Lipschitz regularity for parabolic fractional $p$-Laplace equations

TL;DR

This paper establishes local Lipschitz continuity in space for solutions to nonlocal parabolic p-Laplace equations under certain conditions, extending regularity results to a broad class of kernels.

Contribution

It proves Lipschitz regularity for solutions to nonlocal parabolic p-Laplace equations with minimal kernel assumptions, including discontinuous kernels.

Findings

Solutions are locally Lipschitz continuous in space.

Results hold uniformly in time for all p in (1,∞) and s in (0,1) with sp > p-1.

The proof offers a new approach avoiding blow-up arguments.

Abstract

We prove that local weak solutions to nonlocal parabolic $p$-Laplace equations are locally Lipschitz continuous in space, uniformly in time for every $1<p<\infty$ and $s \in (0,1)$ whenever $sp > p-1$. Our results hold for symmetric, translation-invariant kernels satisfying standard ellipticity bounds, including kernels that may be discontinuous and require only that the tail of the solution be bounded. In the linear case, our proof provides a different route avoiding blow up arguments and Liouville theorems.