  While there are numerous results on minimizers or stable solutions of the
Bernoulli problem proving regularity of the free boundary and analyzing
singularities, much less in known about critical points of the corresponding
energy. Saddle points of the energy (or of closely related energies) and
solutions of the corresponding time-dependent problem occur naturally in
applied problems such as water waves and combustion theory.
  For such critical points $u$ -- which can be obtained as limits of classical
solutions or limits of a singular perturbation problem -- it has been open
since [Weiss03] whether the singular set can be large and what equation the
measure $\Delta u$ satisfies, except for the case of two dimensions. In the
present result we use recent techniques such as a frequency formula for the
Bernoulli problem as well as the celebrated Naber-Valtorta procedure to answer
this more than 20 year old question in an affirmative way:
  For a closed class we call variational solutions of the Bernoulli problem, we
show that the topological free boundary $\partial \{u > 0\}$ (including
degenerate singular points $x$, at which $u(x + r \cdot)/r \rightarrow 0$ as
$r\rightarrow 0$) is countably $\mathcal{H}^{n-1}$-rectifiable and has locally
finite $\mathcal{H}^{n-1}$-measure, and we identify the measure $\Delta u$
completely. This gives a more precise characterization of the free boundary of
$u$ in arbitrary dimension than was previously available even in dimension two.
  We also show that limits of (not necessarily minimizing) classical solutions
as well as limits of critical points of a singularly perturbed energy are
variational solutions, so that the result above applies directly to all of
them.
