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The obstacle problem for the fractional Laplacian with critical drift

Abstract

We study the obstacle problem for the fractional Laplacian with drift, \(\min \{(-\varDelta )^su + b \cdot \nabla u,\,u -\varphi \} = 0\) in \(\mathbb {R}^n\), in the critical regime \(s = \frac{1}{2}\). Our main result establishes the \(C^{1,\alpha }\) regularity of the free boundary around any regular point \(x_0\), with an expansion of the form

where \(e \in \mathbb {S}^{n-1}\) is the normal vector to the free boundary, \(\sigma >0\), and \(c_0> 0\). We also establish an analogous result for more general nonlocal operators of order 1. In this case, the exponent \(\tilde{\gamma }(x_0)\) also depends on the operator.

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Correspondence to Xavier Fernández-Real.

Additional information

Communicated by Y. Giga.

X. Fernández-Real has been supported by ERC grant “Regularity and Stability in Partial Differential Equations (RSPDE)” and by a fellowship from “Obra Social la Caixa”. X. Ros-Oton has been supported by NSF Grant DMS-1565186 and by MINECO Grant MTM-2014-52402-C3-1-P (Spain).

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Fernández-Real, X., Ros-Oton, X. The obstacle problem for the fractional Laplacian with critical drift. Math. Ann. 371, 1683–1735 (2018). https://doi.org/10.1007/s00208-017-1600-9

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  • DOI: https://doi.org/10.1007/s00208-017-1600-9

Mathematics Subject Classification

  • 35R35
  • 47G20
  • 35B65