A regularity result for the p-laplacian near uniform ellipticity

TL;DR

This paper proves enhanced regularity of solutions to p-Laplace boundary value problems near the linear case p=2, showing second derivatives in high L^q spaces and improved C^{1,eta} regularity, especially relevant in higher dimensions.

Contribution

It establishes that second derivatives of p-Laplace solutions are in L^q for large q when p is close to 2, and improves C^{1,eta} regularity, linking regularity to the classical Calderón-Zygmund constant.

Findings

Second derivatives in L^q for p near 2

C^{1,eta} regularity improves to C^{1,1^-} near p=2

Regularity results are significant for higher dimensions n>2

Abstract

We consider weak solutions to a class of Dirichlet boundary value problems invloving the $p$-Laplace operator, and prove that the second weak derivatives are in $L^{q}$ with $q$ as large as it is desirable, provided $p$ is sufficiently close to $p_0=2$. We show that this phenomenon is driven by the classical Calder\'on-Zygmund constant. As a byproduct of our analysis we show that $C^{1,\alpha}$ regularity improves up to $C^{1,1^-}$, when p is close enough to 2. This result we believe it is particularly interesting in higher dimensions $n>2,$ when optimal $C^{1,\alpha}$ regularity is related to the optimal regularity of $p$-harmonic mappings, which is still open.