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Proceedings of the American Mathematical Society

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A uniqueness theorem for superharmonic functions in $ {\bf R}\sp{n}$

Author: J. L. Schiff
Journal: Proc. Amer. Math. Soc. 84 (1982), 362-364
MSC: Primary 31B05
Erratum: Proc. Amer. Math. Soc. 87 (1983), 378.
MathSciNet review: 640231
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Abstract: Let $ s(x)$ be a nonnegative superharmonic function defined on the $ n$-ball $ {B^n}(y;r)$ in $ {{\mathbf{R}}^n}, n \geqslant 3$. If $ s(x)$ tends to zero "too rapidly" as $ x$ tends to a single point $ \xi $ on the boundary of $ {B^n}(y;r)$, then we prove that $ s \equiv 0$. The same result can then be extended to domains $ D \subseteq {{\mathbf{R}}^n}$, whose boundary $ \partial D$ is locally $ {C^1}$ at $ \xi \in \partial D$. These results generalize some earlier work of the author and Ü. Kuran for $ n = 2$.

References [Enhancements On Off] (What's this?)

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  • [3] Ü. Kuran and J. L. Schiff, A uniqueness theorem for non-negative superharmonic functions in planar domains, J. Math. Anal. Appl. (to appear).
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Keywords: Superharmonic function, $ n$-ball, uniqueness theorem, Stolz domain
Article copyright: © Copyright 1982 American Mathematical Society