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| Content Provider | Springer Nature Link |
|---|---|
| Author | Caffarelli, Luis Serra, Joaquim Ros Oton, Xavier |
| Copyright Year | 2016 |
| Abstract | We study the obstacle problem for integro-differential operators of order 2s, with $$s\in (0,1)$$ . Our main result establish that the free boundary is $$C^{1,\gamma }$$ and $$u\in C^{1,s}$$ near all regular points. Namely, we prove the following dichotomy at all free boundary points $$x_0\in \partial \{u=\varphi \}$$ : (i) either $$u(x)-\varphi (x)=c\,d^{1+s}(x)+o(|x-x_0|^{1+s+\alpha })$$ u ( x ) - φ ( x ) = c d 1 + s ( x ) + o ( | x - x 0 | 1 + s + α ) for some $$c>0$$ c > 0 , (ii) or $$u(x)-\varphi (x)=o(|x-x_0|^{1+s+\alpha })$$ u ( x ) - φ ( x ) = o ( | x - x 0 | 1 + s + α ) , 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 $$C^{1,\gamma }$$ and $$u\in C^{1,s}$$ near those points. These results were only known for the fractional Laplacian [2], 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. |
| Ending Page | 1211 |
| Page Count | 57 |
| Starting Page | 1155 |
| File Format | |
| ISSN | 00209910 |
| e-ISSN | 14321297 |
| Journal | Inventiones mathematicae |
| Issue Number | 3 |
| Volume Number | 208 |
| Language | English |
| Publisher | Springer Berlin Heidelberg |
| Publisher Date | 2016-11-14 |
| Publisher Place | Berlin, Heidelberg |
| Access Restriction | One Nation One Subscription (ONOS) |
| Subject Keyword | Smoothness and regularity of solutions Integro-differential operators Free boundary problems Mathematics |
| Content Type | Text |
| Resource Type | Article |
| Subject | Mathematics |
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