$C^\infty$ partial regularity of the singular set in the obstacle problem

TL;DR

This paper proves that the singular set in the classical obstacle problem is mostly smooth and can be covered by a smooth hypersurface, except for a small exceptional set, with implications for free boundary regularity and stability.

Contribution

It establishes $C^ abla$ regularity of the singular set in the obstacle problem, introducing a detailed structure and stability analysis of singular points.

Findings

Singular set can be covered by a $C^ abla$ hypersurface outside a small exceptional set.

Exceptional set has Hausdorff dimension at most $n-2$, countable in 2D.

Singular points are unstable under boundary perturbations, with applications to Hele-Shaw flow.

Abstract

We show that the singular set $\Sigma$ in the classical obstacle problem can be locally covered by a $C^\infty$ hypersurface, up to an "exceptional" set $E$, which has Hausdorff dimension at most $n-2$ (countable, in the $n=2$ case). Outside this exceptional set, the solution admits a polynomial expansion of arbitrarily large order. We also prove that $\Sigma\setminus E$ is extremely unstable with respect to monotone perturbations of the boundary datum. We apply this result to the planar Hele-Shaw flow, showing that the free boundary can have singular points for at most countable many times.