Equivalent Relations with the J. L. Lions Lemma in a Variable Exponent Sobolev Space and Their Applications

Abstract

We consider the equivalent conditions with $W^{-m, p(\cdot )} $-version of the J. L. Lions Lemma, where $p(\cdot )$ is a variable exponent satisfying some condition. As applications with $m=0$, we first derive the Korn inequality and furthermore, we consider the relation to other fundamental results. One of the purpose of this paper is an application to the existence of a weak solution for the Maxwell-Stokes type problem.