Local second order regularity of solutions to elliptic Orlicz-Laplace equation

TL;DR

This paper establishes local second order regularity results for solutions to the Orlicz--Laplace equation, showing that certain nonlinear expressions involving the gradient belong to Sobolev spaces under specific conditions.

Contribution

It introduces new second order regularity results for solutions to the Orlicz--Laplace equation, extending the understanding of regularity in nonlinear elliptic PDEs.

Findings

Solutions exhibit local second order Sobolev regularity.

Regularity holds when comparing Orlicz functions close to each other.

Advances the quantitative understanding of second order regularity in nonlinear PDEs.

Abstract

We consider Orlicz--Laplace equation $-div(\frac{\varphi'(|\nabla u|)}{|\nabla u|}\nabla u)=f$ where $\varphi$ is an Orlicz function and either $f=0$ or $f\in L^\infty$. We prove local second order regularity results for the weak solutions $u$ of the Orlicz--Laplace equation. More precisely, we show that if $\psi$ is another Orlicz function that is close to $\varphi$ in a suitable sense, then $\frac{\psi'(|\nabla u|)}{|\nabla u|}\nabla u\in W^{1,2}_{loc}$. This work contributes to the building up of quantitative second order Sobolev regularity for solutions of nonlinear equations.