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

Abstract

We show that the singular set Σ in the classical obstacle problem can be locally covered by a C ∞ 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 Σ \ 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.

Free boundary partial regularity in the thin obstacle problem

“….. where 𝑄 is a (𝜆 + 𝑚)-homogeneous function which vanishes on the set {𝑥 𝑛 = 0} ∩ {𝑞 • = 0} and is harmonic outside of it. Such 𝑄 will give precisely the next order in the expansion, and we expect to be able to apply similar methods, combined with ideas from [15], to 𝑢 − 𝑞 • − 𝑄 (instead of 𝑢 − 𝑞 • ). One of the complications that will make this more enhanced result significantly more technical than the present one is the necessity to construct and work with refined Anstätze of every order, similarly to [12,15] in the case of the obstacle and Stefan problem, respectively.…”