Free Boundary Regularity for Almost Every Solution to the Signorini Problem

We investigate the regularity of the free boundary for the Signorini problem in Rn+1. It is known that regular points are (n−1)-dimensional and C∞. However, even for C∞ obstacles φ, the set of non-regular (or degenerate) points could be very large—e.g. with infinite Hn−1 measure. The only two assumptions under which a nice structure result for degenerate points has been established are when φ is analytic, and when Δφ0. Finally, we construct some new examples of free boundaries with degenerate points.