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 $\phi$, 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 $\phi$ is analytic, and when $\Delta \phi <0$. 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.

References in corpus (6)

• An extension problem related to the fractional Laplacian
• Regularity estimates for the solution and the free boundary to the obstacle problem for the fractional Laplacian
• Fractional Laplacians on domains, a development of Hörmander's theory of mu-transmission pseudodifferential operators
• Some New Monotonicity Formulas and the Singular Set in the Lower Dimensional Obstacle Problem
• Direct epiperimetric inequalities for the thin obstacle problem and applications
• On the measure and the structure of the free boundary of the lower dimensional obstacle problem

Cited by in corpus (2)

• On the singular set in the thin obstacle problem: higher order blow-ups and the very thin obstacle problem
• $C^{\infty }$ partial regularity of the singular set in the obstacle problem