Analysis of a free boundary at contact points with Lipschitz data

Abstract

In this paper we consider a minimization problem for the functional $$ J(u)=\int_{B_1^+}|\nabla u|\sp 2+λ_{+}^2χ_{\{u>0\}}+λ_{-}^2χ_{\{u\leq0\}}, $$ in the upper half ball $B_1^{+}\subset {\mathbb{R}}^n,n\ge 2$ 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 $n>3$ 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 $$\gamma x_1=\pm x_2,$$ 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.

Accepted: Transactions of AMS, 2013