A Maximum Principle for Bounded Harmonic Functions on Riemannian Spaces

Kwon, Y. K. ; Sario, L.

Canadian journal of mathematics, Tome 22 (1970) no. 4, pp. 847-854

Résumé

Harmonic functions with certain boundedness properties on a given open Riemann surface R attain their maxima and minima on the harmonic boundary ΔB of R. The significance of such maximum principles lies in the fact that the classification theory of Riemann surfaces related to harmonic functions reduces to a study of topological properties of Δ(cf. [11; 8; 3; 12].For the corresponding problem in higher dimensions we shall first show that the complement of ΔR with respect to the Royden boundary ΓR of a Riemannian N-space R is harmonically negligible: given any non-empty compact subset E of ΓR – ΔR there exists an Evans superharmonic function v, i.e., a positive continuous function on R* = R ∪ ΓR, superharmonic on R, with v = 0 on ΔR, v ≡ ∞ on E, and with a finite Dirichlet integral over R.

Détail

DOI : 10.4153/CJM-1970-096-0

Bibliographie

[1] 1. Chang, J., Royden's compactification of Riemannian spaces, Doctoral dissertation, University of California, Los Angeles, 1968. Google Scholar

[2] 2. Constantinescu, C., Dirichletsche Abbildungen, Nagoya Math. J. 20 (1962), 75–89. Google Scholar

[3] 3. Constantinescu, C. and Cornea, A., Idéale Rànder Riemannscher Fldchen, Ergebnisse der Mathematik und ihrer Grenzgebiete, N.F., Band 32 (Springer-Verlag, Berlin, 1963). Google Scholar

[4] 4. Duff, G. F. D., Partial differential equations (Univ. Toronto Press, Toronto, Ontario, 1956). Google Scholar

[5] 5. Feller, W., Ùber die Lösungen der linearen partiellen Differentialgleichungen zweiter Ordnung vont elliptischen Typus, Math. Ann. 102 (1930), 633–649. Google Scholar

[6] 6. Kwon, Y. K., Integral representations of harmonie functions on Riemannian spaces, Doctoral dissertation, University of California, Los Angeles, 1969. Google Scholar

[7] 7. Kwon, Y. K. and Sario, L., A maximum principle for Dirichlet-finite harmonie functions on Riemannian spaces, Can. J. Math. 22 (1970), 855–862. Google Scholar

[8] 8. Nakai, M., A measure on the harmonie boundary of a Riemann surface, Nagoya Math. J. 17 (1960), 181–218. Google Scholar

[9] 9. Nakai, M. and Sario, L., Classification and deformation of Riemannian spaces, Math. Scand. 20 (1967), 193–208. Google Scholar

[10] 10. Rodin, B. and Sario, L., Principal functions (Van Nostrand, Princeton, N.J., 1968). Google Scholar

[11] 11. Royden, H., On the ideal boundary of a Riemann surface, Ann. of Math. (2) 30 (1953), 107–109. Google Scholar

[12] 12. Sario, L. and Nakai, M., Classification theory of Riemann surfaces (Springer-Verlag, New York, 1970). Google Scholar

[13] 13. Sario, L., Schiffer, M., and Glasner, M., The span and principal functions in Riemannian spaces, J. Analyse Math. 15 (1965), 115–134. Google Scholar