Structure and regularity of the singular set in the obstacle problem for the fractional Laplacian

  • Nicola Garofalo

    Università di Padova, Italy
  • Xavier Ros-Oton

    Universität Zürich, Switzerland
Structure and regularity of the singular set in the obstacle problem for the fractional Laplacian cover
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Abstract

We study the singular part of the free boundary in the obstacle problem for the fractional Laplacian, min{(Δ)su,uφ}=0\min\big\{(-\Delta)^su,\,u-\varphi\big\}=0 in Rn\mathbb{R}^n, for general obstacles φ\varphi. Our main result establishes the complete structure and regularity of the singular set. To prove it, we construct new monotonicity formulas of Monneau-type that extend those in those of Garofalo–Petrosyan to all s(0,1)s\in(0,1).

Cite this article

Nicola Garofalo, Xavier Ros-Oton, Structure and regularity of the singular set in the obstacle problem for the fractional Laplacian. Rev. Mat. Iberoam. 35 (2019), no. 5, pp. 1309–1365

DOI 10.4171/RMI/1087