Geodesic
A Maximum Principle for Dirichlet-Finite Harmonic Functions on Riemannian Spaces
Canadian journal of mathematics, Tome 22 (1970) no. 4, pp. 855-862

Voir la notice de l'article provenant de la source Cambridge University Press

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). 
Kwon, Y. K.; Sario, L. A Maximum Principle for Dirichlet-Finite Harmonic Functions on Riemannian Spaces. Canadian journal of mathematics, Tome 22 (1970) no. 4, pp. 855-862. doi: 10.4153/CJM-1970-097-8
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