Boundary regularity of mixed local-nonlocal operators and its application

Let $\mathrm{\Omega }$ 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 $0\le a\le A_0$, $C_0, K\geq 0$,and$I$isasuitablenonlocaloperator\mathrm{.}Weshowthat$u/\delta$isin$C^{\kappa}(\bar \Omega)$forsome$\kappa\in (0,1)$,where$\delta(x)={\rm dist}(x, \Omega^c)$\mathrm{.}Usingthisresult,wealsoestablishthat$u\in C^{1, \gamma}(\bar\Omega)$. Finally, we apply these results to study an overdetermined problem for mixed local-nonlocal operators.