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

Abstract

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.

1. Introduction

In this article, we present a proof for Korn’s inequality in ${\mathbb{R}}^n$. 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].

Remark 1.

Generically, throughout the text we denote

$${\parallel u\parallel }_{0,2,\mathrm{\Omega }}={\left({\int }_{\mathrm{\Omega }}{\left|u\right|}^{2}dx\right)}^{1/2},\forall u\in L^2\left(\mathrm{\Omega }\right),$$

and

$${\parallel u\parallel }_{0,2,\mathrm{\Omega }}={\left(\sum _{j=1}^n{\parallel u_j\parallel }_{0,2,\mathrm{\Omega }}^2\right)}^{1/2},\forall u=(u_1,\dots ,u_n)\in L^2(\mathrm{\Omega };{\mathbb{R}}^n).$$

Moreover,

$${\parallel u\parallel }_{1,2,\mathrm{\Omega }}={\left({\parallel u\parallel }_{0,2,\mathrm{\Omega }}^2+\sum _{j=1}^n{\parallel u_{x_j}\parallel }_{0,2,\mathrm{\Omega }}^2\right)}^{1/2},\forall u\in W^{1,2}\left(\mathrm{\Omega }\right),$$

where we shall also refer throughout the text to the well-known corresponding analogous norm for $u\in W^{1,2}(\mathrm{\Omega };{\mathbb{R}}^n).$

At this point, we first introduce the following definition.

Definition 1.

Let $\mathrm{\Omega }\subset {\mathbb{R}}^n$ be an open, bounded set. We say that $\partial \mathrm{\Omega }$ is ${\widehat{C}}^1$ if such a manifold is oriented and for each $x_0\in \partial \mathrm{\Omega }$, denoting $\widehat{x}=(x_1,...,x_{n-1})$ for a local coordinate system compatible with the manifold $\partial \mathrm{\Omega }$ orientation, there exist $r>0$ and a function $f(x_1,...,x_{n-1})=f\left(\widehat{x}\right)$ such that

$$W=\overline{\mathrm{\Omega }}\cap B_r\left(x_0\right)=\{x\in B_r\left(x_0\right)|x_n\le f(x_1,...,x_{n-1})\}.$$

Moreover, $f\left(\widehat{x}\right)$ is a Lipschitz continuous function, so that

$$|f\left(\widehat{x}\right)-f\left(\widehat{y}\right)|\le C_1{|\widehat{x}-\widehat{y}|}_2,\mathit{on}\mathit{its}\mathit{domain},$$

for some $C_1>0$. Finally, we assume

$${\left\{\frac{\partial f\left(\widehat{x}\right)}{\partial x_k}\right\}}_{k=1}^{n-1}$$

is classically defined, almost everywhere also on its concerning domain, so that $f\in W^{1,2}$.

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 $\mathrm{\Omega }\subset {\mathbb{R}}^n$ is an open bounded set, and that $\partial \mathrm{\Omega }$ is ${\widehat{C}}^1$. Let $1\le p<\infty ,$ and let V be a bounded open set such that $\mathrm{\Omega }\subset \subset V$. Then there exists a bounded linear operator

$$E:W^{1,p}\left(\mathrm{\Omega }\right)\to W^{1,p}\left({\mathbb{R}}^n\right),$$

such that for each $u\in W^{1,p}\left(\mathrm{\Omega }\right)$ we have:

1. 

$Eu=u,\mathrm{a}.\mathrm{e}.\mathit{in}\mathrm{\Omega },$;

2. 

$Eu$ has support in V;

3. 

${\parallel Eu\parallel }_{1,p,{\mathbb{R}}^n}\le C{\parallel u\parallel }_{1,p,\mathrm{\Omega }},$ where the constant depends only on $p,\mathrm{\Omega },\mathrm{and}V.$

Remark 3.

Considering the proof of such a result, the constant $C>0$ may be also such that

$$\parallel e_{ij}{\left(Eu\right)\parallel }_{0,2,V}\le C(\parallel e_{ij}{\left(u\right)\parallel }_{0,2,\mathrm{\Omega }}+{\parallel u\parallel }_{0,2,\mathrm{\Omega }}),\forall u\in W^{1,2}(\mathrm{\Omega };{\mathbb{R}}^n),\forall i,j\in \{1,\dots ,n\},$$

for the operator $e:W^{1,2}(\mathrm{\Omega };{\mathbb{R}}^n)\to L^2(\mathrm{\Omega };{\mathbb{R}}^{n\times n})$ specified in the next theorem.

Finally, as the meaning is clear, we may simply denote $Eu=u.$

2. The Main Results, the Korn Inequalities

Our main result is summarized by the following theorem.

Theorem 2.

Let $\mathrm{\Omega }\subset {\mathbb{R}}^n$ be an open, bounded and connected set with a ${\widehat{C}}^1$ (Lipschitzian) boundary $\partial \mathrm{\Omega }$.

Define $e:W^{1,2}(\mathrm{\Omega };{\mathbb{R}}^n)\to L^2(\mathrm{\Omega };{\mathbb{R}}^{n\times n})$ by

$$e\left(u\right)=\left\{e_{ij}\left(u\right)\right\}$$

where

$$e_{ij}\left(u\right)=\frac{1}{2}(u_{i,j}+u_{j,i}),\forall i,j\in \{1,\dots ,n\},$$

and where generically, we denote

$$u_{i,j}=\frac{\partial u_i}{\partial x_j},\forall i,j\in \{1,\cdots ,n\}.$$

Define also,

$${\parallel e\left(u\right)\parallel }_{0,2,\mathrm{\Omega }}={\left(\sum _{i=1}^n\sum _{j=1}^n{\parallel e_{ij\left(u\right)}\parallel }_{0,2,\mathrm{\Omega }}^2\right)}^{1/2}.$$

Let $L\in {\mathbb{R}}^{+}$ be such $V={[-L,L]}^n$ is also such that $\overline{\mathrm{\Omega }}\subset V^0.$

Under such hypotheses, there exists $C(\mathrm{\Omega },L)\in {\mathbb{R}}^{+}$ such that

$${\parallel u\parallel }_{1,2,\mathrm{\Omega }}\le C(\mathrm{\Omega },L)\left({\parallel u\parallel }_{0,2,\mathrm{\Omega }}+{\parallel e\left(u\right)\parallel }_{0,2,\mathrm{\Omega }}\right),\forall u\in W^{1,2}(\mathrm{\Omega };{\mathbb{R}}^n).$$

(1)

Proof. 

Suppose, to obtain contradiction, that the concerning claim does not hold.

Thus, we are assuming that there is no positive real constant $C=C(\mathrm{\Omega },L)$ such that (1) is valid.

In particular, $k=1\in \mathbb{N}$ is not such a constant C, so that there exists a function $u_1\in W^{1,2}(\mathrm{\Omega };{\mathbb{R}}^n)$ such that

$$\parallel u_1{\parallel }_{1,2,\mathrm{\Omega }}>1\left(\parallel u_1{\parallel }_{0,2,\mathrm{\Omega }}+{\parallel e\left(u_1\right)\parallel }_{0,2,\mathrm{\Omega }}\right).$$

Similarly, $k=2\in \mathbb{N}$ is not such a constant C, so that there exists a function $u_2\in W^{1,2}(\mathrm{\Omega };{\mathbb{R}}^n)$ such that

$$\parallel u_2{\parallel }_{1,2,\mathrm{\Omega }}>2\left(\parallel u_2{\parallel }_{0,2,\mathrm{\Omega }}+{\parallel e\left(u_2\right)\parallel }_{0,2,\mathrm{\Omega }}\right).$$

Hence, since no $k\in \mathbb{N}$ is such a constant C, reasoning inductively, for each $k\in \mathbb{N}$ there exists a function $u_k\in W^{1,2}(\mathrm{\Omega };{\mathbb{R}}^n)$ such that

$$\parallel u_k{\parallel }_{1,2,\mathrm{\Omega }}>k\left(\parallel u_k{\parallel }_{0,2,\mathrm{\Omega }}+{\parallel e\left(u_k\right)\parallel }_{0,2,\mathrm{\Omega }}\right).$$

In particular, defining

$$v_k=\frac{u_k}{\parallel u_k{\parallel }_{1,2,\mathrm{\Omega }}}$$

we obtain

$$\parallel v_k{\parallel }_{1,2,\mathrm{\Omega }}=1>k\left(\parallel v_k{\parallel }_{0,2,\mathrm{\Omega }}+{\parallel e\left(v_k\right)\parallel }_{0,2,\mathrm{\Omega }}\right),$$

so that

$$\left(\parallel v_k{\parallel }_{0,2,\mathrm{\Omega }}+{\parallel e\left(v_k\right)\parallel }_{0,2,\mathrm{\Omega }}\right)<\frac{1}{k},\forall k\in \mathbb{N}.$$

From this we obtain

$$\parallel v_k{\parallel }_{0,2,\mathrm{\Omega }}<\frac{1}{k},$$

and

$$\parallel e_{ij}\left(v_k\right){\parallel }_{0,2,\mathrm{\Omega }}<\frac{1}{k},\forall k\in \mathbb{N},$$

so that

$$\parallel v_k{\parallel }_{0,2,\mathrm{\Omega }}\to 0,\mathrm{as}k\to \infty ,$$

and

$$\parallel e_{ij}\left(v_k\right){\parallel }_{0,2,\mathrm{\Omega }}\to 0,\mathrm{as}k\to \infty .$$

In particular,

$$\parallel {\left(v_k\right)}_{j,j}{\parallel }_{0,2,\mathrm{\Omega }}\to 0,\forall j\in \{1,\dots ,n\}.$$

At this point, we recall the following identity in the distributional sense, found in [3], page 12:

$${\partial }_j\left({\partial }_l{v}_i\right)={\partial }_j{e}_{il}\left(v\right)+{\partial }_l{e}_{ij}\left(v\right)-{\partial }_i{e}_{jl}\left(v\right),\forall i,j,l\in \{1,\dots ,n\}.$$

(2)

Fix $j\in \{1,\dots ,n\}$ and observe that

$$\parallel {\left(v_k\right)}_j{\parallel }_{1,2,V}\le C{\parallel {\left(v_k\right)}_j\parallel }_{1,2,\mathrm{\Omega }},$$

so that

$$\frac{C}{\parallel {\left(v_k\right)}_j{\parallel }_{1,2,V}}\ge \frac{1}{\parallel {\left(v_k\right)}_j{\parallel }_{1,2,\mathrm{\Omega }}},\forall k\in \mathbb{N}.$$

Hence,

$$\begin{array}{ccc}& & \parallel {\left(v_k\right)}_j{\parallel }_{1,2,\mathrm{\Omega }}\hfill \\ & =& \underset{\phi \in C^1\left(\mathrm{\Omega }\right)}{sup}\left\{{⟨\nabla {\left(v_k\right)}_j,\nabla \phi ⟩}_{L^2\left(\mathrm{\Omega }\right)}+{⟨{\left(v_k\right)}_j,\phi ⟩}_{L^2\left(\mathrm{\Omega }\right)}:{\parallel \phi \parallel }_{1,2,\mathrm{\Omega }}\le 1\right\}\hfill \\ & =& {⟨\nabla {\left(v_k\right)}_j,\nabla \left(\frac{{\left(v_k\right)}_j}{\parallel {\left(v_k\right)}_j{\parallel }_{1,2,\mathrm{\Omega }}}\right)⟩}_{L^2\left(\mathrm{\Omega }\right)}\hfill \\ & & +{⟨{\left(v_k\right)}_j,\left(\frac{{\left(v_k\right)}_j}{\parallel {\left(v_k\right)}_j{\parallel }_{1,2,\mathrm{\Omega }}}\right)⟩}_{L^2\left(\mathrm{\Omega }\right)}\hfill \\ & \le & C\left({⟨\nabla {\left(v_k\right)}_j,\nabla \left(\frac{{\left(v_k\right)}_j}{\parallel {\left(v_k\right)}_j{\parallel }_{1,2,V}}\right)⟩}_{L^2\left(V\right)}+{⟨{\left(v_k\right)}_j,\left(\frac{{\left(v_k\right)}_j}{\parallel {\left(v_k\right)}_j{\parallel }_{1,2,V}}\right)⟩}_{L^2\left(V\right)}\right)\hfill \\ & =& C\underset{\phi \in C_c^1\left(V\right)}{sup}\left\{{⟨\nabla {\left(v_k\right)}_j,\nabla \phi ⟩}_{L^2\left(V\right)}+{⟨{\left(v_k\right)}_j,\phi ⟩}_{L^2\left(V\right)}:{\parallel \phi \parallel }_{1,2,V}\le 1\right\}.\hfill \end{array}$$

(3)

Here, we recall that $C>0$ is the constant concerning the extension Theorem 1. From such results and (2), we have that

$$\begin{array}{ccc}& & \underset{\phi \in C^1\left(\mathrm{\Omega }\right)}{sup}\left\{{⟨\nabla {\left(v_k\right)}_j,\nabla \phi ⟩}_{L^2\left(\mathrm{\Omega }\right)}+{⟨{\left(v_k\right)}_j,\phi ⟩}_{L^2\left(\mathrm{\Omega }\right)}:{\parallel \phi \parallel }_{1,2,\mathrm{\Omega }}\le 1\right\}\hfill \\ & \le & C\underset{\phi \in C_c^1\left(V\right)}{sup}\left\{{⟨\nabla {\left(v_k\right)}_j,\nabla \phi ⟩}_{L^2\left(V\right)}+{⟨{\left(v_k\right)}_j,\phi ⟩}_{L^2\left(V\right)}:{\parallel \phi \parallel }_{1,2,V}\le 1\right\}\hfill \\ & =& C\underset{\phi \in C_c^1\left(V\right)}{sup}\left\{{⟨e_{jl}\left(v_k\right),{\phi }_{,l}⟩}_{L^2\left(V\right)}+{⟨e_{jl}\left(v_k\right),{\phi }_{,l}⟩}_{L^2\left(V\right)}\right.\hfill \\ & & \left.-{⟨e_{ll}\left(v_k\right),{\phi }_{,j}⟩}_{L^2\left(V\right)}+{⟨{\left(v_k\right)}_j,\phi ⟩}_{L^2\left(V\right)},:{\parallel \phi \parallel }_{1,2,V}\le 1\right\}.\hfill \end{array}$$

(4)

Therefore,

$$\begin{array}{ccc}& & \parallel {\left(v_k\right)}_j{\parallel }_{\left(W^{1,2}\left(\mathrm{\Omega }\right)\right)}\hfill \\ & =& \underset{\phi \in C^1\left(\mathrm{\Omega }\right)}{sup}\{{⟨\nabla {\left(v_k\right)}_j,\nabla \phi ⟩}_{L^2\left(\mathrm{\Omega }\right)}+{⟨{\left(v_k\right)}_j,\phi ⟩}_{L^2\left(\mathrm{\Omega }\right)}{:\parallel \phi \parallel }_{1,2,\mathrm{\Omega }}\le 1\}\hfill \\ & \le & C\left(\sum _{l=1}^n\left\{\parallel e_{jl}\left(v_k\right){\parallel }_{0,2,V}+{\parallel e_{ll}\left(v_k\right)\parallel }_{0,2,V}\right\}+{\parallel {\left(v_k\right)}_j\parallel }_{0,2,V}\right)\hfill \\ & \le & C_1\left(\sum _{l=1}^n\left\{\parallel e_{jl}\left(v_k\right){\parallel }_{0,2,\mathrm{\Omega }}+{\parallel e_{ll}\left(v_k\right)\parallel }_{0,2,\mathrm{\Omega }}\right\}+{\parallel {\left(v_k\right)}_j\parallel }_{0,2,\mathrm{\Omega }}\right)\hfill \\ & <& \frac{C_2}{k},\hfill \end{array}$$

(5)

for appropriate $C_1>0$ and $C_2>0.$

Summarizing,

$$\parallel {\left(v_k\right)}_j{\parallel }_{\left(W^{1,2}\left(\mathrm{\Omega }\right)\right)}<\frac{C_2}{k},\forall k\in \mathbb{N}.$$

From this we obtain

$$\parallel v_k{\parallel }_{1,2,\mathrm{\Omega }}\to 0,\mathrm{as}k\to \infty ,$$

which contradicts

$$\parallel v_k{\parallel }_{1,2,\mathrm{\Omega }}=1,\forall k\in \mathbb{N}.$$

The proof is complete. □

Corollary 1.

Let $\mathrm{\Omega }\subset {\mathbb{R}}^n$ be an open, bounded and connected set with a ${\widehat{C}}^1$ boundary $\partial \mathrm{\Omega }$. Define $e:W^{1,2}(\mathrm{\Omega };{\mathbb{R}}^n)\to L^2(\mathrm{\Omega };{\mathbb{R}}^{n\times n})$ by

$$e\left(u\right)=\left\{e_{ij}\left(u\right)\right\}$$

where

$$e_{ij}\left(u\right)=\frac{1}{2}(u_{i,j}+u_{j,i}),\forall i,j\in \{1,\dots ,n\}.$$

Define also,

$${\parallel e\left(u\right)\parallel }_{0,2,\mathrm{\Omega }}={\left(\sum _{i=1}^n\sum _{j=1}^n{\parallel e_{ij\left(u\right)}\parallel }_{0,2,\mathrm{\Omega }}^2\right)}^{1/2}.$$

Let $L\in {\mathbb{R}}^{+}$ be such $V={[-L,L]}^n$ is also such that $\overline{\mathrm{\Omega }}\subset V^0.$

Moreover, define

$${\widehat{H}}_0=\{u\in W^{1,2}(\mathrm{\Omega };{\mathbb{R}}^n):u=\mathbf{0},\mathrm{on}{\Gamma }_0\},$$

where ${\Gamma }_0\subset \partial \mathrm{\Omega }$ is a measurable set such that the Lebesgue measure $m_{{\mathbb{R}}^{n-1}}\left({\Gamma }_0\right)>0.$

Assume also ${\Gamma }_0$ is such that for each $j\in \{1,\cdots ,n\}$ and each $x=(x_1,\cdots ,x_n)\in \mathrm{\Omega }$ there exists $x_0=({\left(x_0\right)}_1,\cdots ,{\left(x_0\right)}_n)\in {\Gamma }_0$ such that

$${\left(x_0\right)}_l=x_l,\forall l\ne j,l\in \{1,\cdots ,n\},$$

and the line

$$A_{x_0,x}\subset \overline{\mathrm{\Omega }}$$

where

$$A_{x_0,x}=\{(x_1,\cdots ,(1-t){\left(x_0\right)}_j+t{x}_j,\cdots ,x_n):t\in [0,1]\}.$$

Under such hypotheses, there exists $C(\mathrm{\Omega },L)\in {\mathbb{R}}^{+}$ such that

$${\parallel u\parallel }_{1,2,\mathrm{\Omega }}\le C(\mathrm{\Omega },L){\parallel e\left(u\right)\parallel }_{0,2,\mathrm{\Omega }},\forall u\in {\widehat{H}}_0.$$

Proof. 

Suppose, to obtain contradiction, that the concerning claim does not hold.

Hence, for each $k\in \mathbb{N}$ there exists $u_k\in {\widehat{H}}_0$ such that

$$\parallel u_k{\parallel }_{1,2,\mathrm{\Omega }}>k{\parallel e\left(u_k\right)\parallel }_{0,2,\mathrm{\Omega }}.$$

In particular, defining

$$v_k=\frac{u_k}{\parallel u_k{\parallel }_{1,2,\mathrm{\Omega }}}$$

similarly to the proof of the last theorem, we may obtain

$$\parallel {\left(v_k\right)}_{j,j}{\parallel }_{0,2,\mathrm{\Omega }}\to 0,\mathrm{as}k\to \infty ,\forall j\in \{1,\dots ,n\}.$$

From this, the hypotheses on ${\Gamma }_0$ and from the standard Poincaré inequality proof we obtain

$$\parallel {\left(v_k\right)}_j{\parallel }_{0,2,\mathrm{\Omega }}\to 0,\mathrm{as}k\to \infty ,\forall j\in \{1,\dots ,n\}.$$

Thus, also similarly as in the proof of the last theorem, we may infer that

$$\parallel v_k{\parallel }_{1,2,\mathrm{\Omega }}\to 0,\mathrm{as}k\to \infty ,$$

which contradicts

$$\parallel v_k{\parallel }_{1,2,\mathrm{\Omega }}=1,\forall k\in \mathbb{N}.$$

The proof is complete. □

3. 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 $\mathrm{\Omega }\subset {\mathbb{R}}^2$ be an open, bounded, connected set which represents the middle surface of a plate of thickness h. The boundary of $\mathrm{\Omega }$, which is assumed to be regular (Lipschitzian), is denoted by $\partial \mathrm{\Omega }$. The vectorial basis related to the cartesian system $\{x_1,x_2,x_3\}$ is denoted by $({\mathbf{a}}_{\alpha },{\mathbf{a}}_3)$, where $\alpha =1,2$ (in general, Greek indices stand for 1 or 2), and where ${\mathbf{a}}_3$ is the vector normal to $\mathrm{\Omega }$, whereas ${\mathbf{a}}_1$ and ${\mathbf{a}}_2$ are orthogonal vectors parallel to $\mathrm{\Omega }.$ Furthermore, $\mathbf{n}$ is the outward normal to the plate surface.

The displacements will be denoted by

$$\widehat{\mathbf{u}}=\{{\widehat{u}}_{\alpha },{\widehat{u}}_3\}={\widehat{u}}_{\alpha }{\mathbf{a}}_{\alpha }+{\widehat{u}}_3{\mathbf{a}}_3.$$

The Kirchhoff–Love relations are

$$\begin{array}{ccc}& & {\widehat{u}}_{\alpha }(x_1,x_2,x_3)=u_{\alpha }(x_1,x_2)-x_{3}w{(x_1,x_2)}_{,\alpha }\hfill \\ & & \mathrm{and}{\widehat{u}}_3(x_1,x_2,x_3)=w(x_1,x_2).\hfill \end{array}$$

(6)

Here, $-h/2\le x_3\le h/2$ so that we have $u=(u_{\alpha },w)\in U$ where

$$\begin{array}{ccc}\hfill U& =& \left\{(u_{\alpha },w)\in W^{1,2}(\mathrm{\Omega };{\mathbb{R}}^2)\times W^{2,2}\left(\mathrm{\Omega }\right),\right.\hfill \\ & & u_{\alpha }=w=\frac{\partial w}{\partial \mathbf{n}}=0\left.\mathrm{on}\partial \mathrm{\Omega }\right\}\hfill \\ & =& W_0^{1,2}(\mathrm{\Omega };{\mathbb{R}}^2)\times W_0^{2,2}(\Omega ).\hfill \end{array}$$

It is worth emphasizing that the boundary conditions here specified refer to a clamped plate.

We define the operator $\Lambda :U\to Y\times Y$, where $Y=Y^{*}=L^2(\mathrm{\Omega };{\mathbb{R}}^{2\times 2})$, by

$$\Lambda \left(u\right)=\left\{\gamma \right(u),\kappa (u\left)\right\},$$

$${\gamma }_{\alpha \beta }\left(u\right)=\frac{u_{\alpha ,\beta }+u_{\beta ,\alpha }}{2}+\frac{w_{,\alpha }{w}_{,\beta }}{2},$$

$${\kappa }_{\alpha \beta }\left(u\right)=-w_{,\alpha \beta }.$$

The constitutive relations are given by

$$N_{\alpha \beta }\left(u\right)=H_{\alpha \beta \lambda \mu }{\gamma }_{\lambda \mu }\left(u\right),$$

(7)

$$M_{\alpha \beta }\left(u\right)=h_{\alpha \beta \lambda \mu }{\kappa }_{\lambda \mu }\left(u\right),$$

(8)

where $\left\{H_{\alpha \beta \lambda \mu }\right\}$ and $\{h_{\alpha \beta \lambda \mu }=\frac{h^2}{12}{H}_{\alpha \beta \lambda \mu }\}$, are symmetric positive definite fourth-order tensors. From now on, we denote $\left\{{\overline{H}}_{\alpha \beta \lambda \mu }\right\}={\left\{H_{\alpha \beta \lambda \mu }\right\}}^{-1}$ and $\left\{{\overline{h}}_{\alpha \beta \lambda \mu }\right\}={\left\{h_{\alpha \beta \lambda \mu }\right\}}^{-1}$.

Furthermore, $\left\{N_{\alpha \beta }\right\}$ denote the membrane force tensor and $\left\{M_{\alpha \beta }\right\}$ the moment one. The plate stored energy, represented by $(G\circ \Lambda ):U\to \mathbb{R}$, is expressed by

$$(G\circ \Lambda )\left(u\right)=\frac{1}{2}{\int }_{\mathrm{\Omega }}{N}_{\alpha \beta }\left(u\right){\gamma }_{\alpha \beta }\left(u\right)dx+\frac{1}{2}{\int }_{\mathrm{\Omega }}{M}_{\alpha \beta }\left(u\right){\kappa }_{\alpha \beta }\left(u\right)dx$$

(9)

and the external work, represented by $F:U\to \mathbb{R}$, is given by

$$F\left(u\right)={⟨w,P⟩}_{L^2\left(\mathrm{\Omega }\right)}+{⟨u_{\alpha },P_{\alpha }⟩}_{L^2\left(\mathrm{\Omega }\right)},$$

(10)

where $P,P_1,P_2\in L^2\left(\mathrm{\Omega }\right)$ are external loads in the directions ${\mathbf{a}}_3$, ${\mathbf{a}}_1$, and ${\mathbf{a}}_2$, respectively. The potential energy, denoted by $J:U\to \mathbb{R}$, is expressed by

$$J\left(u\right)=(G\circ \Lambda )\left(u\right)-F\left(u\right)$$

Finally, we also emphasize from now on, as their meaning are clear, we may denote $L^2\left(\mathrm{\Omega }\right)$ and $L^2(\mathrm{\Omega };{\mathbb{R}}^{2\times 2})$ simply by $L^2$, and the respective norms by ${\parallel ·\parallel }_2.$ Moreover, derivatives are always understood in the distributional sense, $\mathbf{0}$ may denote the zero vector in appropriate Banach spaces, and the following and relating notations are used:

$$w_{,\alpha }=\frac{\partial w}{\partial x_{\alpha }},$$

$$w_{,\alpha \beta }=\frac{{\partial }^{2}w}{\partial x_{\alpha }\partial x_{\beta }},$$

$$u_{\alpha ,\beta }=\frac{\partial u_{\alpha }}{\partial x_{\beta }},$$

$$N_{\alpha \beta ,1}=\frac{\partial N_{\alpha \beta }}{\partial x_1},$$

and

$$N_{\alpha \beta ,2}=\frac{\partial N_{\alpha \beta }}{\partial x_2}.$$

4. 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 $\left\{P_{\alpha }\right\}\in L^{\infty }(\mathrm{\Omega };{\mathbb{R}}^2)$. We may easily obtain by appropriate Lebesgue integration $\{{\tilde{T}}_{\alpha \beta }\}$ symmetric and such that

$${\tilde{T}}_{\alpha \beta ,\beta }=-P_{\alpha },\mathrm{in}\mathrm{\Omega }.$$

Indeed, extending $\left\{P_{\alpha }\right\}$ to zero outside Ω if necessary, we may set

$${\tilde{T}}_{11}(x,y)=-{\int }_0^x{P}_1(\xi ,y)d\xi ,$$

$${\tilde{T}}_{22}(x,y)=-{\int }_0^y{P}_2(x,\xi )d\xi ,$$

and

$${\tilde{T}}_{12}(x,y)={\tilde{T}}_{21}(x,y)=0,\mathrm{in}\mathrm{\Omega }.$$

Thus, we may choose a $C>0$ sufficiently big, such that

$$\left\{T_{\alpha \beta }\right\}=\{{\tilde{T}}_{\alpha \beta }+C{\delta }_{\alpha \beta }\}$$

is positive definite $\mathrm{in}\mathrm{\Omega }$, so that

$$T_{\alpha \beta ,\beta }={\tilde{T}}_{\alpha \beta ,\beta }=-P_{\alpha },$$

where

$$\left\{{\delta }_{\alpha \beta }\right\}$$

is the Kronecker delta.

Therefore, for the kind of boundary conditions of the next theorem, we do not have any restriction for the $\left\{P_{\alpha }\right\}$ 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 $\left\{T_{\alpha \beta }\right\}$ are new, even though the final part of the proof is established through a standard procedure in the calculus of variations.

Finally, more details on the Sobolev spaces involved may be found in [5,6,7,8]. Related duality principles are addressed in [5,7,9].

At this point, we present the main theorem in this section.

Theorem 3.

Let $\mathrm{\Omega }\subset {\mathbb{R}}^2$ be an open, bounded, connected set with a Lipschitzian boundary denoted by $\partial \mathrm{\Omega }=\Gamma .$ Suppose $(G\circ \Lambda ):U\to \mathbb{R}$ is defined by

$$G(\Lambda u)=G_1\left(\gamma \left(u\right)\right)+G_2\left(\kappa \left(u\right)\right),\forall u\in U,$$

where

$$G_1\left(\gamma u\right)=\frac{1}{2}{\int }_{\mathrm{\Omega }}{H}_{\alpha \beta \lambda \mu }{\gamma }_{\alpha \beta }\left(u\right){\gamma }_{\lambda \mu }\left(u\right)dx,$$

and

$$G_2\left(\kappa u\right)=\frac{1}{2}{\int }_{\mathrm{\Omega }}{h}_{\alpha \beta \lambda \mu }{\kappa }_{\alpha \beta }\left(u\right){\kappa }_{\lambda \mu }\left(u\right)dx,$$

where

$$\Lambda \left(u\right)=(\gamma \left(u\right),\kappa \left(u\right))=(\left\{{\gamma }_{\alpha \beta }\left(u\right)\right\},\left\{{\kappa }_{\alpha \beta }\left(u\right)\right\}),$$

$${\gamma }_{\alpha \beta }\left(u\right)=\frac{u_{\alpha ,\beta }+u_{\beta ,\alpha }}{2}+\frac{w_{,\alpha }{w}_{,\beta }}{2},$$

$${\kappa }_{\alpha \beta }\left(u\right)=-w_{,\alpha \beta },$$

and where

$$\begin{array}{ccc}\hfill J\left(u\right)& =& W(\gamma \left(u\right),\kappa \left(u\right))-{⟨P_{\alpha },u_{\alpha }⟩}_{L^2\left(\mathrm{\Omega }\right)}\hfill \\ & & -{⟨w,P⟩}_{L^2\left(\mathrm{\Omega }\right)}-{⟨P_{\alpha }^t,u_{\alpha }⟩}_{L^2\left({\Gamma }_t\right)}\hfill \\ & & -{⟨P^t,w⟩}_{L^2\left({\Gamma }_t\right)},\hfill \end{array}$$

(11)

where,

$$\begin{array}{ccc}\hfill U& =& \{u=(u_{\alpha },w)=(u_1,u_2,w)\in W^{1,2}(\mathrm{\Omega };{\mathbb{R}}^2)\times W^{2,2}\left(\mathrm{\Omega }\right):\hfill \\ & & u_{\alpha }=w=\frac{\partial w}{\partial \mathbf{n}}=0,\mathrm{on}{\Gamma }_0\},\hfill \end{array}$$

(12)

where $\partial \mathrm{\Omega }={\Gamma }_0\cup {\Gamma }_t$ and the Lebesgue measures

$$m_{\Gamma }({\Gamma }_0\cap {\Gamma }_t)=0,$$

and

$$m_{\Gamma }\left({\Gamma }_0\right)>0.$$

We also define

$$\begin{array}{ccc}\hfill F_1\left(u\right)& =& -{⟨w,P⟩}_{L^2\left(\mathrm{\Omega }\right)}-{⟨u_{\alpha },P_{\alpha }⟩}_{L^2\left(\mathrm{\Omega }\right)}-{⟨P_{\alpha }^t,u_{\alpha }⟩}_{L^2\left({\Gamma }_t\right)}\hfill \\ & & -{⟨P^t,w⟩}_{L^2\left({\Gamma }_t\right)}+{⟨{\epsilon }_{\alpha },u_{\alpha }^2⟩}_{L^2\left({\Gamma }_t\right)}\hfill \\ & \equiv & -{⟨u,\mathbf{f}⟩}_{L^2}+{⟨{\epsilon }_{\alpha },u_{\alpha }^2⟩}_{L^2\left({\Gamma }_t\right)}\hfill \\ & \equiv & -{⟨u,{\mathbf{f}}_1⟩}_{L^2}-{⟨u_{\alpha },P_{\alpha }⟩}_{L^2\left(\mathrm{\Omega }\right)}+{⟨{\epsilon }_{\alpha },u_{\alpha }^2⟩}_{L^2\left({\Gamma }_t\right)},\hfill \end{array}$$

(13)

where

$${⟨u,{\mathbf{f}}_1⟩}_{L^2}={⟨u,\mathbf{f}⟩}_{L^2}-{⟨u_{\alpha },P_{\alpha }⟩}_{L^2\left(\mathrm{\Omega }\right)},$$

${\epsilon }_{\alpha }>0,\forall \alpha \in \{1,2\}$ and

$$\mathbf{f}=(P_{\alpha },P)\in L^{\infty }(\mathrm{\Omega };{\mathbb{R}}^3).$$

Let $J:U\to \mathbb{R}$ be defined by

$$J\left(u\right)=G(\Lambda u)+F_1\left(u\right),\forall u\in U.$$

Assume there exists $\left\{c_{\alpha \beta }\right\}\in {\mathbb{R}}^{2\times 2}$ such that $c_{\alpha \beta }>0,\forall \alpha ,\beta \in \{1,2\}$ and

$$G_2\left(\kappa \left(u\right)\right)\ge c_{\alpha \beta }{\parallel w_{,\alpha \beta }\parallel }_2^2,\forall u\in U.$$

Under such hypotheses, there exists $u_0\in U$ such that

$$J\left(u_0\right)=\underset{u\in U}{min}J\left(u\right).$$

Proof. 

Observe that we may find ${\mathbf{T}}_{\alpha }=\{{\left(T_{\alpha }\right)}_{\beta }\}$ such that

$$div{\mathbf{T}}_{\alpha }=T_{\alpha \beta ,\beta }=-P_{\alpha },$$

and also such that $\left\{T_{\alpha \beta }\right\}$ is positive, definite, and symmetric (please see Remark 4).

Thus, defining

$$v_{\alpha \beta }\left(u\right)=\frac{u_{\alpha ,\beta }+u_{\beta ,\alpha }}{2}+\frac{1}{2}{w}_{,\alpha }{w}_{,\beta },$$

(14)

we obtain

$$\begin{array}{ccc}\hfill J\left(u\right)& =& G_1\left(\left\{v_{\alpha \beta }\left(u\right)\right\}\right)+G_2\left(\kappa \left(u\right)\right)-{⟨u,\mathbf{f}⟩}_{L^2}+{⟨{\epsilon }_{\alpha },u_{\alpha }^2⟩}_{L^2\left({\Gamma }_t\right)}\hfill \\ & =& G_1\left(\left\{v_{\alpha \beta }\left(u\right)\right\}\right)+G_2\left(\kappa \left(u\right)\right)+{⟨T_{\alpha \beta ,\beta },u_{\alpha }⟩}_{L^2\left(\mathrm{\Omega }\right)}-{⟨u,{\mathbf{f}}_1⟩}_{L^2}+{⟨{\epsilon }_{\alpha },u_{\alpha }^2⟩}_{L^2\left({\Gamma }_t\right)}\hfill \\ & =& G_1\left(\left\{v_{\alpha \beta }\left(u\right)\right\}\right)+G_2\left(\kappa \left(u\right)\right)-{⟨T_{\alpha \beta },\frac{u_{\alpha ,\beta }+u_{\beta ,\alpha }}{2}⟩}_{L^2\left(\mathrm{\Omega }\right)}\hfill \\ & & +{⟨T_{\alpha \beta }{n}_{\beta },u_{\alpha }⟩}_{L^2\left({\Gamma }_t\right)}-{⟨u,{\mathbf{f}}_1⟩}_{L^2}+{⟨{\epsilon }_{\alpha },u_{\alpha }^2⟩}_{L^2\left({\Gamma }_t\right)}\hfill \\ & =& G_1\left(\left\{v_{\alpha \beta }\left(u\right)\right\}\right)+G_2\left(\kappa \left(u\right)\right)-{⟨T_{\alpha \beta },v_{\alpha \beta }\left(u\right)-\frac{1}{2}{w}_{,\alpha }{w}_{,\beta }⟩}_{L^2\left(\mathrm{\Omega }\right)}-{⟨u,{\mathbf{f}}_1⟩}_{L^2}+{⟨{\epsilon }_{\alpha },u_{\alpha }^2⟩}_{L^2\left({\Gamma }_t\right)}\hfill \\ & & +{⟨T_{\alpha \beta }{n}_{\beta },u_{\alpha }⟩}_{L^2\left({\Gamma }_t\right)}\hfill \\ & \ge & c_{\alpha \beta }{\parallel w_{,\alpha \beta }\parallel }_2^2+\frac{1}{2}{⟨T_{\alpha \beta },w_{,\alpha }{w}_{,\beta }⟩}_{L^2\left(\mathrm{\Omega }\right)}-{⟨u,{\mathbf{f}}_1⟩}_{L^2}+{⟨{\epsilon }_{\alpha },u_{\alpha }^2⟩}_{L^2\left({\Gamma }_t\right)}+G_1\left(\left\{v_{\alpha \beta }\left(u\right)\right\}\right)\hfill \\ & & -{⟨T_{\alpha \beta },v_{\alpha \beta }\left(u\right)⟩}_{L^2\left(\mathrm{\Omega }\right)}+{⟨T_{\alpha \beta }{n}_{\beta },u_{\alpha }⟩}_{L^2\left({\Gamma }_t\right)}.\hfill \end{array}$$

(15)

From this, since $\left\{T_{\alpha \beta }\right\}$ is positive definite, clearly J is bounded below.

Let $\left\{u_n\right\}\in U$ be a minimizing sequence for J. Thus, there exists ${\alpha }_1\in \mathbb{R}$ such that

$$\underset{n\to \infty }{lim}J\left(u_n\right)=\underset{u\in U}{inf}J\left(u\right)={\alpha }_1.$$

From (15), there exists $K_1>0$ such that

$$\parallel {\left(w_n\right)}_{,\alpha \beta }{\parallel }_2<K_1,\forall \alpha ,\beta \in \{1,2\},n\in \mathbb{N}.$$

Therefore, there exists $w_0\in W^{2,2}\left(\mathrm{\Omega }\right)$ such that, up to a subsequence not relabeled,

$${\left(w_n\right)}_{,\alpha \beta }\rightharpoonup {\left(w_0\right)}_{,\alpha \beta },\mathrm{weakly}\mathrm{in}{L}^2,$$

$\forall \alpha ,\beta \in \{1,2\},\mathrm{as}n\to \infty .$

Moreover, also up to a subsequence not relabeled,

$${\left(w_n\right)}_{,\alpha }\to {\left(w_0\right)}_{,\alpha },\mathrm{strongly}\mathrm{in}{L}^2\mathrm{and}{L}^4,$$

(16)

$\forall \alpha ,\in \{1,2\},\mathrm{as}n\to \infty .$

Furthermore, from (15), there exists $K_2>0$ such that,

$$\parallel {\left(v_n\right)}_{\alpha \beta }\left(u\right){\parallel }_2<K_2,\forall \alpha ,\beta \in \{1,2\},n\in \mathbb{N},$$

and thus, from this, (14) and (16), we may infer that there exists $K_3>0$ such that

$$\parallel {\left(u_n\right)}_{\alpha ,\beta }+{\left(u_n\right)}_{\beta ,\alpha }{\parallel }_2<K_3,\forall \alpha ,\beta \in \{1,2\},n\in \mathbb{N}.$$

From this and Korn’s inequality, there exists $K_4>0$ such that

$$\parallel u_n{\parallel }_{W^{1,2}(\mathrm{\Omega };{\mathbb{R}}^2)}\le K_4,\forall n\in \mathbb{N}.$$

Therefore, up to a subsequence not relabeled, there exists $\left\{{\left(u_0\right)}_{\alpha }\right\}\in W^{1,2}(\mathrm{\Omega },{\mathbb{R}}^2),$ such that

$${\left(u_n\right)}_{\alpha ,\beta }+{\left(u_n\right)}_{\beta ,\alpha }\rightharpoonup {\left(u_0\right)}_{\alpha ,\beta }+{\left(u_0\right)}_{\beta ,\alpha },\mathrm{weakly}\mathrm{in}{L}^2,$$

$\forall \alpha ,\beta \in \{1,2\},\mathrm{as}n\to \infty ,$ and

$${\left(u_n\right)}_{\alpha }\to {\left(u_0\right)}_{\alpha },\mathrm{strongly}\mathrm{in}{L}^2,$$

$\forall \alpha \in \{1,2\},\mathrm{as}n\to \infty .$

Moreover, the boundary conditions satisfied by the subsequences are also satisfied for $w_0$ and $u_0$ in a trace sense, so that

$$u_0=({\left(u_0\right)}_{\alpha },w_0)\in U.$$

From this, up to a subsequence not relabeled, we obtain

$${\gamma }_{\alpha \beta }\left(u_n\right)\rightharpoonup {\gamma }_{\alpha \beta }\left(u_0\right),\mathrm{weakly}\mathrm{in}{L}^2,$$

$\forall \alpha ,\beta \in \{1,2\},$ and

$${\kappa }_{\alpha \beta }\left(u_n\right)\rightharpoonup {\kappa }_{\alpha \beta }\left(u_0\right),\mathrm{weakly}\mathrm{in}{L}^2,$$

$\forall \alpha ,\beta \in \{1,2\}.$

Therefore, from the convexity of $G_1$ in $\gamma$ and $G_2$ in $\kappa$, we obtain

$$\begin{array}{ccc}\hfill \underset{u\in U}{inf}J\left(u\right)& =& {\alpha }_1\hfill \\ & =& \underset{n\to \infty }{lim\; inf}J\left(u_n\right)\hfill \\ & \ge & J\left(u_0\right).\hfill \end{array}$$

(17)

Thus,

$$J\left(u_0\right)=\underset{u\in U}{min}J\left(u\right).$$

The proof is complete. □

5. 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 ${\mathbb{R}}^n$.