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

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.