Sobolev regularity of the symmetric gradient of solutions to a class of ϕ-Laplacian systems

The paper deals with the second order regularity properties of the weak solutions u∈ W1,ϕ(Ω, ⁿ) } of systems of the form {equation*}{equareg} - A(x, u)=f, {equation*} in a bounded domain Ω⊂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 $Φ$. We prove the Sobolev regularity of a function of the symmetric gradient 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.