We consider the minimization of the functional
$$J\left[u\right]:={\int}_{\Omega}\left(\Delta u{}^{2}+{\chi}_{\left\{u>0\right\}}\right)$$ 
in the admissible class of functions
$$\mathcal{\mathcal{A}}:=\left\{u\in {W}^{2,2}\left(\Omega \right):u{u}_{0}\in {W}_{0}^{1,2}\left(\Omega \right)\right\}.$$ 
Here,
$\Omega $ is a smooth
and bounded domain of ${\mathbb{R}}^{n}$
and
${u}_{0}\in {W}^{2,2}\left(\Omega \right)$ is
a given function defining the Navier type boundary condition.
When
$n=2$, the
functional $J$
can be interpreted as a sum of the linearized Willmore energy of the graph
of $u$ and the
area of
$\left\{u>0\right\}$ on
the $xy$plane.
The regularity of a minimizer $u$
and that of the free boundary
$\partial \left\{u>0\right\}$
are very complicated problems. The most intriguing part of this is to study the structure of
$\partial \left\{u>0\right\}$ near singular
points, where
$\nabla u=0$
(of course, at the nonsingular free boundary points
where $\nabla u\ne 0$, the free
boundary is locally
${C}^{1}$
smooth).
The scale invariance of the problem suggests that, at
the singular points of the free boundary, quadratic growth
of $u$ is expected.
We prove that
$u$
is quadratically nondegenerate at the singular free boundary points using a
refinement of Whitney’s cube decomposition, which applies, if, for instance, the
set $\left\{u>0\right\}$ is
a John domain.
The optimal growth is linked with the approximate symmetries of the free
boundary. More precisely, if at small scales the free boundary can be approximated
by zero level sets of a quadratic degree two homogeneous polynomial, then we say
that
$\partial \left\{u>0\right\}$
is rank2 flat.
Using a dichotomy method for nonlinear free boundary problems, we also show that, at the free
boundary points $x\in \Omega $,
where $\nabla u\left(x\right)=0$,
the free boundary is either well approximated by zero sets of quadratic polynomials, i.e.,
$\partial \left\{u>0\right\}$ is rank2
flat, or
$u$
has quadratic growth.
More can be said if
$n=2$,
in which case we obtain a monotonicity formula and show that, at the singular points
of the free boundary where the free boundary is not well approximated by level sets
of quadratic polynomials, the blowup of the minimizer is a homogeneous function of
degree two.
In particular, if $n=2$
and
$\left\{u>0\right\}$ is a John
domain, then we get that the blowup of the free boundary is a cone; and in the onephase case,
it follows that
$\partial \left\{u>0\right\}$
possesses a tangent line in the measure theoretic sense.
Differently from the classical free boundary problems driven by the Laplacian
operator, the onephase minimizers present structural differences with respect to the
minimizers, and one notion is not included into the other. In addition, onephase
minimizers arise from the combination of a volume type free boundary problem and
an obstacle type problem, hence their growth condition is influenced in a
nonstandard way by these two ingredients.
