An Extension of a Variational Inequality in the Simader Theorem to a Variable Exponent Sobolev Space and Applications: The Dirichlet Case

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DOI: https://doi.org/10.28924/2291-8639-20-2022-13
Published: Feb 16, 2022
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Junichi Aramaki, An Extension of a Variational Inequality in the Simader Theorem to a Variable Exponent Sobolev Space and Applications: The Dirichlet Case, Int. J. Anal. Appl., 20 (2022), 13.

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Junichi Aramaki

Abstract

In this paper, we shall extend a fundamental variational inequality which is developed by Simader in W1,p to a variable exponent Sobolev space W1,p(·). The inequality is very useful for the existence theory to the Poisson equation with the Dirichlet boundary conditions in Lp(·)-framework, where Lp(·) denotes a variable exponent Lebesgue space. Furthermore, we can also derive the existence of weak solutions to the Stokes problem in a variable exponent Lebesgue space.

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