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Higher regularity of the free boundary in the elliptic Signorini problem

In this paper we study the higher regularity of the free boundary for the elliptic Signorini problem. By using a partial hodograph-Legendre transformation we show that the regular part of the free boundary is real analytic. The first complication in the study is the invertibility of the hodograph transform (which is only $C^{0,1/2}$) which can be overcome by studying the precise asymptotic behavior of the solutions near regular free boundary points. The second and main complication in the study is that the equation satisfied by the Legendre transform is degenerate. However, the equation has a subelliptic structure and can be viewed as a perturbation of the Baouendi-Grushin operator. By using the $L^p$ theory available for that operator, we can bootstrap the regularity of the Legendre transform up to real analyticity, which imples the real analyticity of the free boundary.