On a Free Boundary Problem, the Starlike Case

Abstract

Let $\mathcal{D}$ be a doubly connected region limited by the infinite point and a starlike boundary component $\Gamma $ which does not reduce to a point. If $\lambda $ is a given positive number, we show there exists a unique annulus $\omega _\lambda \subset \mathcal{D}$ having $\Gamma $ as one boundary component and another boundary component $\gamma _\lambda $ such that there is a harmonic function V in $\omega _\lambda $ satisfying $V \equiv 0$ on $\Gamma $, $V \equiv 1$ on $\gamma _\lambda $ and $| {{\operatorname{grad}}V_\lambda } | \equiv \lambda $ on $\gamma _\lambda $. We also show that $\gamma _\lambda $ is starlike.