Lipschitz potential estimates for diffusion with jumps

Abstract

For $p\in (1,\infty )$ and $s\in (0,1)$, we consider the following mixed local-nonlocal equation $$-\hat{I}{”}_{p}u+(-\hat{I}{”}_p{)}^{s}u=f\text{in}\hat{I}\text{c◯},$$ where $\hat{I}\text{c◯}\subset {\mathbb{R}}^d$ is a bounded domain and the function $f\in L_{loc}^1(\hat{I}\text{c◯})$. 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 $p=2$.

20 pages