H\"{o}lder regularity for the parabolic perturbed fractional 1-Laplace equations

TL;DR

This paper establishes local Hölder continuity for weak solutions to a class of parabolic perturbed fractional 1-Laplace equations, combining energy estimates and integral decomposition techniques.

Contribution

It is the first to develop a regularity theory for nonlocal parabolic equations of this type, providing quantitative Hölder estimates based on structural parameters.

Findings

Solutions are spatially α-Hölder continuous with 0<α<min{1, s_p p/(p-1)}.

Solutions are γ-Hölder continuous in time, with γ depending on fractional differentiability indexes.

Sobolev regularity of solutions is established for both super- and sub-quadratic cases.

Abstract

This paper studies the regularity of weak solutions to a class of parabolic perturbed fractional $1$-Laplace equations. Our analysis combines finite difference quotients, energy estimates, and iterative arguments, with a key step being the decomposition of the nonlocal integral into local and nonlocal components to handle their contributions separately. We aim to show the local H\"{o}lder continuity of weak solutions within the parabolic domain. More precisely, the solutions are spatially $\alpha$-H\"{o}lder continuous with $0<\alpha<\min\left\lbrace1, \frac{s_p p}{p-1} \right\rbrace$ and $\gamma$-H\"{o}lder continuous in time, where the value of $\gamma$ is determined by the fractional differentiability indexes $s_1$, $s_p$ and the exponent $p$. For both the super-quadratic case ($p\ge 2$) and the sub-quadratic case ($1<p<2$), we establish the Sobolev regularity of solutions, which underpins the derivation of H\"{o}lder continuity. All estimates are quantitative and depend only on the structural parameters of the equation. To the best of our knowledge, this is the first attempt to develop a regularity theory for such nonlocal parabolic equations.