Regularity theory for general stable operators

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• 作者：Xavier Ros-Oton^a ; ^{ros.oton@math.utexas.edu" class="auth_mail" title="E-mail the corresponding author} ; Joaquim Serra^b ; ^{joaquim.serra@upc.edu" class="auth_mail" title="E-mail the corresponding author}
• 关键词：35B65 ; 60G52 ; 47G30
• 刊名：Journal of Differential Equations
• 出版年：2016
• 出版时间：15 June 2016
• 年：2016
• 卷：260
• 期：12
• 页码：8675-8715
• 全文大小：517 K

文摘

We establish sharp regularity estimates for solutions to 131bf" title="Click to view the MathML source">Lu=f$Lu=f$ in Ω⊂Rⁿ$\mathrm{\Omega }\subset {\mathbb{R}}^n$, L being the generator of any stable and symmetric Lévy process. Such nonlocal operators L   depend on a finite measure on Sⁿ⁻¹$S^{n-1}$, called the spectral measure.

First, we study the interior regularity of solutions to 131bf" title="Click to view the MathML source">Lu=f$Lu=f$ in B₁$B_1$. We prove that if f   is C^α$C^{\alpha }$ then u   belong to C^α+2s$C^{\alpha +2s}$ whenever α+2s$\alpha +2s$ is not an integer. In case f∈L^∞$f\in L^{\infty }$, we show that the solution u   is C^2s$C^{2s}$ when s≠1/2$s\ne 1/2$, and C^2s−ϵ$C^{2s-ϵ}$ for all ϵ>0$ϵ>0$ when s=1/2$s=1/2$.

Then, we study the boundary regularity of solutions to 131bf" title="Click to view the MathML source">Lu=f$Lu=f$ in Ω, u=0$u=0$ in Rⁿ∖Ω${\mathbb{R}}^n\setminus \mathrm{\Omega }$, in C^1,1$C^{1,1}$ domains Ω. We show that solutions u   satisfy $u/d^s\in C^{s-ϵ}(\bar{\mathrm{\Omega }})$ for all ϵ>0$ϵ>0$, where d is the distance to ∂Ω.

Finally, we show that our results are sharp by constructing two counterexamples.