The variable coefficient thin obstacle problem: Optimal regularity and regularity of the regular free boundary

  • Wenhui Shi

    Mathematisches Institut, Universität Bonn, Endenicher Allee 60, 53115 Bonn, Germany
  • Herbert Koch

    Mathematisches Institut, Universität Bonn, Endenicher Allee 60, 53115 Bonn, Germany
  • Angkana Rüland

    Mathematical Institute of the University of Oxford, Andrew Wiles Building, Radcliffe Observatory Quarter, Woodstock Road, OX2 6GG Oxford, United Kingdom

Abstract

This article deals with the variable coefficient thin obstacle problem in dimensions. We address the regular free boundary regularity, the behavior of the solution close to the free boundary and the optimal regularity of the solution in a low regularity set-up.

We first discuss the case of zero obstacle and metrics with . In this framework, we prove the regularity of the regular free boundary and derive the leading order asymptotic expansion of solutions at regular free boundary points. We further show the optimal regularity of solutions. New ingredients include the use of the Reifenberg flatness of the regular free boundary, the construction of an (almost) optimal barrier function and the introduction of an appropriate splitting of the solution. Important insights depend on the consideration of various intrinsic geometric structures.

Based on variations of the arguments in [18] and the present article, we then also discuss the case of non-zero and interior thin obstacles. We obtain the optimal regularity of the solutions and the regularity of the regular free boundary for metrics and obstacles with .

Cite this article

Wenhui Shi, Herbert Koch, Angkana Rüland, The variable coefficient thin obstacle problem: Optimal regularity and regularity of the regular free boundary. Ann. Inst. H. Poincaré Anal. Non Linéaire 34 (2017), no. 4, pp. 845–897

DOI 10.1016/J.ANIHPC.2016.08.001