  In this paper we study the two-phase Bernoulli type free boundary problem
arising from the minimization of the functional $$ J(u):=\int_{\Omega}|\nabla
u|^p +\lambda_+^p\,\chi_{\{u>0\}} +\lambda_-^p\,\chi_{\{u\le 0\}}, \quad
1<p<\infty. $$ Here $\Omega \subset \R^N$ 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 $p=2$.
