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Analysis of a free boundary at contact points with Lipschitz data

In this paper we consider a minimization problem for the functional $$ J(u)=\int_{B_1^+}|\nabla u|\sp 2+λ_{+}^2χ_{\{u>0\}}+λ_{-}^2χ_{\{u\leq0\}}, $$ in the upper half ball $B_1^+\subset\R^n, n\geq 2$ subject to a Lipschitz continuous Dirichlet data on $\partial B_1^+$. More precisely we assume that $0\in \partial \{u>0\}$ and the derivative of the boundary data has a jump discontinuity. If $0\in \bar{\partial(\{u>0\} \cap B_1^+)}$ then (for $n=2$ or $n>3$ and one-phase case) we prove, among other things, that the free boundary $\partial \{u>0\}$ approaches the origin along one of the two possible planes given by $$ γx_1 = \pm x_2, $$ where $γ$ is an explicit constant given by the boundary data and $λ_\pm$ the constants seen in the definition of $J(u)$. Moreover the speed of the approach to $γx_1=x_2$ is uniform.