论文标题
混合本地非局部操作员及其应用的边界规律性
Boundary regularity of mixed local-nonlocal operators and its application
论文作者
论文摘要
让$ω$为$ \ m athbb {r}^n $和$ u \ in C（\ Mathbb {r}^n）$ solves \ solves \ begin {equation*} \ begin {aligned}ΔU + a iu + a iu + a + c_0 | du | du | du \ geq -k \ quad \ text {in} \; ω，\quadΔU + a iu -c_0 | du | \ leq k \ quad \ text {in} \; ω，\ Quad U = 0 \ Quad \ text {in} \; ω^c，\ end {Aligned} \ end {equation*}在粘度意义上，其中$ 0 \ leq a \ leq a \ leq a_0 $，$ c_0，k \ geq 0 $和$ i $是合适的非局部运算符。我们表明，对于某些$κ\ in（0,1）$，$ u/δ$在$ c^κ（\barΩ）$中，其中$δ（x）= {\ rm dist}（x，x，ω^c）$。使用此结果，我们还确定了c^{1，γ}（\barΩ）$中的$ u \。最后，我们将这些结果应用于研究局部非局部运营商的过度确定问题。
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.
