  In this paper we prove uniqueness of blow-ups and $C^{1,\log}$-regularity for
the free-boundary of minimizers of the Alt-Caffarelli functional at points
where one blow-up has an isolated singularity. We do this by establishing a
(log-)epiperimetric inequality for the Weiss energy for traces close to that of
a cone with isolated singularity, whose free-boundary is graphical and smooth
over that of the cone in the sphere. With additional assumptions on the cone,
we can prove a classical epiperimetric inequality which can be applied to
deduce a $C^{1,\alpha}$ regularity result. We also show that these additional
assumptions are satisfied by the De Silva-Jerison-type cones, which are the
only known examples of minimizing cones with isolated singularity. Our approach
draws a connection between epiperimetric inequalities and the \L ojasiewicz
inequality, and, to our knowledge, provides the first regularity result at
singular points in the one-phase Bernoulli problem.
