H\"older regularity for parabolic fractional $p$-Laplacian

TL;DR

This paper proves local H"older regularity for weak solutions to parabolic fractional p-Laplace equations with measurable kernels, using refined DeGiorgi iteration and intrinsic scaling methods, even in the linear case.

Contribution

It introduces a novel proof technique that avoids logarithmic estimates and comparison principles, advancing understanding of regularity in nonlocal parabolic equations.

Findings

Establishes H"older continuity for solutions with measurable kernels.

Refines intrinsic scaling methods for nonlocal equations.

Provides a new proof approach even for linear cases.

Abstract

Local H\"older regularity is established for certain weak solutions to a class of parabolic fractional $p$-Laplace equations with merely measurable kernels. The proof uses DeGiorgi's iteration and refines DiBenedetto's intrinsic scaling method. The control of a nonlocal integral of solutions in the reduction of oscillation plays a crucial role and entails delicate analysis in this intrinsic scaling scenario. Dispensing with any logarithmic estimate and any comparison principle, the proof is new even for the linear case.