Contact of a thin free boundary with a fixed one in the Signorini   problem
  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: $1/2, 3/2, \ldots, m-1/2, \ldots$. 
Norayr Matevosyan and Arshak Petrosyan
math.AP