# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
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. $u \ge \varphi$ in $\Bbb R^n$; 2. $(\Delta)^s u \ge 0$ in $\Bbb R^n$; 3. $(\Delta)^s u= 0$ for those $x$ such that $u(x) >\varphi(x)$; 4. $\lim_{\vert x\vert \rightarrow +\infty} u(x)=0$. 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.

##### MSC:
 49N60 Regularity of solutions in calculus of variations 35B65 Smoothness and regularity of solutions of PDE 93B07 Observability
##### Keywords:
optimal regularity; Signorini problem
Full Text: