Optimal second-order regularity for the p-Laplace system

TL;DR

This paper proves optimal second-order regularity results for solutions to the p-Laplace system with L^2 right-hand side, establishing equivalence of norms and applicability to convex domains with minimal boundary regularity.

Contribution

It introduces new second-order regularity estimates for the p-Laplace system with minimal boundary regularity assumptions, extending previous results.

Findings

Gradient expression belongs to W^{1,2}

Norm equivalence between W^{1,2} and L^2 right-hand side

Results apply to convex domains and include local estimates

Abstract

Second-order estimates are established for solutions to the $p$-Laplace system with right-hand side in $L^2$. The nonlinear expression of the gradient under the divergence operator is shown to belong to $W^{1,2}$, and hence to enjoy the best possible degree of regularity. Moreover, its norm in $W^{1,2}$ is proved to be equivalent to the norm of the right-hand side in $L^2$. Our global results apply to solutions to both Dirichlet and Neumann problems, and entail minimal regularity of the boundary of the domain. In particular, our conclusions hold for arbitrary bounded convex domains. Local estimates for local solutions are provided as well.