A Remark on the J. L. Lions Lemma and Its Applications in a Variable Exponent Sobolev Space

Abstract

In the author's previous paper, we considered the equivalent conditions with $W^{-m, p(\cdot )} $-version ($m\ge 0$ integer) of the J. L. Lions Lemma,  where $p(\cdot)$ is a variable exponent.  In this paper, we directly derive $W^{-m,p(\cdot)}$-version of the J. L. Lions Lemma. Therefore, we can use all of the equivalent conditions. As an application, we derive the generalized  Korn inequality. Furthermore, we consider the relation to other fundamental results.