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

Abstract

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

References in corpus (1)

• Campanato estimates for the generalized Stokes System