Nous démontrons des théorèmes de type Liouville pour les applications à -distorsion bornée entre variétés riemanniennes. En plus de ces applications, nous introduisons et étudions une nouvelle classe d’applications : les applications à -co-distorsion bornée.
We obtain Liouville type theorems for mappings with bounded -distorsion between Riemannian manifolds. Besides these mappings, we introduce and study a new class, which we call mappings with bounded -codistorsion.
@article{AIF_2002__52_6_1753_0,
author = {Troyanov, Marc and Vodop'yanov, Sergei},
title = {Liouville type theorems for mappings with bounded (co)-distortion},
journal = {Annales de l'Institut Fourier},
volume = {52},
year = {2002},
pages = {1753-1784},
doi = {10.5802/aif.1933},
mrnumber = {1952530},
zbl = {1019.30022},
language = {en},
url = {http://dml.mathdoc.fr/item/AIF_2002__52_6_1753_0}
}
Troyanov, Marc; Vodop'yanov, Sergei. Liouville type theorems for mappings with bounded (co)-distortion. Annales de l'Institut Fourier, Tome 52 (2002) pp. 1753-1784. doi : 10.5802/aif.1933. http://gdmltest.u-ga.fr/item/AIF_2002__52_6_1753_0/
[1] A Mountain Pass to the Jacobian Conjecture, Canad. Math. Bull, Tome Vol. 41 (1998), pp. 442-451 | Article | MR 1658243 | Zbl 0947.14030
[2] Sobolev Spaces and Hypoelliptic Equations, II, Siberian Advances in Mathematics, Tome 6 (1996), pp. 64-96 | MR 1469044 | Zbl 0925.46011
[3] Harnack Inequality and hyperbolicity for subelliptic -Laplacians with applications to Picard type theorems, Geom. Funct. Anal, Tome 11 (2001), pp. 1139-1191 | Article | MR 1878317 | Zbl 1005.58013
[4] Measure Theory and Fine properties of Functions, CRC press, Studies in Advanced Mathematics (1992) | MR 1158660 | Zbl 0804.28001
[5] Geometric Measure Theory, Grundlehren der Mathematik, Tome 153 (1969) | MR 257325 | Zbl 0176.00801
[6] Étude d'une classe d'applications liées à des homomorphismes d'algèbres de fonctions et généralisant les quasi-conformes, Duke Math. J, Tome 40 (1973), pp. 163-186 | MR 315628 | Zbl 0272.30025
[7] Algèbre de Royden et Homéomorphismes -dilatation bornée entre espaces métriques mesurés (2001) (Thèse, EPFL Lausanne)
[8] Homeomorphisms that induce Monomorphisms of Sobolev Spaces, Israel J. of Math, Tome 91 (1995), pp. 31-60 | Article | MR 1348304 | Zbl 0836.46021
[9] The Kelvin-Nevanlinna-Royden criterion for p-parabolicity, Math. Zeitschrift, Tome 232 (1999), pp. 607-619 | Article | MR 1727544 | Zbl 0939.31011
[10] Analytic and Geometric Background of Recurence and Non-Explosion of the Brownian Motion on Riemannian Manifolds, Bull. Amer. Math. Soc, Tome 36 (1999), pp. 135-242 | Article | MR 1659871 | Zbl 0927.58019
[11] Change of Variables Formula under Minimal Assumptions, Colloquium Mathematicum, Tome 64 (1993), pp. 93-101 | MR 1201446 | Zbl 0840.26009
[12] Non Linear Potential Theory of Degenerate Elliptic Equations, Oxford Math. Monographs (1993) | MR 1207810
[13] Sobolev Mappings with Integrable Dilatation, Arch. Rational Mech.Anal, Tome 125 (1993), pp. 81-97 | Article | MR 1241287 | Zbl 0792.30016
[14] Non linear Potential Theory and Quasiregular Mappings on Riemannian Manifolds, Annales Academiae Scientiarum Fennicae Series A, Tome 74 (1990) | MR 1052971 | Zbl 0698.31010
[15] Rough isometries and p-harmonic functions with finite Dirichlet integral, Revista Mathemática Iberoamericana, Tome 10 (1994), pp. 143-175 | MR 1271760 | Zbl 0797.31008
[16] Mappings of finite distortion: Monotonicity and continuity (Preprint) | MR 1833892 | Zbl 1006.30016
[17] Rough isometries and the parabolicity of Riemannian manifolds, J. Math. Soc. Japan, Tome 38 (1986) no. 2, pp. 227-238 | Article | MR 833199 | Zbl 0577.53031
[18] Mappings of finite distortion: discreteness and openess (2000) (preprint 226, University of Jyv\"vaskylä) | MR 1864838 | Zbl 0998.30024
[19] Absolutely continuous functions of several variables, J. Math. Anal. Appl, Tome 231 (1999), pp. 492-508 | Article | MR 1669167 | Zbl 0924.26008
[20] Lusin's condition N and mappings of the class , J. Reine Angew. Math, Tome 458 (1995), pp. 19-36 | Article | MR 1310951 | Zbl 0812.30007
[21] An extension of Reshetnyak's theorem, Indiana Univ. Math. J, Tome 47 (1998) no. 3, pp. 1131-1145 | MR 1665761 | Zbl 0931.30014
[22] Definitions for Quasiregular Mappings, Ann. Acad. Sc. Fenn, Tome 448 (1969), pp. 5-40 | MR 259114 | Zbl 0189.09204
[23] Lusin's condition (N) and mappings of non negative Jacobian, Michigan. Math. J, Tome 39 (1992), pp. 495-508 | Article | MR 1182504 | Zbl 0807.46032
[24] Sobolev Spaces, Springer Verlag (1985) | MR 817985 | Zbl 0692.46023
[25] Theory of Multipliers in Spaces of Differentiable Functions, Pitman (1985) | MR 881055
[26] On a new class of elastic deformations not allowing for cavitation, Ann. Inst. Henri Poincaré, Tome 11 (1994), pp. 217-243 | Numdam | MR 1267368 | Zbl 0863.49002
[27] An existence theory for nonlinear elasticity that allows for cavitation, Arch. Rational Mech. Anal., Tome 131 (1995) no. 1, pp. 1-66 | Article | MR 1346364 | Zbl 0836.73025
[28] Difféomorphismes de p-dilatation bornées, Ann. Acad. Sc. Fenn, Tome 223 (1997), pp. 475-506 | MR 1469804 | Zbl 0890.22005
[29] Cohomologie , espaces homogènes et pincement (1999) (preprint, Orsay)
[30] Über harmonishe Kapazität und quasikonforme Abbildungen in Raum., Comm. Math. Helv, Tome 44 (1969), pp. 284-307 | MR 252637 | Zbl 0176.03504
[31] Space Mappings with Bounded Distortion, Translation of Math. Monographs, Tome 73 (1989) | MR 994644 | Zbl 0667.30018
[32] Mappings with bounded distortion as extremals of integrals of Dirichlet type, Siberian Math. Journal, Tome 9 (1968), pp. 652-666 | MR 230900 | Zbl 0162.38202
[33] Sobolev classes of functions with values in a metric space, Siberian Math. Journal, Tome 38 (1997), pp. 657-675 | MR 1457485 | Zbl 0944.46024
[34] Quasi-regular Mappings, Springer, Ergebnisse der Mathematik, Tome 26 (1993) | Zbl 0816.30017
[35] Topics in the Theory of Quasi-regular Mappings, Aspekte der Mathematik, Tome E 12 (1988) | Zbl 0658.30015
[36] The extension of interiority, with some applications, Trans. Amer. Math. Soc, Tome 103 (1962), pp. 329-340 | Article | MR 137103 | Zbl 0113.38001
[37] Parabolicity of Manifolds, Siberian Adv. Math, Tome 9 (1999) no. 4, pp. 1-25 | MR 1749853 | Zbl 0991.31008
[38] Taylor Formula and Function Spaces (in Russian), Novosibirsk University (1988) | MR 996639 | Zbl 0709.46009
[39] Geometric Properties of Function Spaces (in Russian) (1992) (D. Sc. Thesis, Novosibirsk, Sobolev Institute of Mathematics)
[40] Monotone Functions and Quaiconformal Mappings on Carnot Groups, Siberian Math. J, Tome 37 (1996) no. 6, pp. 1269-1295 | MR 1440383 | Zbl 0876.30020
[41] Topological and Geometric Properties of Mappings of Sobolev Spaces with Integrable Jacobian I, Siberian Math. J, Tome 41 (2000) no. 1, pp. 23-48 | MR 1756474 | Zbl 0983.30009
[42] Mappings with Bounded Distortion and with Finite Distortion on Carnot Groups, Siberian Math. J, Tome 40 (1999) no. 4, pp. 764-804 | MR 1721674 | Zbl 0973.30021
[43] Quasi-conformal mapping and spaces of functions with generalized first derivatives, Siberian Math. J, Tome 17 (1977) no. 4, pp. 515-531 | Zbl 0353.30019
[44] Sobolev Spaces and (P,Q)-Quasiconformal Mappings of Carnot Groups, Siberian Math. J, Tome 39 (1998) no. 4, pp. 665-682 | Article | MR 1654140 | Zbl 0917.46023
[45] On the Conformal Type of a Riemannian Manifold, Func. Anal. and Appl, Tome 30 (1996), pp. 106-117 | Article | MR 1402080 | Zbl 0873.53025
[46] A theorem of M.A. Lavrent'ev on quasiconformal space maps, Mat.Sb., Tome 74 (1967) no. 116, pp. 417-433 | Zbl 0181.08701
[46] A theorem of M.A. Lavrent'ev on quasiconformal space maps, Math. USSR Sb. (English transl.), Tome 3 (1967) | Zbl 0184.10801