Revista Matemática Iberoamericana


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Volume 35, Issue 5, 2019, pp. 1309–1365
DOI: 10.4171/rmi/1087

Published online: 2019-06-04

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

Nicola Garofalo[1] and Xavier Ros-Oton[2]

(1) Università di Padova, Italy
(2) Universität Zürich, Switzerland

We study the singular part of the free boundary in the obstacle problem for the fractional Laplacian, $\min\big\{(-\Delta)^su,\,u-\varphi\big\}=0$ in $\mathbb{R}^n$, for general obstacles $\varphi$. 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 $s\in(0,1)$.

Keywords: Obstacle problem, fractional Laplacian, free boundary, monotonicity formulas

Garofalo Nicola, Ros-Oton Xavier: Structure and regularity of the singular set in the obstacle problem for the fractional Laplacian. Rev. Mat. Iberoam. 35 (2019), 1309-1365. doi: 10.4171/rmi/1087