New York Journal of Mathematics
Volume 25 (2019), 745-838


Herbert Koch, Angkana Rüland, and Wenhui Shi

Higher regularity for the fractional thin obstacle problem

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Published: September 5, 2019.
Keywords: Variable coefficient fractional Signorini problem, variable coefficient fractional thin obstacle problem, thin free boundary, Hodograph-Legendre transform.
Subject: Primary 35R35.

In this article we investigate the higher regularity properties of the regular free boundary in the fractional thin obstacle problem. Relying on a Hodograph-Legendre transform, we show that for smooth or analytic obstacles the regular free boundary is smooth or analytic, respectively. This leads to the analysis of a fully nonlinear, degenerate (sub)elliptic operator which we identify as a (fully nonlinear) perturbation of the fractional Baouendi-Grushin Laplacian. Using its intrinsic geometry and adapted function spaces, we invoke the analytic implicit function theorem to deduce analyticity of the regular free boundary.


H.K. acknowledges support by the DFG through SFB 1060, Bonn.
A.R. acknowledges a Junior Research Fellowship at Christ Church, Oxford University.
W.S. is supported by the Hausdorff Center for Mathematics, Bonn.

Author information

Herbert Koch:
Mathematisches Institut
Universität Bonn
Endenicher Allee 60, 53115 Bonn, Germany


Angkana Rüland:
Max-Planck Institute for Mathematics in the Sciences
Inselstrasse 22
04105 Leipzig, Germany


Wenhui Shi:
School of Mathematics
Monash University
9 Rainforest Walk, 3168 Clayton, Australia