Regularity of the solution of the scalar Signorini problem in polygonal domains

TL;DR

This paper analyzes the regularity and structure of solutions to the Signorini problem in polygonal domains, revealing how boundary geometry influences solution singularities and the coincidence set.

Contribution

It provides a detailed description of the solution's regularity and singularities, including explicit formulas related to boundary angles, in polygonal domains.

Findings

The coincidence set consists of boundary parts and isolated points.

Leading singularities depend on boundary angles and transition types.

Explicit singularity exponents are derived for different boundary features.

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 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.