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**The variable coefficient thin obstacle problem: higher regularity.**
*(English)*
Zbl 1377.35289

Summary: In this article, we continue our investigation of the variable coefficients thin obstacle problem which was initiated in [H. Koch et al., Adv. Math. 301, 820–866 (2016; Zbl 1346.35240); “The variable coefficient thin obstacle problem: optimal regularity, free boundary regularity and first order asymptotics”, Ann. Inst. Henri Poincaré (to appear)]. Using a partial Hodograph-Legendre transform and the implicit function theorem, we prove the higher order Hölder 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. We also discuss the set-up of \(W^{1,p}\) coefficients with \(p>n+1\) and \(L^p\) inhomogeneities. Key ingredients in our analysis are the introduction of generalized Hölder 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.

### MSC:

35R35 | Free boundary problems for PDEs |

35J20 | Variational methods for second-order elliptic equations |