Second-order estimates for the $p$-Laplacian in RCD spaces

TL;DR

This paper proves new second-order regularity results for the p-Laplacian in RCD spaces, including Lipschitz regularity in finite dimensions, advancing understanding of nonlinear PDEs in metric measure spaces.

Contribution

It establishes quantitative second-order Sobolev regularity for p-Laplacian functions in RCD spaces, extending known results to a broader class of functions and spaces.

Findings

Second-order Sobolev regularity for p-Laplacian functions in RCD spaces

Lipschitz regularity in finite-dimensional RCD spaces under integrability conditions

Applicability to p-Laplacian eigenfunctions and p-harmonic functions with compact level sets

Abstract

We establish quantitative second-order Sobolev regularity for functions having a $2$-integrable $p$-Laplacian in bounded RCD spaces, with $p$ in a suitable range. In the finite-dimensional case, we also obtain Lipschitz regularity under the assumption that $p$-Laplacian is sufficiently integrable. Our results cover both $p$-Laplacian eigenfunctions and $p$-harmonic functions having relatively compact level sets.