Regularity for one-phase Bernoulli problems with discontinuous weights and applications

TL;DR

This paper establishes regularity results for one-phase Bernoulli free boundary problems with discontinuous weights, showing smoothness outside a small singular set and constructing examples of singular free boundaries.

Contribution

It proves $C^{1, eta}$ regularity of free boundaries with discontinuous weights and constructs singular solutions in two dimensions for weights close to constant.

Findings

Free boundaries are $C^{1, eta}$ outside a small singular set.

In 2D, free boundaries are fully $C^{1, eta}$ regular.

Constructs singular free boundaries in 2D with weights near constant.

Abstract

We study a one-phase Bernoulli free boundary problem with weight function admitting a discontinuity along a smooth jump interface. In any dimension $N\ge 2$, we show the $C^{1, \alpha}$ regularity of the free boundary outside of a singular set of Hausdorff dimension at most $N-3$. In particular, we prove that the free boundaries are $C^{1, \alpha}$ regular in dimension $N=2$, while in dimension $N=3$ the singular set can contain at most a finite number of points. We use this result to construct singular free boundaries in dimension $N=2$, which are minimizing for one-phase functionals with weight functions in $L^\infty$ that are arbitrarily close to a positive constant.