The variable coefficient thin obstacle problem: Higher regularity

In this article, we continue our investigation of the variable coefficients thin obstacle problem which was initiated in [20], [21]. Us­ing a partial Hodograph-Legendre transform and the implicit function theorem, we prove the higher order Holder regularity for the regular free boundary, if the associated coefficients are of the corresponding regular­ity. For the zero obstacle, this yields an improvement of a full derivative  for the free boundary regularity compared to the regularity of the co­efficients. In the presence of inhomogeneities, we gain three halves of a derivative for the free boundary regularity with respect to the regularity of the inhomogeneity. Further, we show analyticity of the regular free boundary for analytic coefficients. vVe also discuss the set-up of W1 ·P coefficients with p > n + 1 and LP inhomogeneities. Key ingredients in our analysis are the introduction of generalized Holder spaces, which al­low to interpret the transformed fully nonlinear, degenerate (sub)elliptic equation as a perturbation of the Baouendi-Grushin operator, various  uses of intrinsic geometries associated with appropriate operators, the application of the implicit function theorem to deduce (higher) regular­ity.