Bounded in the mean solutions of $\triangle u=Pu$ on Riemannian manifolds
Authors:
Kwang-nan Chow and Moses Glasner
Journal:
Proc. Amer. Math. Soc. 26 (1970), 261-265
MSC:
Primary 53.72; Secondary 30.00
DOI:
https://doi.org/10.1090/S0002-9939-1970-0271871-8
MathSciNet review:
0271871
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Abstract | References | Similar Articles | Additional Information
Abstract: Let $\Phi$ be a convex positive increasing function and $d = {\lim _{t \to \infty }}\Phi (t)/t$. A harmonic function $u$ on a Riemann surface $R$ is called $\Phi$-bounded if $\Phi (|u|)$ is majorized by a harmonic function on R. M. Parreau has shown that if $d < \infty$ ($d = \infty$, resp.), then every positive (bounded, resp.) harmonic function on $R$ reduces to a constant if and only if every $\Phi$-bounded harmonic function does. In this paper analogues of these results are given for the equation $\Delta u = Pu(P \geqq 0)$ on a Riemannian manifold.
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Additional Information
Keywords:
Riemannian manifold,
Riemann surface,
harmonic space,
harmonic function,
solution of <!– MATH $\Delta u = Pu$ –> <IMG WIDTH="87" HEIGHT="20" ALIGN="BOTTOM" BORDER="0" SRC="images/img1.gif" ALT="$\Delta u = Pu$">,
bounded in the mean,
Dirichlet problem,
<IMG WIDTH="22" HEIGHT="38" ALIGN="MIDDLE" BORDER="0" SRC="images/img2.gif" ALT="$Q$">-compactification,
relatively hyperbolic,
Harnack principle
Article copyright:
© Copyright 1970
American Mathematical Society