Regularity of the Solution of the Scalar Signorini Problem in Polygonal Domains

The Signorini problem for the Laplace operator is considered in a general polygonal domain. It is proved that the coincidence set consists of a finite number of boundary parts plus a finite number of isolated points. The regularity of the solution is described. In particular, we show that the leading singularity is in general riπ/(2αi)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r_i^{\pi /(2\alpha _i)}$$\end{document} at transition points of Signorini to Dirichlet or Neumann conditions but riπ/αi\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r_i^{\pi /\alpha _i}$$\end{document} at kinks of the Signorini boundary, with αi\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha _i$$\end{document} being the internal angle of the domain at these critical points.