Free boundary regularity for almost every solution to the Signorini problem

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 $\Delta \varphi0$. Finally, we construct some new examples of free boundaries with degenerate points