Free Boundary Regularity for Almost Every Solution to the Signorini Problem.

We investigate the regularity of the free boundary for the Signorini problem in . It is known that regular points are -dimensional and . However, even for obstacles , the set of non-regular (or degenerate) points could be very large-e.g. with infinite measure. The only two assumptions under which a nice structure result for degenerate points has been established are when is analytic, and when . However, even in these cases, the set of degenerate points is in general -dimensional-as large as the set of regular points. In this work, we show for the first time that, "usually", the set of degenerate points is . Namely, we prove that, given any obstacle, for solution the non-regular part of the free boundary is at most -dimensional. This is the first result in this direction for the Signorini problem. Furthermore, we prove analogous results for the obstacle problem for the fractional Laplacian , and for the parabolic Signorini problem. In the parabolic Signorini problem, our main result establishes that the non-regular part of the free boundary is -dimensional for almost all times , for some . Finally, we construct some new examples of free boundaries with degenerate points.