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

Abstract

We consider Orlicz–Laplace equation $-div(\frac{{\phi }^{\prime }\left(|\nabla u|\right)}{|\nabla u|}\nabla u)=f$ where $\phi$ 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 $\phi$ in a suitable sense, then $\frac{{\psi }^{\prime }\left(|\nabla u|\right)}{|\nabla u|}\nabla u\in W_{\text{loc}}^{1,2}$. This work contributes to the building up of quantitative second order Sobolev regularity for solutions of nonlinear equations.