Global second order optimal regularity for the vectorial $p$-Laplacian

TL;DR

This paper establishes optimal global second order regularity results for solutions to vectorial p-Laplace equations, advancing understanding of the stress field regularity in nonlinear PDEs.

Contribution

It provides the first optimal global second order regularity estimates for vectorial p-Laplace equations, extending previous partial results.

Findings

Proves global second order regularity for solutions

Establishes optimal estimates for the stress field

Advances regularity theory for nonlinear PDEs

Abstract

We obtain optimal regularity results for solutions to vectorial $p$-Laplace equations $$ -{\boldsymbol \Delta}_p{\boldsymbol u}=-\operatorname{\bf div}(|D{\boldsymbol u}|^{p-2}D{\boldsymbol u}) = {\boldsymbol f}(x)\,\, \mbox{ in $\Omega$}\,.$$ More precisely we address the issue of global second order estimates for the stress field.