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

Junichi Aramaki

doi:10.28924/2291-8639-20-2022-13

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.

Highlights

In Simader [25], the author derived a variational inequality of a bilinear form
We show that the Stokes problem in a variable exponent Sobolev space has a unique weak solution by a new approach which is an application of Theorem 3.1
Let G be a bounded domain of Rd (d ≥ 2) with a C1-boundary and p ∈ P+log(G)

Summary

Introduction

In Simader [25], the author derived a variational inequality of a bilinear form. More precisely, let G is a bounded domain of Rd (d ≥ 2) with a C1-boundary ∂G and 1 < p < ∞. Where ∇u, ∇v G = G ∇u · ∇v dx, ∇ denotes the gradient operator and p is the conjugate exponent of p, that is, He considered the case where G is an exterior domain and got a variational inequality like as in (1.1). Variational inequality; Dirichlet problem for the Poisson equation; the Helmholtz decomposition; the Stokes problem; variable exponent Sobolev spaces. We show that the Stokes problem in a variable exponent Sobolev space has a unique weak (strong) solution by a new approach which is an application of Theorem 3.1.

Preliminaries

Dirichlet problem for the Poisson equation

An approach to the Stokes problem