$L^{\Phi(A(x,\cdot))}$ estimate for the gradient in $W^{1,A(x,\cdot)}$

TL;DR

This paper establishes an $L^{ ext{Phi}(A(x,ullet))}$ estimate for gradients of minimizers in Musielak-Orlicz-Sobolev spaces, advancing regularity theory using Calderón-Zygmund decomposition under specific assumptions.

Contribution

It introduces a novel $L^{ ext{Phi}(A(x,ullet))}$ gradient estimate for minimizers in Musielak-Orlicz-Sobolev spaces, extending existing regularity results.

Findings

Gradient estimates in Musielak-Orlicz-Sobolev spaces established.

Uses Calderón-Zygmund decomposition technique.

Provides conditions under which the estimates hold.

Abstract

Under appropriate assumptions on the $N(\Omega)$-fucntion, the $L^{\Phi(A(x,\cdot))}$ estimate for the gradient of the minimizers of a class of energy functional in Musielak-Orlicz-Sobolev space $W^{1,A(x,\cdot)}$ is presented by using Calder\'{o}n-Zygmund decomposition.