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

Higher Regularity for the fractional thin obstacle problem