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

Abstract

In this paper we study the two-phase Bernoulli type free boundary problem arising from the minimization of the functional $$J(u):={\int }_{\Omega }|\nabla u{|}^p+{\lambda }_{+}^p{\chi }_{\{u>0\}}+{\lambda }_{-}^p{\chi }_{\{u\le 0\}},1<p<\infty .$$ Here $\Omega \subset {\mathbb{R}}^N$ is a bounded smooth domain and ${\lambda }_{\pm }$ are positive constants such that ${\lambda }_{+}^p-{\lambda }_{-}^p>0$. We prove the following dichotomy: if $x_0$ is a free boundary point then either the free boundary is smooth near $x_0$ or $u$ has linear growth at $x_0$. 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 $p=2$.