Maximal Sobolev regularity of the stress tensor for the symmetric gradient p-Laplace system

TL;DR

This paper establishes maximal Sobolev regularity for the stress tensor in the symmetric p-Laplace system, advancing understanding of second-order differentiability in models of non-Newtonian fluids and elastic materials.

Contribution

It proves second-order differentiability properties and maximal Sobolev regularity of solutions to the symmetric p-Laplace system, a novel result in the analysis of nonlinear PDEs.

Findings

Maximal Sobolev regularity of the stress tensor is achieved.

Second-order differentiability of solutions is established.

Results apply to models in mathematical physics like non-Newtonian fluids.

Abstract

The symmetric $p$-Laplace operator enters various models in mathematical physics, such as incompressible materials with power-type hardening and non-Newtonian fluids. In this work, second-order differentiability properties of solutions to the symmetric $p$-Laplace system are established. They are formulated as maximal Sobolev regularity of the nonlinear stress tensor for locally square integrable right-hand sides.