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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.
}, issn = {2617-8702}, doi = {https://doi.org/10.4208/jms.v56n3.23.03}, url = {http://global-sci.org/intro/article_detail/jms/21874.html} }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.