Harmonic morphisms and non-linear potential theory

• Autorzy: Ilpo Laine
• Języki publikacji: EN

Abstrakty

EN

Originally, harmonic morphisms were defined as continuous mappings φ:X → X' between harmonic spaces such that h'∘φ remains harmonic whenever h' is harmonic, see [1], p. 20. In general linear axiomatic potential theory, one has to replace harmonic functions h' by hyperharmonic functions u' in this definition, in order to obtain an interesting class of mappings, see [3], Remark 2.3. The modified definition appears to be equivalent with the original one, provided X' is a Bauer space, i.e., a harmonic space with a base consisting of regular sets, see [3], Theorem 2.4. To extend the linear proof of this result directly into the recent non-linear theories fails, even in the case of semi-classical non-linear considerations [6]. The aim of this note is to give a modified proof which settles such difficulties in the quasi-linear theories [4], [5].

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Banach Center Publications
• Rocznik: 1992
• Tom: 27
• Numer: 1
• Strony: 271-275

Daty

wydano

1992

Twórcy

autor

Ilpo Laine

• Department of Mathematics, University of Joensuu, P.O. Box 111, SF-80101 Joensuu, Finland

Bibliografia

• [1] C. Constantinescu and A. Cornea, Compactifications of harmonic spaces, Nagoya Math. J. 25 (1965), 1-57.
• [2] C. Constantinescu and A. Cornea, Potential Theory on Harmonic Spaces, Springer, 1972.
• [3] I. Laine, Covering properties of harmonic Bl-mappings III, Ann. Acad. Sci. Fenn. Ser. AI Math. 1 (1975), 309-325.
• [4] I. Laine, Introduction to a quasi-linear potential theory, ibid. 10 (1985), 339-348.
• [5] I. Laine, Axiomatic non-linear potential theories, in: Lecture Notes in Math. 1344, Springer, 1988, 118-132.
• [6] O. Martio, Potential theoretic aspects of non-linear elliptic partial differential equations, Univ. of Jyväskylä, Dept. of Math., Report 44, 1989.