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Regularity estimates for the solution and the free boundary of the obstacle problem for the fractional Laplacian

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Abstract

We use a characterization of the fractional Laplacian as a Dirichlet to Neumann operator for an appropriate differential equation to study its obstacle problem. We write an equivalent characterization as a thin obstacle problem. In this way we are able to apply local type arguments to obtain sharp regularity estimates for the solution and study the regularity of the free boundary.

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Authors and Affiliations

  1. Department of Mathematics, University of Texas at Austin, Austin, TX, 78712-1082, USA

    Luis A. Caffarelli

  2. Department of Mathematics, Polytechnic Institute of Milan, Milan, 20133, Italy

    Sandro Salsa

  3. Courant Institute of Mathematical Sciences, New York, NY, 10012, USA

    Luis Silvestre

Authors
  1. Luis A. Caffarelli
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  2. Sandro Salsa
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  3. Luis Silvestre
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Correspondence to Luis A. Caffarelli.

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Caffarelli, L., Salsa, S. & Silvestre, L. Regularity estimates for the solution and the free boundary of the obstacle problem for the fractional Laplacian . Invent. math. 171, 425–461 (2008). https://doi.org/10.1007/s00222-007-0086-6

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  • DOI: https://doi.org/10.1007/s00222-007-0086-6

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