Regularity of Lipschitz free boundaries for the thin one-phase problem

Abstract

We study regularity properties of the free boundary for the thin one-phase problem which consists of minimizing the energy functionalamong all functions u ≥ 0 which are fixed on ∂Ω.We prove that the free boundary F (u) = ∂ R n {u > 0} of a minimizer u has locally finite H n−1 measure and is a C 2,α surface except on a small singular set of Hausdorff dimension n − 3. We also obtain C 2,α regularity of Lipschitz free boundaries of viscosity solutions associated to this problem.

C^\infty regularity of certain thin free boundaries

“…This was proved in [DS1], and its proof is similar to the proof above. We sketch below a slightly different proof that uses Theorem 3.1.…”

“…The free boundary condition forw is understood in the viscosity sense defined in [DS1], i.e.w cannot be touched on L say at 0 by below by a function of the form…”

“…In [DS1], [DS2] we continued investigating this regularity issue. These results parallel the regularity theory for the free boundary in the classical one-phase problem and in the theory of minimal surfaces.…”

Thin one-phase almost minimizers

“…The study of the regularity of thin one-phase free boundaries was initiated by the first author and Roquejoffre in [DR], where it was shown that "flat" free boundaries are C 1,α via a viscosity approach. In [DS1,DS2,DS3] we investigated further properties of minimizers by combining variational and nonvariational techniques. We showed that Lipschitz free boundaries are of class C ∞ and local minimizers of E have smooth free boundary except possibly for a small singular set of Hausdorff dimension n − 3.…”

“…We follow here a similar approach, by showing that almost minimizers of E are "viscosity solutions" in this more general sense (see Subsection 3.5). Roughly, our viscosity solutions satisfy comparison in a neighborhood of a touching point whose size depends on the properties of the test functions.Once this is established, then we employ the techniques developed in [DR,DS1,DS2] to study the regularity of the free boundary of viscosity solutions.…”