Natural second order regularity for systems in the case $1<p\leq 2$ using the $A$-approximation

In this paper we consider nonlinear problems with an operator depending only on the deformation tensor. We consider the class of operators derived from a potential and with $(p,\delta)$ structure, for $1<p\leq 2$ and for all $\delta\geq0$. We apply the so called $A$-approximation method to approximate the operator by another one with linear growth. This allows us to prove the "natural" second order regularity (up to the boundary) in the case of homogeneous Dirichlet boundary conditions. We focus on the steady (elliptic) case, but results are given also in the time-dependent (parabolic) case. Results presented are not completely new, but the method we apply was not used before in this setting.