The Dirichlet Problem on Ends

  • Teruo Ikegami

Abstract

In the recent investigations of the Dirichlet problem in an axiomatic framework we focus our interest on the following two problems: (1) the boundary behavior of the normalized PWB-solutions, originally studied by O. Frostman[3] and generalized by Constantinescu-Cornea[2] and Lukeš-Malý[11] and (2) the unicity of Keldych operators formulated by J. Lukeš[10], given by Bliedtner-Hansen[1] its essential contribution. The normalized PWB-solutions H f U,X on an open set U is formed by the Perron-Wiener-Brelot’s method taking the closure of U in the one-point compactified space and putting f to be 0 at the infinity, and the effect of the ideal boundary is neglected. In this article we consider above problem under the full influence of the ideal boundary and in the conclusing result we reflect that our result is a generalized version of that obtained formerly.

Keywords

Radon 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    J. Bliedtner-W. Hansen: Simplicial cone in potential theory, Invent. Math. 29 (1975) 83–110.MathSciNetMATHCrossRefGoogle Scholar
  2. [2]
    C. Constantinescu-A. Cornea: Potential theory in harmonic spaces, Grundlehr. der mat. Wiss. 158 Berlin-Heidelberg-New York 1972.Google Scholar
  3. [3]
    O. Frostman: Les points irréguliers dans la théorie du potentiel et la citère de Wiener, Medd. Lunds Univ. Mat. Sem. 4 (1936) 1–10.Google Scholar
  4. [4]
    T. Ikegami: A note on axiomatic Dirichlet problem, Osaka J. Math. 6 (1969) 39–47.MathSciNetMATHGoogle Scholar
  5. [5]
    T. Ikegami: On the regularity of boundary points in a resolutive compactification of a harmonic space, Osaka J. Math. 14 (1977) 271–289.MathSciNetMATHGoogle Scholar
  6. [6]
    T. Ikegami: Remarks on the reguality of boundary points in a resolutive compactification, Osaka J. Math. 17 (1980) 177–186.MathSciNetMATHGoogle Scholar
  7. [7]
    T. Ikegami: On a generalization of Lukeš theorem, Osaka J. Math. 18 (1981) 699–702.MathSciNetMATHGoogle Scholar
  8. [8]
    T. Ikegami: On the simplicial cone of superharmonic functions in a resolutive compactification of a harmonic space, Osaka J. Math. 20 (1983) 881–898.MathSciNetMATHGoogle Scholar
  9. [9]
    T. Ikegami: On the boundary behavior of the Dirichlet solutions at an irregular boundary point, Osaka J. Math. 21 (1984) 851–858.MathSciNetMATHGoogle Scholar
  10. [10]
    J. Lukeš: Théorème de Keldych dans la théorie axiomatique de Bauer des fonctions harmoniques, Czechoslovak Math. J. 24 (1974) 114–125.MathSciNetGoogle Scholar
  11. [11]
    J. Lukeš-J. Malý On the boundary behavior of the Perron generalized solution, Math Ann. 257 (1981) 355–366.MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Plenum Press, New York 1988

Authors and Affiliations

  • Teruo Ikegami
    • 1
  1. 1.Department of MathematicsOsaka City UniversityOsakaJapan

Personalised recommendations