×

zbMATH — the first resource for mathematics

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.

MSC:
49N60 Regularity of solutions in optimal control
35B65 Smoothness and regularity of solutions to PDEs
93B07 Observability
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] ; Optimal regularity of lower dimensional obstacle problems. Preprint. · Zbl 1108.35038
[2] Boyarchenko, SIAM J Control Optim 40 pp 1663– (2002)
[3] Caffarelli, Comm Partial Differential Equations 4 pp 1067– (1979)
[4] Caffarelli, J Fourier Anal Appl 4 pp 383– (1998)
[5] Caffarelli, J Analyse Math 37 pp 285– (1980)
[6] ; Financial modelling with jump processes. Chapman & Hall/CRC Financial Mathematics Series. Chapman & Hall/CRC, Boca Raton, Fla., 2004.
[7] Frehse, Boll Un Mat Ital (4) 6 pp 312– (1972)
[8] ; Elliptic partial differential equations of second order. Reprint of the 1998 edition. Classics in Mathematics. Springer, Berlin, 2001.
[9] Foundations of modern potential theory. Die Grundlehren der mathematischen Wissenschaften, Band 180. Translated from the Russian by A. P. Doohovskoy. Springer, New York, 1972.
[10] Levendorski??, Int J Theor Appl Finance 7 pp 303– (2004)
[11] Pham, Appl Math Optim 35 pp 145– (1997)
[12] Doctoral dissertation. University of British Columbia, Vancouver, 1978.
[13] Hölder estimates for solutions of integro differential equations like the fractional Laplace. Preprint. · Zbl 1101.45004
[14] Singular integrals and differentiability properties of functions. Princeton Mathematical Series, No. 30. Princeton University Press, Princeton, N.J., 1970.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.