Poincar\'e Inequalities and Neumann Problems for the Variable Exponent Setting

TL;DR

This paper establishes a link between Poincaré inequalities and the existence of solutions to Neumann problems involving the variable exponent p(x)-Laplacian, extending previous results to nonstandard growth conditions.

Contribution

It generalizes the equivalence between Poincaré inequalities and Neumann problem solutions to the variable exponent setting with degenerate p(x)-Laplacian.

Findings

Proves equivalence between Poincaré inequalities and weak solutions for p(x)-Laplacian.

Extends previous results from constant to variable exponent spaces.

Provides theoretical foundation for analysis of nonstandard growth elliptic equations.

Abstract

We extend the results of [5], where we proved an equivalence between weighted Poincar\'e 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 Poincar\'e inequalities in variable exponent spaces and solutions to a degenerate $p(x)$-Laplacian, a non-linear elliptic equation with nonstandard growth conditions.