Free Boundary Problem

Abstract

Let $\mathcal{D}$ be a doubly connected region in the complex plane limited by the infinite point and a convex set $\Gamma $. If $\lambda > 0$, then we study the existence, uniqueness and geometry of annuli $\omega \subset \mathcal{D}$ having $\Gamma as one boundary component and another boundary component $\gamma $, such that there exists a harmonic function V in $\omega satisfying : (a) $V = 0$ on $\Gamma $, (b) $V = 1$ on $\gamma $ and (c) $| \operatorname{grad}V | = \lambda on $\gamma $.