Regularity of the obstacle problem for a fractional power of the Laplace operator. (English) Zbl 1141.49035

Given a function \(\varphi\) and \(s \in (0,1)\), in this paper is considered the following obstacle problem: 1. \(u \geq \varphi\) in \(\mathbb R^n\); 2. \((\Delta)^s u \geq 0\) in \(\mathbb R^n\); 3. \((\Delta)^s u= 0\) for those \(x\) such that \(u(x) >\varphi(x)\); 4. \(\lim_{| x| \rightarrow +\infty} u(x)=0\).
The author proves that if \(\varphi\) is in \(C^{1,s}\) then the solution \(u\) is in \(C^{1,\alpha}\) for all \(\alpha <s\). In the case where the contact set \(u=\varphi\) is convex, the optimal regularity \(u \in C^{1,s}\) is obtained. Moreover, when \(\varphi\) is in \(C^{1,\beta}\) with \(\beta < s\) the solution is in \(C^{1,\alpha}\) for all \(\alpha <\beta\). Finally some applications of the results are presented. In particular, interesting considerations on a well known Signorini problem are given.


49N60 Regularity of solutions in optimal control
35B65 Smoothness and regularity of solutions to PDEs
93B07 Observability
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