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

Given a function φ and s(0,1), in this paper is considered the following obstacle problem: 1. uφ in n ; 2. (Δ) s u0 in n ; 3. (Δ) s u=0 for those x such that u(x)>φ(x); 4. lim |x|+ u(x)=0.

The author proves that if φ is in C 1,s then the solution u is in C 1,α for all α<s. In the case where the contact set u=φ is convex, the optimal regularity uC 1,s is obtained. Moreover, when φ is in C 1,β with β<s the solution is in C 1,α for all α<β. Finally some applications of the results are presented. In particular, interesting considerations on a well known Signorini problem are given.

MSC:
49N60Regularity of solutions in calculus of variations
35B65Smoothness and regularity of solutions of PDE
93B07Observability