Hostname: page-component-cd9895bd7-dzt6s Total loading time: 0 Render date: 2024-12-24T00:01:32.014Z Has data issue: false hasContentIssue false
  • English
  • Français

A test for Picard principle

Published online by Cambridge University Press:  22 January 2016

Mitsuru Nakai
Mitsuru Nakai*
Affiliation:
Nagoya University
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

A nonnegative locally Hölder continuous function P(z) on 0 < | z | ≤ 1 will be referred to as a density on 0 < | z | ≤ 1. The elliptic dimension of a density P(z) at z = 0, dim P in notation, is defined to be the dimension of the half module of nonnegative solutions of the equation Δu(z) = P(z)u(z) on the punctured unit disk Ω : 0 < | z | < 1 with boundary values zero on | z | = 1. After Bouligand we say that the Picard principle is valid for a density P at z = 0 if dim P = 1.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 56 , January 1975 , pp. 105 - 119
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1975

References

[1] Brelot, M.: Étude de l’équation de la chaleur Γu = c(M)u(M), c(M), au voisinage d’un point singulier du coefficient, Ann. Ec. N. Sup., 48 (1931), 153–246.Google Scholar
[2] Cornea, C. Constantinescu-A.: Über einige Problem von M. Heins, Rev. math. pures appl., 4 (1959), 277–281.Google Scholar
[3] Hayashi, K.: Les solutions positives de l’équation Γu = Pu sur une surface de Riemann, Kōdai Math. Sem. Rep., 13 (1961), 20–24.Google Scholar
[4] Heins, M.: Riemann surfaces of infinite genus, Ann. Math., 55 (1952), 296–317.CrossRefGoogle Scholar
[5] Itô, S.: Martin boundary for linear elliptic differential operators of second order in a manifold, J. Math. Soc. Japan, 16 (1964), 307–334.CrossRefGoogle Scholar
[6] Kuramochi, Z.: An example of a null-boundary Riemann surface, Osaka Math. J., 6 (1954), 83–91.Google Scholar
[7] Lahtinen, A.: On the equation Γu = Pu and the classification of acceptable densities on Riemann surfaces, Ann. Acad. Sci. Fenn., 533 (1973), 1–26.Google Scholar
[8] Martin, R.: Minimal positive harmonic functions, Trans. Amer. Math. Soc. 49 (1941).CrossRefGoogle Scholar
[9] Miranda, C.: Partial Differential Equations of Elliptic Type, Springer, 1970.Google Scholar
[10] Myrberg, L.: Über die Existenz der Greenschen Funktion der Gleichung Γu = c(P)u auf Riemannschen Flächen, Ann. Acad. Sci. Fenn., 170 (1954).Google Scholar
[11] Nakai, M.: The space of nonnegative solutions of the equation Γu = Pu on a Riemann surface, Kōdai Math. Sem. Rep., 12 (1960), 151–178.Google Scholar
[12] Nakai, M.: Order comparisons on canonical isomorphisms, Nagoya Math. J., 50 (1973), 67–87.CrossRefGoogle Scholar
[13] Nakai, M.: Martin boundary over an isolated singularity of rotation free density, J. Math. Soc. Japan, 26 (1974), 483–507.CrossRefGoogle Scholar
[14] Ozawa, M.: Classification of Riemann surfaces, Kōdai Math. Sem. Rep., 4 (1952), 63–76.Google Scholar
[15] Ozawa, M.: Some classes of positive solutions of Γu = Pu on Riemann surfaces, I, Kōdai Math. Sem. Rep., 6 (1954), 121–126.Google Scholar
[16] Ozawa, M.: Some classes of positive solutions of Γu = Pu on Riemann surfaces, II. Kōdai Math. Sem. Rep., 7 (1955), 15–20.Google Scholar
[17] Royden, H.: The equation Γu = Pu, and the classification of open Riemann surfaces, Ann. Acad. Sci. Fenn., 271 (1959).Google Scholar
[18] Tsuji, M.: Potential Theory in Mordern Function Theory, Maruzen, 1959.Google Scholar
[19] Yosida, K.: Functional Analysis, Springer, 1965.Google Scholar