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A Maximum Principle for Dirichlet-Finite Harmonic Functions on Riemannian Spaces

Published online by Cambridge University Press:  20 November 2018

Y. K. Kwon  and
L. Sario
Y. K. Kwon
Affiliation:
University of California, Los Angeles, California
L. Sario
Affiliation:
University of California, Los Angeles, California
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Representations of harmonic functions by means of integrals taken over the harmonic boundary ΔR of a Riemann surface R enable one to study the classification theory of Riemann surfaces in terms of topological properties of ΔR (cf. [6; 4; 1; 7]). In deducing such integral representations, essential use is made of the fact that the functions in question attain their maxima and minima on ΔR.

The corresponding maximum principle in higher dimensions was discussed for bounded harmonic functions in [3]. In the present paper we consider Dirichlet-finite harmonic functions. We shall show that every such function on a subregion G of a Riemannian N-space R attains its maximum and minimum on the set , where ∂G is the relative boundary of G in R and the closures are taken in Royden's compactification R*. As an application we obtain the harmonic decomposition theorem relative to a compact subset K of R* with a smooth ∂(K ∩ R).

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 22 , Issue 4 , 01 August 1970 , pp. 855 - 862
Copyright
Copyright © Canadian Mathematical Society 1970

References

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