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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Energy-finite solutions of $ \Delta u=Pu$ and Dirichlet mappings


Author: Moses Glasner
Journal: Proc. Amer. Math. Soc. 29 (1971), 553-556
MSC: Primary 53.72; Secondary 30.00
MathSciNet review: 0279734
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Abstract: Let R, S be noncompact Riemannian m-manifolds and let $ T:R \to S$ be a Dirichlet mapping. Consider a nonnegative locally bounded measurable m-form P on R and set $ Q = T_\ast ^{ - 1}P$, the pull-back of P under $ {T^{ - 1}}$. Denote by $ PE(R)\;(QE(S)$ resp.) the space of energy-finite solutions of $ \Delta u = Pu$ on R ( $ \Delta u = Qu$ on S resp.). The spaces $ PE(R)$ and $ QE(S)$ are isomorphic, the isomorphism being bicontinuous with respect to the energy norms and preserves the sup norm of bounded solutions.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1971-0279734-X
PII: S 0002-9939(1971)0279734-X
Keywords: Dirichlet mappings, quasi-conformal mapping, quasiisometry, solution of $ \Delta u = Pu$, Riemannian manifold, energy integral, Royden compactification
Article copyright: © Copyright 1971 American Mathematical Society