Free Boundary Regularity for Almost Every Solution to the Signorini Problem

We investigate the regularity of the free boundary for the Signorini problem in [Formula: see text] . It is known that regular points are [Formula: see text] -dimensional and [Formula: see text] . However, even for [Formula: see text] obstacles [Formula: see text] , the set of non-regular (or degenerate) points could be very large—e.g. with infinite [Formula: see text] measure. The only two assumptions under which a nice structure result for degenerate points has been established are when [Formula: see text] is analytic, and when [Formula: see text] . However, even in these cases, the set of degenerate points is in general [Formula: see text] -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 small. Namely, we prove that, given any [Formula: see text] obstacle, for almost every solution the non-regular part of the free boundary is at most [Formula: see text] -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 [Formula: see text] , 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 [Formula: see text] -dimensional for almost all times t, for some [Formula: see text] . Finally, we construct some new examples of free boundaries with degenerate points.