Global gradient estimates for the $p(\cdot)$-Laplacian

TL;DR

This paper establishes Calderón-Zygmund type estimates for the variable exponent p-Laplacian, showing that integrability of the data implies integrability of the gradient, with new techniques for variable exponent analysis.

Contribution

It provides the first global gradient estimates for the non-homogeneous p(x)-Laplacian with variable exponents, including domain-independent local estimates and novel analytical methods.

Findings

$|G|^{p( ext{·})} otin L^q$ implies $|D u|^{p( ext{·})} otin L^q$ for all $q \\geq 1$

Established domain-independent local estimates for the p(x)-Laplacian

Developed new techniques for variable exponent analysis

Abstract

We consider Calder\'on-Zygmund type estimates for the non-homogeneous $p(\cdot)$-Laplacian system $ -\text{div}(|D u|^{p(\cdot)-2} Du) = -\text{div}(|G|^{p(\cdot)-2} G),$ where $p$ is a variable exponent. We show that $|G|^{p(\cdot)} \in L^q(\mathbb{R}^n)$ implies $|D u|^{p(\cdot)} \in L^q(\mathbb{R}^n)$ for any $q \geq 1$. We also prove local estimates independent of the size of the domain and introduce new techniques to variable analysis.