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 $C^{1/2}$ 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 $\kappa \ge 3/2$, while at the non-contact points the homogeneity must take one of the values: $1/2,3/2,\dots ,m-1/2,\dots$.

16 pages, 2 figues