Boundary regularity of mixed local-nonlocal operators and its application

TL;DR

This paper investigates the boundary regularity of solutions to mixed local-nonlocal PDEs in smooth domains, establishing Hölder continuity of the normalized solution and its gradient, and applies these findings to overdetermined problems.

Contribution

It proves boundary regularity results for solutions to mixed local-nonlocal PDEs and applies these to overdetermined boundary value problems.

Findings

$u/\delta$ is in $C^{\kappa}(ar \Omega)$ for some $\kappa\in(0,1)$

$u$ belongs to $C^{1, \gamma}(ar \Omega)$

Application to overdetermined problems with mixed operators

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} \Delta u + a Iu + C_0|Du| \geq -K\quad \text{in}\; \Omega, \quad \Delta u + a Iu - C_0|Du|\leq K \quad \text{in}\; \Omega, \quad u=0\quad \text{in}\; \Omega^c, \end{aligned} \end{equation*} in the viscosity sense, where $0\leq a\leq A_0$, $C_0, K\geq 0$, 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 $\delta(x)={\rm dist}(x, \Omega^c)$. 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.