Hostname: page-component-745bb68f8f-g4j75 Total loading time: 0 Render date: 2025-02-04T09:47:33.786Z Has data issue: false hasContentIssue false
  • English
  • Français

Harmonic morphisms in nonlinear potential theory*

Published online by Cambridge University Press:  22 January 2016

J. Heinonen ,
T. Kilpeläinen  and
O. Martio
J. Heinonen
Affiliation:
University of Michigan Department of Mathematics, Ann Arbor, MI 48109, U.S.A.
T. Kilpeläinen
Affiliation:
University of Jyväskylä Department of Mathematics, P.O. Box 35, 40351 Jyväskylä, Finland
O. Martio
Affiliation:
University of Michigan Department of Mathematics, Ann Arbor, MI 48109, U.S.A.
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

This article concerns the following problem: given a family of partial differential operators with similar structure and given a continuous mapping f from an open set Ω in Rn into Rn, then when does f pull back the solutions of one equation in the family to solutions of another equation in that family? This problem is typical in the theory of differential equations when one wants to use a coordinate change to study solutions in a different environment.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 125 , March 1992 , pp. 115 - 140
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1992

Footnotes

2

Supported by an NSF grant 89-02749.

1

The authors acknowledge the hospitality of the Mittag-Leffler Institute where part of this research was conducted.

References

[CC] Constantinescu, C. and Cornea, A., Compactifications of harmonic spaces, Nagoya Math. J., 25 (1965), 1–57.CrossRefGoogle Scholar
[Ch1] Chernavskii, A. V., Discrete and open mappings on manifolds, (in Russian), Mat. Sbornik, 65 (1964), 357–369.Google Scholar
[Ch2] Chernavskii, A. V., Continuation to “Discrete and open mappings on manifolds”, (in Russian), Mat. Sbornik, 66 (1965), 471–472.Google Scholar
[EL] Eremenko, A. and Lewis, J. L., Uniform limits of certain A-harmonic functions with applications to quasiregular mappings, Ann. Acad. Sci. Fenn. Ser. A I. Math., 16 (1991), 361–375.Google Scholar
[F1] Fuglede, B., Harmonic morphisms between Riemannian manifolds, Ann. Inst. Fourier, Grenoble, 28.2 (1978), 107–144.CrossRefGoogle Scholar
[F2] Fuglede, B., Harnack sets and openness of harmonic morphisms, Math. Ann., 241 (1979), 181–186.CrossRefGoogle Scholar
[G] Gehring, F. W., Symmetrization of rings in space, Trans. Amer. Math. Soc, 101 (1961), 499–519.Google Scholar
[GH] Gehring, F. W. and Haahti, H., The transformations which preserve the harmonic functions, Ann. Acad. Sci. Fenn. Ser. A I. Math., 293 (1960), 1–12.Google Scholar
[GLM1] Granlund, S., Lindqvist, P., and Martio, O., Conformally invariant variational integrals, Trans. Amer. Math. Soc, 277 (1983), 43–73.CrossRefGoogle Scholar
[GLM2] Granlund, S., Lindqvist, P., and Martio, O., F-harmonic measure in space, Ann. Acad. Sci. Fenn. Ser. A I. Math., 7 (1982), 233–247.CrossRefGoogle Scholar
[HK1] Heinonen, J. and Kilpeläinen, T., A-superharmonic functions and supersolutions of degenerate elliptic equations, Ark. Mat, 26 (1988), 87–105.Google Scholar
[HK2] Heinonen, J. and Kilpeläinen, T., Polar sets for supersolutions of degenerate elliptic equations, Math. Scand., 63 (1988), 136–150.Google Scholar
[HK3] Heinonen, J. and Kilpeläinen, T., On the Wiener criterion and quasilinear obstacle problems, Trans. Amer. Math. Soc, 310 (1988), 239–255.Google Scholar
[HKM1] Heinonen, J., Kilpeläinen, T., and Martio, O., Fine topology and quasilinear elliptic equations, Ann. Inst. Fourier, Grenoble, 39.2 (1989), 293–318.Google Scholar
[HKM2] Heinonen, J., Kilpeläinen, T., and Martio, O., Nonlinear Potential Theory of Degenerate Elliptic Equations, Oxford University Press (To appear).Google Scholar
[H] Holopainen, I., Nonlinear potential theory and quasiregular mappings on Riemannian manifolds, Ann. Acad. Sci. Fenn. Ser. A I. Math. Dissertationes, 74 (1990), 1–45.Google Scholar
[I] Ishihara, T., A mapping of Riemannian manifolds which preserves harmonic functions, J. Math. Kyoto Univ., 19 (1979), 215–229.Google Scholar
[K] Kilpeläinen, T., Potential theory for supersolutions of degenerate elliptic equations, Indiana Univ. Math. J., 38 (1989), 253–275.Google Scholar
[L] Laine, I., Harmonic morphisms and non-linear potential theory. (To appear).Google Scholar
[LV] Lehto, O. and Virtanen, K. I., Quasiconformal mappings in the plane, Springer-Verlag 1973.Google Scholar
[LM] Lindqvist, P. and Martio, O., Two theorems of N. Wiener for solutions of quasilinear elliptic equations, Acta Math., 155 (1985), 153–171.Google Scholar
[Mar] Martio, O., F-harmonic measures, quasihyperbolic distance and Milloux’s problem, Ann. Acad. Sci. Fenn. Ser. A I. Math., 1 (1987), 151–162.CrossRefGoogle Scholar
[MRVl] Martio, O., Rickman, S., and Väisälä, J., Definitions for quasiregular mappings, Ann. Acad. Sci. Fenn. Ser. A I. Math., 448 (1969), 1–40.Google Scholar
[MRV2] Martio, O., Rickman, S., and Väisälä, J., Distortion and singularities of quasiregular mappings, Ann. Acad. Sci. Fenn. Ser. A I. Math., 465 (1970), 1–13.Google Scholar
[MV] Martio, O. and Väisälä, J., Elliptic equations and maps of bounded length distortion, Math. Ann., 282 (1988), 423–443.CrossRefGoogle Scholar
[Maz] Maz’ya, V. G., On the continuity at a boundary point of the solution of quasilinear elliptic equations, Vestnik Leningrad Univ. Mat. Mekh. Astronom., 25 (1970), 42–55 (in Russian).Google Scholar
[RR] Radó, T. and Reichelderfer, P. V., Continuous transformations in analysis, Springer-Verlag, 1955.Google Scholar
[Re] Reshetnyak, Yu. G., Space mappings with bounded distortion, Amer. Math. Soc, Trans. Math Monographs Vol. 73 (1989).Google Scholar
[Ril] Rickman, S., On the number of omitted values of entire quasiregular mappings, J. Analyse Math., 37 (1980), 100–117.Google Scholar
[Ri2] Rickman, S., Quasiregular mappings. (To appear).Google Scholar
[S] Serrin, J., Local behavior of solutions of quasi-linear equations, Acta Math., III (1964), 247–302.Google Scholar
[TY] Titus, C. J. and Young, G. S., The extension of interiority, with some applications, Trans. Amer. Math. Soc, 103 (1962), 329–340.Google Scholar
[VI] Väisälä, J., Discrete open mappings on manifolds, Ann. Acad. Sci. Fenn. Ser. A I. Math., 392 (1966), 1–10.Google Scholar
[V2] Väisälä, J., Capacity and measure, Michigan Math. J., 22 (1975), 1–3.Google Scholar