Higher regularity of the free boundary in the elliptic Signorini problem

Herbert Koch, Arshak Petrosyan, Wenhui Shi

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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 C0,1/2) 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 Lp theory available for that operator, we can bootstrap the regularity of the Legendre transform up to real analyticity, which implies the real analyticity of the free boundary.

Original languageEnglish
Article number10442
Pages (from-to)3-44
Number of pages42
JournalNonlinear Analysis, Theory, Methods and Applications
Publication statusPublished - 26 Oct 2015
Externally publishedYes


  • Almgren's frequency
  • Baouendi-Grushin operator
  • Higher regularity
  • Partial hodograph-Legendre transform
  • Real analyticity
  • Signorini problem
  • Smoothness
  • Subelliptic equations
  • Thin free boundary
  • Thin obstacle problem

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