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A Note on Korn’s Inequality in an N-Dimensional Context and a Global Existence Result for a Non-Linear Plate Model
A Note on Korn’s Inequality in an N-Dimensional Context and a Global Existence Result for a...
Botelho, Fabio Silva
Article A Note on Korn’s Inequality in an N-Dimensional Context and a Global Existence Result for a Non-Linear Plate Model Fabio Silva Botelho Department of Mathematics, Federal University of Santa Catarina, UFSC, Florianópolis 88040-900, Brazil; firstname.lastname@example.org 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. Keywords: Korn’s inequality; global existence result; non-linear plate model MSC: 35Q74; 35J58 1. Introduction In this article, we present a proof for Korn’s inequality in R . 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 . Other important classical results on Korn’s inequality and concerning applications to models in elasticity may be found in [2–4]. Citation: Botelho, F.S. A Note on Remark 1. Generically, throughout the text we denote Korn’s Inequality in an N-Dimensional Context and a Global 1/2 2 2 Existence Result for a Non-Linear kuk = juj dx , 8u 2 L (W), 0,2,W Plate Model. AppliedMath 2023, 3, 406–416. https://doi.org/10.3390/ and appliedmath3020021 1/2 2 2 n kuk = ku k , 8u = (u , . . . , u ) 2 L (W;R ). Received: 15 March 2023 0,2,W å j 1 0,2,W j=1 Revised: 12 April 2023 Accepted: 17 April 2023 Moreover, Published: 2 May 2023 1/2 2 2 1,2 kuk = kuk + ku k , 8u 2 W (W), 1,2,W å 0,2,W j 0,2,W j=1 Copyright: © 2023 by the authors. where we shall also refer throughout the text to the well-known corresponding analogous norm for Licensee MDPI, Basel, Switzerland. 1,2 n u 2 W (W;R ). This article is an open access article distributed under the terms and At this point, we ﬁrst introduce the following deﬁnition. conditions of the Creative Commons Attribution (CC BY) license (https:// n 1 Deﬁnition 1. Let W R be an open, bounded set. We say that ¶W is C if such a manifold is creativecommons.org/licenses/by/ oriented and for each x 2 ¶W, denoting x ˆ = (x , ..., x ) for a local coordinate system compatible 0 1 n 1 4.0/). AppliedMath 2023, 3, 406–416. https://doi.org/10.3390/appliedmath3020021 https://www.mdpi.com/journal/appliedmath AppliedMath 2023, 3 407 with the manifold ¶W orientation, there exist r > 0 and a function f (x , ..., x ) = f (x ˆ) such 1 n 1 that W = W\ B (x ) = fx 2 B (x ) j x f (x , ..., x )g. r 0 r 0 n 1 n 1 Moreover, f (x ˆ) is a Lipschitz continuous function, so that j f (x ˆ) f (y ˆ)j C jx ˆ