Regularity theory for mixed local and nonlocal parabolic p-Laplace equations

TL;DR

This paper develops a regularity theory for solutions to mixed local and nonlocal parabolic p-Laplace equations, proving local boundedness and Hölder continuity using advanced inequalities and iterative methods.

Contribution

It introduces a novel approach combining Caccioppoli inequalities and De Giorgi-Nash-Moser iteration for mixed local and nonlocal equations, establishing regularity results.

Findings

Proved local boundedness of weak solutions.

Established Hölder continuity of solutions.

Extended regularity theory to mixed local and nonlocal operators.

Abstract

We investigate the mixed local and nonlocal parabolic $p$-Laplace equation \begin{align*} \partial_t u(x,t)-\Delta_p u(x,t)+\mathcal{L}u(x,t)=0, \end{align*} where $\Delta_p$ is the local $p$-Laplace operator and $\mathcal{L}$ is the nonlocal $p$-Laplace operator. Based on the combination of suitable Caccioppoli-type inequality and Logarithmic Lemma with a De Giorgi-Nash-Moser iteration, we establish the local boundedness and H\"{o}lder continuity of weak solutions for such equations.