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

$$\begin{aligned} \begin{aligned} \Delta u + a Iu + C_0|Du| \ge -K\quad \text {in}\; \Omega , \quad \Delta u + a Iu - C_0|Du|\le K \quad \text {in}\; \Omega , \quad u=0\quad \text {in}\; \Omega ^c, \end{aligned} \end{aligned}$$

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 $C^{\upkappa }(\bar{\Omega })$ for some $\upkappa \in (0,1)$, where $\delta (x)=\mathop {\mathrm {dist}}\limits (x, \Omega ^c)$. 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.