Contact of a thin free boundary with a fixed one in the Signorini problem

TL;DR

This paper investigates the regularity and blowup behavior of solutions to the Signorini problem near a fixed boundary, establishing sharp regularity results and characterizing homogeneity at contact and non-contact points.

Contribution

It provides the first detailed analysis of the contact between a free boundary and a fixed boundary in the Signorini problem, including regularity and homogeneity classifications.

Findings

Solutions are at least $C^{1/2}$ regular, which is sharp.

Blowup solutions at contact points have homogeneity $/2$ or higher.

At non-contact points, homogeneity takes specific fractional values.

Abstract

We study the Signorini problem near a fixed boundary, where the solution is "clamped down" or "glued." We show that in general the solutions are at least $C^{1/2}$ regular and that this regularity is sharp. We prove that near the actual points of contact of the free boundary with the fixed one the blowup solutions must have homogeneity $\kappa\geq 3/2$, while at the non-contact points the homogeneity must take one of the values: $1/2, 3/2, \ldots, m-1/2, \ldots$.