A regularity result for the p-laplacian near uniform ellipticity

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 Calderón-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.