Obstacle problems for integro-differential operators: regularity of solutions and free boundaries

• 成果类型: Article
• 署名作者: Caffarelli, Luis; Ros-Oton, Xavier; Serra, Joaquim
• 署名单位: University of Texas System; University of Texas Austin; Universitat Politecnica de Catalunya
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-016-0703-3
• 发表日期: 2017
• 页码: 1155-1211
• 关键词: equations

摘要

We study the obstacle problem for integro-differential operators of order 2s, with s is an element of(0,1). Our main results establish the optimal C<^>{1,s} regularity of solutions u, and the C<^>{1,\alpha} regularity of the free boundary near regular points. Namely, we prove the following dichotomy at all free boundary points x_0\in\partial\{u=\varphi\}: (i) either u(x)-\varphi(x)=c\,d<^>{1+s}(x)+o(|x-x_0|<^>{1+s+\alpha}) for some c>0, (ii) or u(x)-\varphi(x)=o(|x-x_0|<^>{1+s+\alpha}), where d is the distance to the contact set \{u=\varphi\}. Moreover, we show that the set of free boundary points x_0 satisfying (i) is open, and that the free boundary is C<^>{1,\alpha} near those points. These results were only known for the fractional Laplacian \cite{CSS}, and are completely new for more general integro-differential operators. The methods we develop here are purely nonlocal, and do not rely on any monotonicity-type formula for the operator. Thanks to this, our techniques can be applied in the much more general context of fully nonlinear integro-differential operators: we establish similar regularity results for obstacle problems with convex operators.