Free boundary regularity for almost every solution to the Signorini problem

TL;DR

This paper demonstrates that for almost every solution to the Signorini problem, the set of degenerate free boundary points is significantly smaller than previously known, and extends these results to fractional and parabolic cases.

Contribution

It proves that typically, the non-regular free boundary points are at most (n-2)-dimensional, a novel result for the Signorini problem and related free boundary problems.

Findings

Non-regular points are at most (n-2)-dimensional for almost every solution.

Results extend to fractional Laplacian obstacle problem and parabolic Signorini problem.

Constructs examples of free boundaries with degenerate points.

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.