Some infinite free boundary problems

Abstract:

Let $\Gamma$ be the boundary of an unbounded simply connected region $\mathcal {D}$, and let $\mathcal {C}(\Gamma )$ denote the family of all simply connected regions $\Delta \subset \mathcal {D}$ such that $\partial \Delta = \Gamma \cup \gamma$ where $\gamma \cap \Gamma$ contains only the infinite point. For $\Delta \in \mathcal {C}(\Gamma )$ we call $\gamma$ the free boundary of $\Delta$. Given a positive constant $\lambda$, we seek to find a region ${\Delta _\lambda } \in \mathcal {C}(\Gamma )$ with free boundary ${\gamma _\lambda }$ such that there is a bounded harmonic function V in ${\Delta _\lambda }$ with the properties that (i) $V = 0$ on $\Gamma$, (ii) $V = 1$ on $\gamma$, (iii) $\left | {{\text {grad }}V(z)} \right | = \lambda$ for $z \in {\gamma _\lambda }$. We give sufficient conditions for existence and uniqueness of ${\Delta _\lambda }$. We also give quantitative properties of ${\gamma _\lambda }$.