On the boundary branching set of the one-phase problem

TL;DR

This paper investigates the structure of the boundary branching set in the one-phase Bernoulli free boundary problem, establishing Hausdorff dimension bounds and deriving boundary unique continuation results using Almgren-type frequency functions.

Contribution

It proves that the boundary branching set has Hausdorff dimension at most d-2 for the one-phase problem and extends similar estimates to the two-phase problem under an analytic separation condition.

Findings

Boundary branching set has Hausdorff dimension at most d-2.

Derived strong boundary unique continuation results.

Established monotonicity of a boundary Almgren-type frequency function.

Abstract

We consider minimizers of the one-phase Bernoulli free boundary problem in domains with analytic fixed boundary. In any dimension $d$, we prove that the branching set at the boundary has Hausdorff dimension at most $d-2$. As a consequence, we also obtain an analogous estimate on the branching set for solutions to the two-phase problem under an analytic separation condition. Moreover, as a byproduct of our analysis we obtain strong boundary unique continuation results for quasilinear operators and thin-obstacle variational inequalities. The approach we use is based on the (almost-)monotonicity of a boundary Almgren-type frequency function, obtained via regularity estimates and a Calder\'on-Zygmund decomposition in the spirit of Almgren-De Lellis-Spadaro.