Lipschitz potential estimates for diffusion with jumps

TL;DR

This paper establishes gradient potential estimates for solutions to mixed local-nonlocal p-Laplacian equations with jumps, connecting local and nonlocal behaviors and extending known results even in the linear case.

Contribution

It provides new gradient potential estimates for a broad class of mixed local-nonlocal equations, including the linear case, across all parameter ranges.

Findings

Derived gradient potential estimates for solutions in the entire parameter range.

Connected local and nonlocal estimates, unifying the theory.

Extended results to the linear case p=2, even in previous gaps.

Abstract

For $p \in (1, \infty)$ and $s \in (0,1)$, we consider the following mixed local-nonlocal equation $$ - \Delta_p u + (-\Delta_p)^s u = f \; \text{in} \; \Omega,$$ where $\Omega \subset \mathbb{R}^d$ is a bounded domain and the function $f \in L_{loc}^1(\Omega)$. Depending on the dimension $d$, we prove gradient potential estimates of weak solutions for the entire ranges of $p$ and $s$. As a byproduct, we recover the corresponding estimates in the purely diffusive setup, providing connections between the local and nonlocal aspects of the equation. Our results are new, even for the linear case $p=2$.