Sobolev regularity of the symmetric gradient of solutions to a class of $\phi$-Laplacian systems

TL;DR

This paper establishes second order Sobolev regularity for solutions to a class of $\

Contribution

It introduces new regularity results for solutions of $\

Findings

Solutions exhibit higher differentiability properties.

Regularity results depend on the nonlinear growth conditions.

Approximating problems with singular perturbations are effective.

Abstract

The paper deals with the second order regularity properties of the weak solutions $u\in W^{1,\phi}(\Omega, \real^n)$ } of systems of the form \begin{equation*}\label{equareg} -\dive A(x,\E u)=f, \end{equation*} in a bounded domain $\Omega\subset\R^n$, $n>2$, where the operator $ A(x,P)$ is Lipschitz continuous with respect to the $x$-variable and satisfies growth conditions with respect to the second variable expressed through a Young function $\Phi$. We prove the Sobolev regularity of a function of the symmetric gradient $\E u$ that takes into account the nonlinear growth of the operator $A(x,P)$, {assuming that the force term $f$ belongs to a suitable Orlicz-Sobolev space. {The main result is achieved through some uniform higher differentiability estimates for solutions to a class of approximating problems, constructed adding singular higher order perturbations to the system.