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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Some infinite free boundary problems


Authors: David E. Tepper and Gerald Wildenberg
Journal: Trans. Amer. Math. Soc. 248 (1979), 135-144
MSC: Primary 31A25
MathSciNet review: 521697
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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\vert {{\text{grad }}V(z)} \right\vert = \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 }$.


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DOI: https://doi.org/10.1090/S0002-9947-1979-0521697-5
Keywords: Harmonic function, Dirichlet problem, free boundary
Article copyright: © Copyright 1979 American Mathematical Society