Higher Regularity for the fractional thin obstacle problem

2016/05/21 by Herbert Koch, Koch, Herbert, Angkana Rüland +3
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.1605.06662

openalex publication_date 2016/05/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

Citations

• Regularity of the obstacle problem for a fractional power of the laplace operator
• Regularity estimates for the solution and the free boundary of the obstacle problem for the fractional Laplacian

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