Boundary regularity of mixed local-nonlocal operators and its application

Let $Ω$ 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 $0\leq a\leq A_0$, $C_0, K\geq 0$, and $I$ is a suitable nonlocal operator. We show that $u/δ$ is in $C^κ(\bar Ω)$ for some $κ\in (0,1)$, where $δ(x)={\rm dist}(x, Ω^c)$. Using this result, we also establish that $u\in C^{1, γ}(\barΩ)$. Finally, we apply these results to study an overdetermined problem for mixed local-nonlocal operators.

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