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Research Article | Open Access

Poincaré inequalities and Neumann problems for the variable exponent setting

David Cruz-Uribe1( ), Michael Penrod1, Scott Rodney2
Department of Mathematics, University of Alabama, Tuscaloosa, AL 35487, USA
Department of Mathematics, Physics and Geology, Cape Breton University, Sydney, NS B1M1A2, CA
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Abstract

In an earlier paper, Cruz-Uribe, Rodney and Rosta proved an equivalence between weighted Poincaré inequalities and the existence of weak solutions to a family of Neumann problems related to a degenerate p-Laplacian. Here we prove a similar equivalence between Poincaré inequalities in variable exponent spaces and solutions to a degenerate p ( ⋅ ) -Laplacian, a non-linear elliptic equation with nonstandard growth conditions.

Keywords

degenerate Sobolev spacesp-LaplacianPoincaréinequalitiesvariable exponentnonstandard growth conditions

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Mathematics in Engineering
Volume 4 Issue 5,
October 2022
Pages 1-22
DOI: 10.3934/mine.2022036

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Cite this article:
Cruz-Uribe D, Penrod M, Rodney S. Poincaré inequalities and Neumann problems for the variable exponent setting. Mathematics in Engineering, 2022, 4(5): 1-22. https://doi.org/10.3934/mine.2022036

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Received: 10 April 2021
Accepted: 17 August 2021
Published: 15 October 2022
©2022 the Author(s), licensee AIMS Press.

This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)