Analysis of a free boundary at contact points with Lipschitz data

TL;DR

This paper studies the behavior of free boundaries in a minimization problem with Lipschitz boundary data, showing they approach specific planes near contact points with a uniform speed.

Contribution

It establishes the asymptotic approach of the free boundary to explicit planes near contact points with Lipschitz data, including the speed of convergence.

Findings

Free boundary approaches planes $\, ext{or}\, -x_2$ near contact points.

The approach is along explicit lines determined by boundary data.

The convergence speed to these lines is uniform.

Abstract

In this paper we consider a minimization problem for the functional $$ J(u)=\int_{B_1^+}|\nabla u|\sp 2+\lambda_{+}^2\chi_{\{u>0\}}+\lambda_{-}^2\chi_{\{u\leq0\}}, $$ 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 $$ \gamma x_1 = \pm x_2, $$ 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.