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Bounded solutions of the equation $ \Delta u=pu$ on a Riemannian manifold


Author: Young K. Kwon
Journal: Proc. Amer. Math. Soc. 45 (1974), 377-382
MSC: Primary 53C20; Secondary 30A48
DOI: https://doi.org/10.1090/S0002-9939-1974-0350654-8
MathSciNet review: 0350654
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Abstract: Given a nonnegative $ {C^1}$-function $ p(x)$ on a Riemannian manifold $ R$, denote by $ {B_p}(R)$ the Banach space of all bounded $ {C^2}$-solutions of $ \Delta u = pu$ with the sup-norm. The purpose of this paper is to give a unified treatment of $ {B_p}(R)$ on the Wiener compactification for all densities $ p(x)$. This approach not only generalizes classical results in the harmonic case $ (p \equiv 0)$, but it also enables one, for example, to easily compare the Banach space structure of the spaces $ {B_p}(R)$ for various densities $ p(x)$. Typically, let $ \beta (p)$ be the set of all $ p$-potential nondensity points in the Wiener harmonic boundary $ \Delta $, and $ {C_p}(\Delta )$ the space of bounded continuous functions $ f$ on $ \Delta $ with $ f\vert\Delta - \beta (p) \equiv 0$.

Theorem. The spaces $ {B_p}(R)$ and $ {C_p}(\Delta )$ are isometrically isomorphic with respect to the sup-norm.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1974-0350654-8
Keywords: Riemannian manifold, $ p$-potential nondensity point, Wiener compactification, Wiener harmonic boundary, Green's potential, Fredholm integral equation, Perron family
Article copyright: © Copyright 1974 American Mathematical Society

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