A Note on Korn’s Inequality in an N-Dimensional Context and a Global Existence Result for a Non-Linear Plate Model

: In the ﬁrst part of this article, we present a new proof for Korn’s inequality in an n-dimensional context. The results are based on standard tools of real and functional analysis. For the ﬁnal result, the standard Poincaré inequality plays a fundamental role. In the second text part, we develop a global existence result for a non-linear model of plates. We address a rather general type of boundary conditions and the novelty here is the more relaxed restrictions concerning the external load magnitude.


Introduction
In this article, we present a proof for Korn's inequality in R n . The results are based on standard tools of functional analysis and on the Sobolev spaces theory.
We emphasize that such a proof is relatively simple and easy to follow since it is established in a very transparent and clear fashion.
About the references, we highlight that related results in a three-dimensional context may be found in [1]. Other important classical results on Korn's inequality and concerning applications to models in elasticity may be found in [2][3][4]. where we shall also refer throughout the text to the well-known corresponding analogous norm for u ∈ W 1,2 (Ω; R n ).
At this point, we first introduce the following definition.

Definition 1.
Let Ω ⊂ R n be an open, bounded set. We say that ∂Ω isĈ 1 if such a manifold is oriented and for each x 0 ∈ ∂Ω, denotingx = (x 1 , ..., x n−1 ) for a local coordinate system compatible with the manifold ∂Ω orientation, there exist r > 0 and a function f (x 1 , ..., x n−1 ) = f (x) such that is classically defined, almost everywhere also on its concerning domain, so that f ∈ W 1,2 .

Remark 2.
This mentioned set Ω is of a Lipschitzian type, so that we may refer to such a kind of sets as domains with a Lipschitzian boundary, or simply as Lipschitzian sets.
At this point, we recall the following result found in [5], at page 222 in its Chapter 11.

Theorem 1.
Assume Ω ⊂ R n is an open bounded set, and that ∂Ω isĈ 1 . Let 1 ≤ p < ∞, and let V be a bounded open set such that Ω ⊂⊂ V. Then there exists a bounded linear operator such that for each u ∈ W 1,p (Ω) we have: Eu has support in V; 3.
Eu 1,p,R n ≤ C u 1,p,Ω , where the constant depends only on p, Ω, and V.
Finally, as the meaning is clear, we may simply denote Eu = u.

The Main Results, the Korn Inequalities
Our main result is summarized by the following theorem.
Proof. Suppose, to obtain contradiction, that the concerning claim does not hold. Thus, we are assuming that there is no positive real constant C = C(Ω, L) such that (1) is valid.
Proof. Suppose, to obtain contradiction, that the concerning claim does not hold. Hence, for each k ∈ N there exists u k ∈Ĥ 0 such that u k 1,2,Ω > k e(u k ) 0,2,Ω .
The proof is complete.

An Existence Result for a Non-Linear Model of Plates
In the present section, as an application of the results on Korn's inequalities presented in the previous sections, we develop a new global existence proof for a Kirchhoff-Love thin plate model. Previous results on the existence of mathematical elasticity and related models may be found in [2][3][4].
At this point we start to describe the primal formulation.
Let Ω ⊂ R 2 be an open, bounded, connected set which represents the middle surface of a plate of thickness h. The boundary of Ω, which is assumed to be regular (Lipschitzian), is denoted by ∂Ω. The vectorial basis related to the cartesian system {x 1 , x 2 , x 3 } is denoted by (a α , a 3 ), where α = 1, 2 (in general, Greek indices stand for 1 or 2), and where a 3 is the vector normal to Ω, whereas a 1 and a 2 are orthogonal vectors parallel to Ω. Furthermore, n is the outward normal to the plate surface.
The displacements will be denoted bŷ The Kirchhoff-Love relations arê Here It is worth emphasizing that the boundary conditions here specified refer to a clamped plate.
We define the operator Λ : The constitutive relations are given by M αβ (u) = h αβλµ κ λµ (u),  (9) and the external work, represented by F : U → R, is given by where P, P 1 , P 2 ∈ L 2 (Ω) are external loads in the directions a 3 , a 1 , and a 2 , respectively. The potential energy, denoted by J : U → R, is expressed by Finally, we also emphasize from now on, as their meaning are clear, we may denote L 2 (Ω) and L 2 (Ω; R 2×2 ) simply by L 2 , and the respective norms by · 2 . Moreover, derivatives are always understood in the distributional sense, 0 may denote the zero vector in appropriate Banach spaces, and the following and relating notations are used: and N αβ,2 = ∂N αβ ∂x 2 .

On the Existence of a Global Minimizer
At this point, we present an existence result concerning the Kirchhoff-Love plate model. We start with the following two remarks.

Remark 4.
Let {P α } ∈ L ∞ (Ω; R 2 ). We may easily obtain by appropriate Lebesgue integration {T αβ } symmetric and such thatT αβ,β = −P α , in Ω. Indeed, extending {P α } to zero outside Ω if necessary, we may set andT 12 (x, y) =T 21 (x, y) = 0, in Ω. Thus, we may choose a C > 0 sufficiently big, such that {δ αβ } is the Kronecker delta. Therefore, for the kind of boundary conditions of the next theorem, we do not have any restriction for the {P α } norm.
In summary, the next result is new and it is really a step forward concerning the previous one in Ciarlet [3]. We emphasize that this result and its proof through such a tensor {T αβ } are new, even though the final part of the proof is established through a standard procedure in the calculus of variations.
At this point, we present the main theorem in this section.
Proof. Observe that we may find and also such that {T αβ } is positive, definite, and symmetric (please see Remark 4).
we obtain From this, since {T αβ } is positive definite, clearly J is bounded below. Let {u n } ∈ U be a minimizing sequence for J. Thus, there exists α 1 ∈ R such that lim n→∞ J(u n ) = inf u∈U J(u) = α 1 .
The proof is complete.

Conclusions
In this article, we have developed a new proof for Korn's inequality in a specific n-dimensional context. In the second text part, we present a global existence result for a non-linear model of plates. Both results represent some new advances concerning the present literature. In particular, the results for Korn's inequality known so far are for a three-dimensional context such as in [1], for example, whereas we have here addressed a more general n-dimensional case.
In a future research, we intend to address more general models, including the corresponding results for manifolds in R n .