Open AccessAn Extension of a Variational Inequality in the Simader Theorem to a Variable Exponent Sobolev Space and Applications: The Dirichlet Case
Author(s) -
Junichi Aramaki
Publication year - 2022
Publication title -
international journal of analysis and applications
Language(s) - English
Resource type - Journals
ISSN - 2291-8639
DOI - 10.28924/2291-8639-20-2022-13
Subject(s) - mathematics , standard probability space , sobolev inequality , sobolev space , exponent , mathematical analysis , lp space , space (punctuation) , variable (mathematics) , extension (predicate logic) , dirichlet distribution , pure mathematics , dirichlet problem , lebesgue integration , boundary value problem , banach space , philosophy , linguistics , computer science , programming language
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.