A free boundary problem driven by the biharmonic operator

In this paper we consider the minimization of the functional $$J[u]:=\int_\Omega |\Delta u|^2+\chi_{\{u>0\}}$$ in the admissible class of functions $$\mathcal A:= \left\{u\in W^{2, 2}(\Omega) {\mbox{ s.t. }} u-u_0\in W^{1,2}_0(\Omega) \right\}.$$ Here, $\Omega$ is a smooth and bounded domain and $u_0\in W^{2,2}(\Omega)$ is a given function defining the Navier type boundary condition. 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 $\{ u>0\}$ 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 $\partial\{ u>0\}$ is rank-2 flat. Using a dichotomy method for nonlinear free boundary problems, we also show that, at the free boundary points $x\in \Omega$ where $\nabla u(x)=0$, the free boundary is either well approximated by zero sets of quadratic polynomials, i.e. $\partial\{ u>0\}$ is rank-2 flat, or $u$ has quadratic growth. Differently from the classical free boundary problems driven by the Laplacian operator, the one-phase minimizers present structural differences with respect to the minimizers, and one notion is not included into the other. In addition, one-phase 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 non-standard way by these two ingredients.