Hostname: page-component-78c5997874-ndw9j Total loading time: 0 Render date: 2024-11-10T09:20:38.972Z Has data issue: false hasContentIssue false
  • English
  • Français

Canonical isomorphisms of energy finite solutions of Δu = Pu on open Riemann surfaces

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.

We call a second order differential P(z)dxdy on a Riemann surface R a density if it is not identically zero and P(z) is a nonnegative Hölder continuous function of the local parameter z = x + iy in each parametric disk. To each density P on R we associate the linear space P(R) of C2 solutions of the equation Δu(z) = P(z)u(z) invariantly defined on R. We also consider subspaces PX(R) of P(R) consisting of solutions with certain boundedness properties X.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 56 , January 1975 , pp. 79 - 84
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1975

References

[1] Cornea, C. Constantinescu-A.: Ideale Ränder Riemannseher Flächen, Springer, 1963.Google Scholar
[2] Katz, M. Glasner-R.: On the behavior of solutions of Δu = Pu at the Royden boundary, J. d’Analyse Math., 22 (1969), 345–354.Google Scholar
[3] Nakai, M.: Dirichlet finite solutions of Δu = Pu on open iRemann Surfaces, Kôdai Math. Sem. Rep., 23 (1971), 385–397.Google Scholar
[4] Nakai, M.: The equation Δu= Pu on the unit disk with almost rotation free P ≥ 0, J. Diff. Eq., 11 (1972), 307–320.Google Scholar
[5] Ozawa, M.: Classification of Riemann surfaces, Kôdai Math. Sem. Rep., 4 (1952), 63–76.Google Scholar
[6] Royden, H.: Harmonic functions on open Riemann surfaces, Trans. Amer. Math. Soc., 73 (1952), 40–94.Google Scholar
[7] Royden, H.: The equation Δu = Pu, and the classification of open Riemann surfaces, Ann. Acad. Sci. Fenn., 271 (1959), 1–27.Google Scholar
[8] Nakai, L. Sario-M.: Classification Theory of Riemann Surfaces, Springer, 1970.Google Scholar
[9] Singer, I.: Boundary isomorphism between Dirichlet finite solutions of Δu = Pu, and harmonic functions, Nagoya Math. J., 50 (1973), 7–20.CrossRefGoogle Scholar
[10] Tsuji, M.: Potential Theory in Modern Function Theory, Maruzen, 1959.Google Scholar