A one-sided two phase Bernoulli free boundary problem

TL;DR

This paper investigates a two-phase Bernoulli free boundary problem with impenetrability conditions, establishing regularity results and characterizing the singular set, advancing understanding of free boundary regularity in multi-phase problems.

Contribution

It introduces a new two-membrane free boundary problem with harmonic functions, proving epsilon-regularity and regularity of flat points, and bounds on the singular set's Hausdorff dimension.

Findings

Flat points have $C^{1,1/2}$ regularity.

Singular set has Hausdorff dimension at most $N-5$.

Established epsilon-regularity theorem with sharp modulus of continuity.

Abstract

We study a two-phase free boundary problem in which the two-phases satisfy an impenetrability condition. Precisely, we have two ordered positive functions, which are harmonic in their supports, satisfy a Bernoulli condition on the one-phase part of the free boundary and a two-phase condition on the collapsed part of the free boundary. For this two-membrane type problem, we prove an epsilon-regularity theorem with sharp modulus of continuity. Precisely, we show that at flat points each of the two boundaries is $C^{1,1/2}$ regular surface. Moreover, we show that the remaining singular set has Hausdorff dimension at most $N-5$ as in the case of the classical one-phase problem, $N$ being the dimension of the space.