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

Abstract

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 $$r_i^{\pi /(2\alpha _i)}$$ at transition points of Signorini to Dirichlet or Neumann conditions but $$r_i^{\pi /\alpha _i}$$ at kinks of the Signorini boundary, with $$\alpha _i$$ being the internal angle of the domain at these critical points.