Fractional $p$-caloric functions are Lipschitz

TL;DR

This paper proves that solutions to the fractional p-Laplace equation are Lipschitz continuous in space and time under certain conditions, and establishes fundamental properties like comparison principles and solution equivalences.

Contribution

It demonstrates Lipschitz regularity for weak solutions of the fractional p-Laplace equation in the degenerate range, and links weak and viscosity solutions.

Findings

Weak solutions are Lipschitz continuous in space.

Solutions are Lipschitz in time if p > 1/(1-s).

Comparison principle and equivalence of solution notions are established.

Abstract

We study the parabolic fractional $p-$Laplace equation $$\p_t u+(-\Delta_p)^su = 0$$ in the degenerate range \(2 \leq p < 2/(1-s)\). We show that weak solutions are Lipschitz continuous in space and, if \(p > 1/(1-s)\), also in time. We also prove a comparison principle for both weak and viscosity solutions, and establish the equivalence between the two notions of solution.