Full and partial regularity for a class of nonlinear free boundary problems

In this paper we classify the nonnegative global minimizers of the functional$J_F\left(u\right)=\underset{\Omega }{\int }F\left(\right|\nabla u{|}^2)+{\lambda }^2{\chi }_{\{u>0\}},$ where F satisfies some structural conditions and ${\chi }_D$ is the characteristic function of a set $D\subset R^n$. We compute the second variation of the energy and study the properties of the stability operator. The free boundary $\partial \{u>0\}$ can be seen as a rectifiable $n-1$ varifold. If the free boundary is a Lipschitz multigraph then we show that the first variation of this varifold is bounded. Hence one can use Allard's monotonicity formula to prove the existence of tangent cones modulo a set of small Hausdorff dimension. In particular, we prove that if $n=3$ and the ellipticity constants of the quasilinear elliptic operator generated by F are close to 1 then the conical free boundary must be flat.