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

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 C1/2 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 κ ≥ 3/2, while at the non-contact points the homogeneity must take one of the values: 1/2, 3/2, . . . , m− 1/2, . . . .