Free Boundary Regularity for Almost Every Solution to the Signorini Problem

Abstract

We investigate the regularity of the free boundary for the Signorini problem in {mathbb {R}}^{n+1}. It is known that regular points are (n-1)-dimensional and C^infty . However, even for C^infty obstacles varphi , the set of non-regular (or degenerate) points could be very large—e.g. with infinite {mathcal {H}}^{n-1} measure. The only two assumptions under which a nice structure result for degenerate points has been established are when varphi is analytic, and when Delta varphi < 0. However, even in these cases, the set of degenerate points is in general (n-1)-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 C^infty obstacle, for almost every solution the non-regular part of the free boundary is at most (n-2)-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 (-Delta )^s, 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 (n-1-alpha _circ )-dimensional for almost all times t, for some alpha _circ > 0. Finally, we construct some new examples of free boundaries with degenerate points.