Nagoya Mathematical Journal

Harmonic morphisms in nonlinear potential theory

J. Heinonen, T. Kilpeläinen, and O. Martio

Full-text: Open access

Article information

Nagoya Math. J., Volume 125 (1992), 115-140.

First available in Project Euclid: 14 June 2005

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 30C65: Quasiconformal mappings in $R^n$ , other generalizations
Secondary: 35J60: Nonlinear elliptic equations


Heinonen, J.; Kilpeläinen, T.; Martio, O. Harmonic morphisms in nonlinear potential theory. Nagoya Math. J. 125 (1992), 115--140.

Export citation


  • [CC] Constantinescu, C. and A. Cornea, Compactifications of harmonic spaces, Na- goya Math. J., 25 (1965), 1-57.
  • [Chi] Chernavskii, A. V., Discrete and open mappings on manifolds, (in Russian), Mat. Sbornik, 65 (1964), 357-369.
  • [Ch2] Chernavskii, Continuation to Discrete and open mappings on manifolds, (in Rus- sian), Mat. Sbornik, 66 (1965), 471-472.
  • [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.
  • [Fl] Fuglede, B., Harmonic morphisms between Riemannian manifolds, Ann. Inst. Fourier, Grenoble, 28.2 (1978), 107-144.
  • [F2] Fuglede, Harnack sets and openness of harmonic morphisms, Math. Ann., 241 (1979), 181-186.
  • [G] Gehring, F. W., Symmetrization of rings in space, Trans. Amer. Math. Soc, 101 (1961), 499-519.
  • [GH] Gehring, F. W. and H. Haahti, The transformations which preserve the har- monic functions, Ann. Acad. Sci. Fenn. Ser. A I. Math., 293 (1960), 1-12.
  • [GLM1] Granlund, S., P. Lindqvist, and 0. Martio, Conformally invariant variational integrals, Trans. Amer. Math. Soc, 277 (1983), 43-73.
  • [GLM2] Granlund, F-harmonic measure in space, Ann. Acad. Sci. Fenn. Ser. A I. Math.,
  • [HK1] Heinonen, J. and T. Kilpelainen, A-superharmonic functions and supersolu- tions of degenerate elliptic equations, Ark. Mat, 26 (1988), 87-105.
  • [HK2] Heinonen, Polar sets for supersolutions of degenerate elliptic equations, Math. Scand., 63 (1988), 136-150.
  • [HK3] Heinonen, On the Wiener criterion and quasilinear obstacle problems, Trans. Amer. Math. Soc, 310 (1988), 239-255.
  • [RKM1] Heinonen, J., T. Kilpelainen, and O. Martio, Fine topology and quasilinear elliptic equations, Ann. Inst. Fourier, Grenoble, 39.2 (1989), 293-318.
  • [HKM2] Heinonen, Nonlinear Potential Theory of Degenerate Elliptic Equations, Oxford University Press (To appear).
  • [H] Holopainen, I., Nonlinear potential theory and quasiregular mappings on Rie- mannian manifolds, Ann. Acad. Sci. Fenn. Ser. A I. Math. Dissertationes, 74 (1990), 1-45.
  • [I] Ishihara, T., A mapping of Riemannian manifolds which preserves harmonic functions, J. Math. Kyoto Univ., 19 (1979), 215-229.
  • [K] Kilpelainen, T., Potential theory for supersolutions of degenerate elliptic equa- tions, Indiana Univ. Math. J., 38 (1989), 253-275.
  • [L] Laine, I., Harmonic morphisms and non-linear potential theory. (To appear).
  • [LV] Lehto, 0. and K. I. Virtanen, Quasiconformal mappings in the plane, Springer- Yerlag 1973.
  • [LM] Lindqvist, P. and O. Martio, Two theorems of N. Wiener for solutions of quasilinear elliptic equations, Acta Math., 155 (1985), 153-171.
  • [Mar] Martio, 0., F-harmonic measures, quasihyperbolic distance and Milloux's prob- lem, Ann. Acad. Sci. Fenn. Ser. A I. Math., 1 (1987), 151-162.
  • [MRV1] Martio, 0., S. Rickman, and J. Vaisala, Definitions for quasiregular mappings, Ann. Acad. Sci. Fenn. Ser. A I. Math., 448 (1969), 1-40.
  • [MRV2] Martio, Distortion and singularities of quasiregular mappings, Ann. Acad. Sci. Fenn. Ser. A I. Math., 465 (1970), 1-13.
  • [MV] Martio, 0. and J. Vaisala, Elliptic equations and maps of bounded length distortion, Math. Ann., 282 (1988), 423-443.
  • [Maz] Maz'ya, V. G., On the continuity at a boundary point of the solution of quasi- linear elliptic equations, Vestnik Leningrad Univ. Mat. Mekh. Astronom., 25 (1970), 42-55 (in Russian).
  • [RR] Rad, T. and P. V. Reichelderfer, Continuous transformations in analysis, Springer-Verlag, 1955.
  • [Re] Reshetnyak, Yu. G., Space mappings with bounded distortion, Amer. Math. Soc, Trans. Math Monographs Vol. 73 (1989).
  • [Ril] Rickman, S., On the number of omitted values of entire quasiregular map- pings, J. Analyse Math., 37 (1980), 100-117.
  • [Ri2] Rickman, Quasiregular mappings. (To appear).
  • [S] Serrin, J., Local behavior of solutions of quasi-linear equations, Acta Math., I l l (1964), 247-302.
  • [TY] Titus, C. J. and G. S. Young, The extension of interiority, with some applica- tions, Trans. Amer. Math. Soc, 103 (1962), 329-340.
  • [VI] Vaisala, J., Discrete open mappings on manifolds, Ann. Acad. Sci. Fenn. Ser. A I. Math., 392 (1966), 1-10.
  • [V2] Vaisala, Capacity and measure, Michigan Math. J., 22 (1975), 1-3.