Second-order regularity for parabolic p-Laplace problems

TL;DR

This paper proves optimal second-order regularity for solutions to nonlinear parabolic p-Laplacian problems, showing solutions are strong under minimal boundary regularity, even in convex domains.

Contribution

It establishes new second-order regularity results for parabolic p-Laplacian problems with minimal boundary regularity assumptions.

Findings

Solutions have optimal second-order regularity in space variables.

Generalized solutions are shown to be strong solutions.

Results hold in arbitrary bounded convex domains.

Abstract

Optimal second-order regularity in the space variables is established for solutions to Cauchy-Dirichlet problems for nonlinear parabolic equations and systems of $p$-Laplacian type, with square-integrable right-hand sides and initial data in a Sobolev space. As a consequence, generalized solutions are shown to be strong solutions. Minimal regularity on the boundary of the domain is required, though the results are new even for smooth domains. In particular, they hold in arbitrary bounded convex domains.