Hardy-type inequality in variable exponent Lebesgue spaces derived from nonlinear problem

TL;DR

This paper establishes a new family of weighted Hardy-type inequalities in variable exponent Lebesgue spaces, derived from solutions to nonlinear PDE inequalities involving the p(x)-Laplacian, with applications to one-dimensional examples.

Contribution

It introduces Hardy-type inequalities in variable exponent spaces based on solutions to nonlinear PDEs, including new Caccioppoli inequalities and explicit examples.

Findings

Derived weighted Hardy inequalities involving measures from PDE solutions.

Established new Caccioppoli-type inequalities for solutions to p(x)-Laplacian inequalities.

Provided illustrative one-dimensional examples demonstrating the inequalities.

Abstract

We derive a family of weighted Hardy-type inequalities in the variable exponent Lebesgue space with an additional term of the form \[ \int_\Omega\ |\xi|^{p(x)} \mu_{1,\beta}(dx)\leqslant \int_\Omega |\nabla \xi|^{p(x)}\mu_{2,\beta}(dx)+\int_\Omega \left|\xi{\log \xi} \right|^{p(x)} \mu_{3,\beta}(dx), \] where $\xi$ is any compactly supported Lipschitz function. The involved measures depend on a certain solution to the partial differential inequality involving $p(x)$-Laplacian ${-}\Delta_{p(x)} u\geqslant \Phi$, where $\Phi$ is a given locally integrable function, and $u$ is defined on an open and not necessarily bounded subset $\Omega\subseteq\mathbb{R}^n $, and a certain parameter $\beta$. We derive new Caccioppoli-type inequality for the solution $u$. As its consequence we get Hardy-type inequality. We illustrate the result by several one-dimensional examples.