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

Abstract

In this paper we classify the nonnegative global minimizers of the functional \[ J_F(u)=\int_\Omega F(|\nabla u|^2)+\lambda^2\chi_{\{u>0\}}, \] where $F$ satisfies some structural conditions and $\chi_D$ is the characteristic function of a set $D\subset \mathbb R^n$. 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 and 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.