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

The paper deals with the second order regularity properties of the weak solutions $u\in W^{1,\phi }(\mathrm{\Omega },{\mathrm{\Re }}^n)$ } of systems of the form \begin{equation*}\label{equareg} -\dive A(x,\E u)=f, \end{equation*} in a bounded domain $\mathrm{\Omega }\subset {\mathbb{R}}^n$, $n>2$, 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 $\mathrm{\Phi }$. We prove the Sobolev regularity of a function of the symmetric gradient $\text{E}u$ 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.