Stratification of free boundary points for a two-phase variational problem

In this paper we study the two-phase Bernoulli type free boundary problem arising from the minimization of the functional J(u):=∫Ω|∇u|p+λp+χ{u>0}+λp−χ{u≤0},1<p<∞. Here $\Omega \subset \R^N$ is a bounded smooth domain and λ± are positive constants such that λp+−λp−>0. We prove the following dichotomy: if x0 is a free boundary point then either the free boundary is smooth near x0 or u has linear growth at x0. Furthermore, we show that for p>1 the free boundary has locally finite perimeter and the set of non-smooth points of free boundary is of zero (N−1)-dimensional Hausdorff measure. Our approach is new even for the classical case p=2.