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On the Dirichlet Problem in the Axiomatic Theory of Harmonic Functions

Published online by Cambridge University Press:  22 January 2016

N. Boboc ,
C. Constantinescu  and
A. Cornea
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In the frame of the recent axiomatic theories of harmonic functions [2], [3], [1], it has been shown that the continuous bounded functions on the boundaries of relatively compact open sets are resolutive [5], [1]. The aim of the present paper is to substitute in these results the continuous functions by Borel-measurable functions and to leave out the restriction that the open sets are relatively compact. H. Bauer has replaced the axiom 3 of Brelot’s axiomatic by two weaker axioms: the axiom of separation (Trennungsaxiom) and the axiom K1. Since the axiom of separation is not fulfilled in some important cases (e.g. the compact Riemann surfaces) we shall weaken this axiom too, substituting it by one of its consequences: the minimum principle for hyperharmonic functions.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 23 , December 1963 , pp. 73 - 96
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1963

References

[1] Bauer, H., Axiomatische Behandlung des Dirichletschen Problems für elliptische und parabolische Differentialgleichungen, Math. Ann., 146, (1962), 1–59.CrossRefGoogle Scholar
[2] Brelot, M., Axiomatique des fonctions harmoniques et surharmoniques dans un espace localement compact, Séminaire de Théorie du Potentiel, 2 (1959), 1.1–1.40.Google Scholar
[3] Brelot, M., Lectures on Potential Theory, Tata Institute of Fund. Reasearch, Bombay (1960).Google Scholar
[4] Brelot, M., Étude comparée de quelques axiomatiques des fonctions harmoniques et surharmoniques, Séminaire de Theorie du Potentiel, 6 (1962), 1.13–1.26.Google Scholar
[5] Hervé, R.-M., Développements sur une théorie axiomatique des fonctions surharmoniques, Comptes rendus Acad. Sci. (Paris), 248 (1959), 179–181.Google Scholar