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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.

Information

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