The structure of the free boundary for lower dimensional obstacle   problems
  We study the regularity of the "free surface" in boundary obstacle problems. We show that near a non-degenerate point the free boundary is a $C^{1,\alpha}$ $(n-2)$-dimensional surface in $\real^{n-1}$. 
I. Athanasopoulos, L. A. Caffarelli, and S. Salsa