Sharp conditions for Korn inequalities in Orlicz spaces
type: Item
instance of: scholarly article
title: Sharp conditions for Korn inequalities in Orlicz spaces
zbMATH Open document ID: 1280.35106
author: D. Breit
author: Lars Diening
published in: Journal of Mathematical Fluid Mechanics
publication date: 2013-01-04T00:00:00Z
review text: As is well-known, Korn's inequality is a crucial tool in the mathematical approach to the behavior of Newtonian fluids. In the paper under consideration, the authors show (Theorem 1.1) that Korn's inequality in Orlicz spaces is valid if and only if the corresponding Orlicz function \(\varphi\) satisfies both the \(\Delta_2\) and the \(\nabla_2\) conditions. We remind the reader that a convex, left-continuous function \(\varphi\) such that \(\varphi(0)= 0\), \(\lim_{t\to 0^+}\varphi(t)= 0\), and \(\lim_{t\to\infty} \varphi(t)=\infty\) verifies the \(\Delta_2\) condition if \(\varphi(2t)\leq K\varphi(t)\), for all \(t\geq 0\), with \(K>0\) a constant. The \(\nabla_2\) condition is the \(\Delta_2\) condition of the conjugate function \(\varphi^*\). It is interesting to point out that Korn's inequality fails on \(L^1\), \(L\log L\), Exp and \(L^\infty\). Another application of Theorem 1.1 is to the solvability of the Poisson equation with zero boundary data in an Orlicz space generated by \(\varphi\).
Mathematics Subject Classification ID: 35Q35
Mathematics Subject Classification ID: 76A05
zbMATH DE Number: 6122077
zbMATH Keywords: Korn's inequality in \(L \log L\)
zbMATH Keywords: Korn's inequality in Orlicz spaces
zbMATH Keywords: counter example
zbMATH Keywords: Prandtl-Eyring fluids
reviewed by: Peter R. Popivanov
MaRDI profile type: Publication
full work available at URL: s00021-011-0082-x
OpenAlex ID: W2049745804
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DOI: 10.1007/S00021-011-0082-X
description: scientific article; zbMATH DE number 6122077