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 $J(u):=\int_Ω|\nabla u|^p +λ_+^p\,χ_{\{u>0\}} +λ_-^p\,χ_{\{u\le 0\}}, \quad 1<p<\infty.$ Here $Ω\subset \R^N$ is a bounded smooth domain and $λ_\pm$ are positive constants such that $λ_+^p-λ^p_->0$. 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 $p=2$.

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