Optimal second order boundary regularity for solutions to $p$-Laplace equations

TL;DR

This paper establishes optimal second order boundary regularity for solutions to p-Laplace equations, showing that under certain conditions, the regularity of a key vector field is sharp up to the boundary.

Contribution

It proves sharp second order boundary regularity estimates for solutions of p-Laplace equations, extending previous results and confirming the optimality of regularity under specific assumptions.

Findings

The vector field | abla u|^{p-2} abla u is in W^{1,2} up to the boundary.

Regularity results are sharp and optimal under the given assumptions.

The study confirms the necessity of L^2 regularity of the source term.

Abstract

Solutions to $p$-Laplace equations are not, in general, of class $C^2$. The study of Sobolev regularity of the second derivatives is, therefore, a crucial issue. An important contribution by Cianchi and Maz'ya shows that, if the source term is in $L^2$, then the field $|\nabla u|^{p-2}\nabla u$ is in $W^{1,2}$. The $L^2$-regularity of the source term is also a necessary condition. Here, under suitable assumptions, we obtain sharp second order estimates, thus proving the optimal regularity of the vector field $|\nabla u|^{p-2}\nabla u$, up to the boundary.