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

We consider Orlicz–Laplace equation $-div(\frac{{\phi }^{\mathrm{\prime }}(\mathrm{\mid }\mathrm{\nabla }u\mathrm{\mid })}{\mathrm{\mid }\mathrm{\nabla }u\mathrm{\mid }}\mathrm{\nabla }u)=f$ where $\phi$ is an Orlicz function and either $f=0$ or $f\in L^{\mathrm{\infty }}$. 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 $\frac{{\psi }^{\mathrm{\prime }}(\mathrm{\mid }\mathrm{\nabla }u\mathrm{\mid })}{\mathrm{\mid }\mathrm{\nabla }u\mathrm{\mid }}\mathrm{\nabla }u\in W_{loc}^{1,2}$. This work contributes to the building up of quantitative second order Sobolev regularity for solutions of nonlinear equations.