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

TL;DR

This paper establishes second order regularity results for nonlinear elliptic and parabolic systems with operators depending on the deformation tensor, using an innovative $A$-approximation method for operators with $(p, abla)$-structure in the range $1<p extless 2$.

Contribution

It introduces the $A$-approximation technique to prove natural second order regularity for systems with $p$-structure, a method not previously applied in this context.

Findings

Proved second order regularity up to the boundary for $p$-structure systems.

Extended results to time-dependent (parabolic) systems.

Demonstrated the effectiveness of the $A$-approximation method in this setting.

Abstract

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.