Analysis of a free boundary at contact points with Lipschitz data

Abstract:

In this paper we consider a minimization problem for the functional \[ J(u)=\int _{B_1^+}|\nabla u|^ 2+\lambda _{+}^2\chi _{\{u>0\}}+\lambda _{-}^2\chi _{\{u\leq 0\}} \] in the upper half ball $B_1^+\subset \mathbb {R}^n, n\geq 2$, 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 $0\in \overline {\partial ( \{u>0\} \cap B_1^+)}$, then (for $n=2$ or $n\geq 3$ and the 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 \[ \gamma x_1 = \pm x_2, \] where $\gamma$ is an explicit constant given by the boundary data and $\lambda _\pm$ the constants seen in the definition of $J(u)$. Moreover the speed of the approach to $\gamma x_1=x_2$ is uniform.