# Harmonic morphisms and non-linear potential theory

Banach Center Publications (1992)

- Volume: 27, Issue: 1, page 271-275
- ISSN: 0137-6934

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topLaine, Ilpo. "Harmonic morphisms and non-linear potential theory." Banach Center Publications 27.1 (1992): 271-275. <http://eudml.org/doc/262739>.

@article{Laine1992,

abstract = {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].},

author = {Laine, Ilpo},

journal = {Banach Center Publications},

keywords = {harmonic morphisms; harmonic functions; hyperharmonic functions},

language = {eng},

number = {1},

pages = {271-275},

title = {Harmonic morphisms and non-linear potential theory},

url = {http://eudml.org/doc/262739},

volume = {27},

year = {1992},

}

TY - JOUR

AU - Laine, Ilpo

TI - Harmonic morphisms and non-linear potential theory

JO - Banach Center Publications

PY - 1992

VL - 27

IS - 1

SP - 271

EP - 275

AB - 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].

LA - eng

KW - harmonic morphisms; harmonic functions; hyperharmonic functions

UR - http://eudml.org/doc/262739

ER -

## References

top- [1] C. Constantinescu and A. Cornea, Compactifications of harmonic spaces, Nagoya Math. J. 25 (1965), 1-57. Zbl0138.36701
- [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. Zbl0328.31014
- [4] I. Laine, Introduction to a quasi-linear potential theory, ibid. 10 (1985), 339-348. Zbl0593.31013
- [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. Zbl0684.35008

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