Higher regularity for the fractional thin obstacle problem

Herbert Koch, Angkana Rüland, Wenhui Shi

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Abstract

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.

Original languageEnglish
Pages (from-to)745-838
Number of pages94
JournalNew York Journal of Mathematics
Volume25
Publication statusPublished - 1 Jan 2019

Keywords

  • Hodograph-legendre transform
  • Thin free boundary
  • Variable coefficient fractional signorini problem
  • Variable coefficient fractional thin obstacle problem

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