论文标题
混合局部和非局部椭圆方程溶液的规律性结果
Regularity results for solutions of mixed local and nonlocal elliptic equations
论文作者
论文摘要
我们认为由布朗和莱维过程的叠加驱动的局部非局部半线性椭圆方程\ begin {equation*} \ left \ left \ {\ oken {arnay} {ll} {ll} -ΔU +（-Δ）^s u = g（x，u） u = 0＆\ hbox {in $ \ mathbb {r}^n \backslashΩ$。} \\ \ end {array} \ right。 \ end {equation*}在非线性术语$ g $的温和假设下，我们显示了Moser Iteration方法的任何弱解决方案（不更改符号或签名）的$ l^\ infty $有界。此外，当$ s \ in（0，\ frac {1} {2}] $中时，我们获得了解决方案是唯一的，实际上属于（0,1）$的任何$α\ for任何$α\。
We consider the mixed local-nonlocal semi-linear elliptic equations driven by the superposition of Brownian and Lévy processes \begin{equation*} \left\{ \begin{array}{ll} - Δu + (-Δ)^s u = g(x,u) & \hbox{in $Ω$,} u=0 & \hbox{in $\mathbb{R}^n\backslashΩ$.} \\ \end{array} \right. \end{equation*} Under mild assumptions on the nonlinear term $g$, we show the $L^\infty$ boundedness of any weak solution (either not changing sign or sign-changing) by the Moser iteration method. Moreover, when $s\in (0, \frac{1}{2}]$, we obtain that the solution is unique and actually belongs to $C^{1,α}(\overlineΩ)$ for any $α\in (0,1)$.
