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

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.

70 pages

References in corpus (3)

• Regularity Theory for Elliptic PDE
• Volume estimates on the critical sets of solutions to elliptic PDEs
• Free boundary regularity for almost every solution to the Signorini problem