which simply consists in deleting the nonlinear term ω (det F)Φ(S) dy, reduces to the linear Donati compatibility conditions of Theorem 2.4.

The intrinsic approach to nonlinear plate theory by means of Donati compatibility conditions

Our analysis of the intrinsic approach applied to the minimization problem of Section 1 in the special case where γ 0 = γ is essentially based on the properties of a specific set T(ω) of admissible tensor fields and of a specific nonlinear mapping acting from T(ω) onto the space H 1 0 (ω) × H 1 0 (ω) × H 1 0 (ω), the definitions and properties of which are the object of the next theorem. For simplicity, we again consider the case where the domain ω is simply-connected, but the extension to general domains (based on Theorem 2.3 instead of Theorem 2.2) is clearly possible.

Since the proof of the next theorem is similar to that of Theorem 5.1 in Ciarlet & S. Mardare [START_REF] Ciarlet | Saint-Venant compatibility conditions and the intrinsic approach for nonlinearly elastic plates[END_REF], it is only sketched.

= (η i ) ∈ H 1 0 (ω) × H 1 0 (ω) × H 2 0 (ω) such that 1 2 (∂ α η β + ∂ β η α + ∂ α η 3 ∂ β η 3 ) = E αβ in L 2 (ω), ∂ αβ η 3 = F αβ in L 2 (ω). Let F : (E, F) ∈ T(ω) → η ∈ H 1 0 (ω) × H 1 0 (ω) × H 2 0 (ω)
denote the mapping defined in this fashion. Then F maps weakly convergent sequences in the set T(ω) into sequences that strongly converge in the space H 1 0 (ω)×H 1 0 (ω)×H 2 0 (ω) endowed with the norm of the space L 2 (ω) × L 2 (ω) × H 1 (ω).

Proof. (i) The proof rests on the following nonlinear Korn's inequality: there exists a constant C such that

η H 1 (ω)×H 1 (ω)×H 2 (ω) ≤ C 1 2 (∂ α η β + ∂ β η α + ∂ α η 3 ∂ β η 3 ) L 2 (ω) + (∂ αβ η 3 ) L 2 (ω) + (∂ αβ η 3 ) 2 L 2 (ω)
for all η = (η 1 ) ∈ H 1 0 (ω) × H 1 0 (ω) × H 2 0 (ω). To prove this inequality, it suffices to use the two-dimensional Korn's inequality for vector fields (η α ) ∈ H 1 0 (ω) × H 1 0 (ω), the inequality

(∂ α η β + ∂ β η α ) L 2 (ω) ≤ ∂ α η β + ∂ β η α + ∂ α η 3 ∂ β η 3 L 2 (ω) + ∂ α η 3 ∂ β η 3 L 2 (ω) , 9
and the continuous injection H 1 (ω) ֒→ L 4 (ω) (which holds since ω is a two-dimensional domain), the combination of which implies that there exists a constant c such that

∂ α η 3 ∂ β η 3 L 2 (ω) ≤ ∂ α η 3 L 4 (ω) ∂ β η 3 L 4 (ω) ≤ c ∂ β η 3 2 H 1 (ω) ≤ c η 3 2 H 2 (ω) .
Since the mapping F : T(ω) → H 1 0 (ω) × H 1 0 (ω) × H 2 0 (ω) is one-to-one and onto by Theorem 3.1, the above Korn's inequality can be immediately converted into an inequality for elements in the set T(ω), viz.,

F(E, F) H 1 (ω)×H 1 (ω)×H 2 (ω) ≤ C E L 2 (ω) + F L 2 (ω) + F 2 L 2 (ω)
for all (E, F) ∈ T(ω).

(ii) Let ((E k , F k )) ∞ k=1 be a sequence of elements (E k , F k ) ∈ T(ω) that weakly converges to (E, F) ∈ L 2 (ω) × L 2 (ω). Since this sequence is then bounded in L 2 (ω) × L 2 (ω), the sequence (η k ) ∞ k=1 , where η k := F(E k , F k ), is bounded in the space H 1 0 (ω) × H 1 0 (ω) × H 2 0 (ω) by (i). Hence there exist a subsequence (η ℓ ) ∞ ℓ=1 and η ∈ H 1 0 (ω) × H 1 0 (ω) × H 2 0 (ω) such that η ℓ ⇀ η in H 1 0 (ω) × H 1 0 (ω) × H 2 0 (ω) as ℓ → ∞
, where ⇀ denotes weak convergence. Therefore,

η ℓ → η in L 2 (ω) × L 2 (ω) × H 1 (ω) as ℓ → ∞,
thanks to the compact injections H 1 (ω) ⋐ L 2 (ω) and H 2 (ω) ⋐ H 1 (ω) (which hold since ω is a two-dimensional domain). From these properties, it is then easy to conclude that η = F(E, F), hence that (E, F) ∈ T(ω). Consequently, the set T(ω) is sequentially weakly closed in L 2 (ω) × L 2 (ω).

Finally, the uniqueness of the limit shows that the whole sequence (η k ) ∞ k=1 converges strongly to η in the space

L 2 (ω) × L 2 (ω) × H 1 (ω).
Thanks to Theorem 4.1, the minimization problem of the nonlinear Kirchhoff-von Kármán-Love theory when γ 0 = γ, i.e., with the displacement field η * = (η * i ) ∈ H 1 0 (ω) × H 1 0 (ω) × H 2 0 (ω) as the unknown (Section 1), can now be recast as one of the intrinsic approach to this theory, i.e., as a minimization problem with

E * αβ := 1 2 (∂ α η * β + ∂ β η * α + ∂ α η * 3 ∂ β η * 3 ) ∈ L 2 (ω) and F * αβ := ∂ αβ η * 3 ∈ L 2 (ω)
as the new unknowns. This is the object of the next theorem, where the existence and uniqueness of the solution to this new minimization problem are also established; this result constitutes the second main result of this paper.

Recall that the constants a αβστ are the components of the two-dimensional elasticity tensor of the plate and that the functions p i ∈ L 2 (ω) and q α ∈ L 2 (ω) designate the resultants and couples acting on the plate (Section 1). Theorem 4.2. Let the space T(ω) and the mapping F : T(ω) → H 1 0 (ω) × H 1 0 (ω) × H 2 0 (ω) be defined as in Theorem 4.1 and let the functional I :

T(ω) → R be defined for each (E, F) = ((E αβ ), (F αβ )) ∈ T(ω) by I(E, F) := 1 2 ω εa αβστ E στ E αβ + ε 3 3 a αβστ F στ F αβ dy -L(F(E, F)) where L(η) := ω p i η i dy - ω q α ∂ α η 3 dy for each η = (η i ) ∈ H 1 0 (ω) × H 1 0 (ω) × H 2 0 (ω).
Then, if the norms p α L 2 (ω) are small enough, there exists at least one element

(E * , F * ) = ((E * αβ ), (F * αβ )) ∈ T(ω) such that I(E * , F * ) = inf (E,F)∈T(ω)

I(E, F).

Besides,

E * αβ := 1 2 (∂ α η * β + ∂ β η * α + ∂ α η * 3 ∂ β η * 3 ) ∈ L 2 (ω) and F * αβ := ∂ αβ η * 3 ∈ L 2 (ω), 10
where the vector field η

* ∈ H 1 0 (ω) × H 1 0 (ω) × H 2 0 (ω) satisfies J(η * ) = inf η∈H 1 0 (ω)×H 1 0 (ω)×H 2 0 (ω) J(η),
and the functional J : (ii) In what follows, the constants C 1 , C 2 , and C 3 are independent of the various functions, vector fields, or tensor fields, appearing in a given inequality. First, it is clear that

H 1 0 (ω) × H 1 0 (ω) × H 2 0 (ω) → R is defined for each η = (η i ) ∈ H 1 0 (ω) × H 1 0 (ω) × H 2 0 (ω) by J(η) := 1 2 ω ε 4 a αβστ (∂ σ η τ + ∂ τ η σ + ∂ σ η 3 ∂ τ η 3 )(∂ α η β + ∂ β η α + ∂ α η 3 ∂ β η 3 ) + ε 3 3
I(E, F) ≥ C 1 E 2 L 2 (ω) + F 2 L 2 (ω) -(p α ) L 2 (ω) (η α ) L 2 (ω) -p 3 L 2 (ω) η 3 L 2 (ω) -(q α ) L 2 (ω) (∂ α η 3 ) L 2 (ω) ,
for all (E, F) ∈ T(ω), where η = F(E, F). Second, as already noted in the proof of Theorem 4.1, (η α ) L 2 (ω) ≤ (η α ) H 1 (ω) ≤ C 2 E L 2 (ω) + F 2 L 2 (ω) , so that

I(E, F) ≥C 1 E 2 L 2 (ω) + C 1 -C 2 (p α ) L 2 (ω) F 2 L 2 (ω) -C 2 (p α ) L 2 (ω) E L 2 (ω) -C 3 (p 3 ) L 2 (ω) + (q α ) L 2 (ω) F L 2 (ω)
for all (E, F) ∈ T(ω). Hence the functional I : T(ω) → R is coercive if the norms p α L 2 (ω) are small enough.

(iii) By a standard result from the calculus of variations (see, e.g., Theorem 3.30 in Dacorogna [START_REF] Dacorogna | Direct Methods in the Calculus of Variations[END_REF] or Theorem 9.3-1 in [START_REF] Ciarlet | Linear and Nonlinear Functional Analysis with Applications[END_REF]), there thus exists at least one minimizer (E * , F * ) of the functional I in the set T(ω).

(iv) Given such a minimizer (E * , F * ) ∈ T(ω), the definitions of the functionals I and J clearly imply that the vector field F(E * , F * ) minimizes the functional J over the space H 1 0 (ω) × H 1 0 (ω) × H 2 0 (ω). Note that a similar intrinsic approach could be a fortiori applied to the Kirchhoff-von Kármán-Love theory for a linearly elastic plate, i.e., where the change of metric tensor

Theorem 2 . 4 .

 24 (a) Let there be given a domain ω ⊂ R 2 and a tensor field (F αβ ) ∈ L 2 (ω).

Theorem 3 . 1 .

 31 Let ω be a simply-connected domain in R 2 . Given a matrix field S ∈ L 2 (ω) that satisfiesdiv S = 0 in H -1 (ω),there exists a unique function w ∈ H 2 (ω) such that ∂ 11 w = S 22 , ∂ 12 w = -S 12 , ∂ 22 w = S 11 in L 2 (ω), and ω w dy = ω ∂ α w dy = 0.LetΦ := S ∈ L 2 (ω); div S = 0 in H -1 (ω) → ψ ∈ H 2 (ω);ω ψ dy = ω ∂ α ψ dy = 0 denote the mapping defined in this fashion, i.e., by Φ(S) := w.

F 11 F

 11 dy = 0. Consequently, for any such tensor field S, ω E αβ S αβ dy = ω {E 11 ∂ 22 w + E 22 ∂ 11 w -2E 12 ∂ 12 w} dy, on the one hand, and ω (det F)Φ(S) dy = ω 22 -(F 12 ) 2 w dy = ω (∂ 11 η 3 ∂ 22 η 3 -∂ 12 η 3 ∂ 22 η 3 )w dy, on the other hand. Using the Green's formula of Theorem 2.5, we can therefore re-write the left-hand side of the second Donati compatibility condition as ω E αβ S αβ + det(F)Φ(S) dy

Theorem 4 . 1 .F

 41 Let ω be a simply-connected domain in R 2 . (a) Define the setT(ω) := {((E αβ , (F αβ )) ∈ L 2 (ω) × L 2 (ω); ω αβ T αβ dy = 0 for all T = (T αβ ) ∈ L 2 (ω) satisfying div div T = 0 in H -2 (ω), ω {E αβ S αβ + (det F)Φ(S)} dy = 0 for all S = (S αβ ) ∈ L 2 (ω) satisfying div S = 0 in H -1 (ω)where Φ is the mapping defined in Theorem 3.1. Then the set T(ω) is sequentially weakly closed in the space L 2 (ω) × L 2 (ω).(b) Given any ((E αβ , F αβ )) ∈ T(ω), there exists by Theorem 3.1 a unique vector field η

1 2 (

 2 ∂ α η β + ∂ β η α + ∂ α η 3 ∂ β η 3 ) is replaced by the linearized change of metric tensor 1 2 (∂ α η β + ∂ β η α ) .In this case, the nonlinear Donati compatibility conditions of Theorem 3.1 are to be replaced by their linearized version found in Theorem 2.4.

  a αβστ ∂ στ η 3 ∂ αβ η 3 dy -L(η).Proof.(i) The function (E, F) ∈ T(ω) → 1 2 ω εa αβστ E στ E αβ + ε 3 3 a αβστ F στ F αβ dy, which is clearly continuous,and convex since the assumed inequalities λ ≥ 0 and µ > 0 imply the existence of a constant α > 0 such that a αβστ t στ t αβ ≥ αt αβ t αβ for all 2 × 2 symmetric matrices (t αβ ), is therefore sequentially weakly lower semi-continuous (see, e.g., Theorem 9.2-3 in[START_REF] Ciarlet | Linear and Nonlinear Functional Analysis with Applications[END_REF]).

The function (E, F) ∈ T(ω) → L(F(E, F)) ∈ R is sequentially weakly continuous since, by Theorem 4.1,

(E k , F k ) ⇀ k→∞ (E, F) in T(ω) implies F(E k , F k ) → k→∞ F(E, F) in L 2 (ω) × L 2 (ω) × H 1 (ω),

and the linear form

L : L 2 (ω) × L 2 (ω) × H 1 (ω) → R is continuous.

Consequently, the functional I : T(ω) → R is sequentially weakly lower semi-continuous.

Acknowledgement

This work was supported by a grant from the Research Grants Council of the Hong Kong Special Administrative Region, China [Project No. 9041738-CityU 100612].


Philippe Leroy
email: p.leroy@brgm.fr

Guillaume Mériguet

Eggon Zimmermann

Johanalexander A Huisman

Low frequency complex

come

The accurate knowledge of the textural and electrochemical properties of clay suspensions is of great importance for many applications in soil and environmental chemistry involving the use of clay-rich porous media. These applications concern, for example,the understanding of the reactive transport properties of bentonites and argillites for the long-term prediction of the diffusivity of ionic species around radioactive waste repositories 1 . Na-montmorillonite suspensions are characterized by an anisotropic surface charge distribution: a permanent high surface charge of their basal planes and a pH and salinity dependent surface charge of their edge planes. Current complex impedance measurements of charged suspensions are limited at low frequencies (typically<1 kHz) by the electrochemical polarization of the current and potential electrodes and by the phase accuracy of the impedancemeter.A new four electrodes sample holder (separate current injection and potential electrodes,Fig. 1a) and a very high accuracy impedancemeter 2 are proposed to improve the low-frequency complex impedance measurements of charged Na-montmorillonitesuspensions. Low-frequency ([0.1 Hz 45 kHz]) complex impedance measurements,at various clay volume fractions (0.1, 0.2, 0.4, 0.6, and 2%), pH (5,7,9) and salinities (NaCl, from 10 -4 to 5x10 -2 mol L -1 ), were done to elucidate the textural and electrochemical properties of Na-montmorillonite suspensions. Our results show that the electrochemical polarization of the electrical double layer around the Namontmorilloniteparticles can also be observed at low frequency and high salinity (Fig. 2a). Nevertheless, our measurements are also disturbed by the electromagnetic coupling due to the short triaxial cables, which connect the electrodes to the impedancemeter.

Fig. 1 .

 1 Fig. 1.a. The four electrodes sample holder for the complex impedance measurements and b. the recorded phase of Na-montmorillonite suspensions