Boundary regularity of mixed local-nonlocal operators and its application

Abstract

Let $\Omega$ be a bounded $C^2$ domain in ${\mathbb{R}}^n$ and $u\in C({\mathbb{R}}^n)$ solves \begin{equation*} \begin{aligned} Δu + a Iu + C_0|Du| \geq -K\quad \text{in}\; Ω, \quad Δu + a Iu - C_0|Du|\leq K \quad \text{in}\; Ω, \quad u=0\quad \text{in}\; Ω^c, \end{aligned} \end{equation*} in the viscosity sense, where $0\le a\le A_0$, $C_0,K\ge 0$, and $I$ is a suitable nonlocal operator. We show that $u/\delta$ is in $C^{\kappa }(\bar{\Omega })$ for some $\kappa \in (0,1)$, where $\delta (x)=\mathrm{dist}(x,{\Omega }^c)$. Using this result, we also establish that $u\in C^{1,\gamma }(\bar{\Omega })$. Finally, we apply these results to study an overdetermined problem for mixed local-nonlocal operators.

26 pages