Higher Regularity for the fractional thin obstacle problem

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.

arXiv comment79 pages

References in corpus (5)

• Regularity estimates for the solution and the free boundary to the obstacle problem for the fractional Laplacian
• Obstacle problems for integro-differential operators: Regularity of solutions and free boundaries
• Higher regularity of the free boundary in the elliptic Signorini problem
• The Variable Coefficient Thin Obstacle Problem: Higher Regularity
• The Variable Coefficient Thin Obstacle Problem: Optimal Regularity and Regularity of the Regular Free Boundary

Cited by in corpus (2)

• Strong unique continuation for the higher order fractional Laplacian
• Boundary estimates and a Wiener criterion for the fractional Laplacian