On the boundary branching set of the one-phase problem

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.