We present inversion results for Lipschitz maps f : Ω ⊂ ℝN → (Y, d) and stability of inversion for uniformly convergent sequences. These results are based on the Area Formula and on the l.s.c. of metric Jacobians.
@article{bwmeta1.element.doi-10_2478_agms-2014-0008,
author = {Luca Granieri},
title = {Inverse Function Theorems and Jacobians over Metric Spaces},
journal = {Analysis and Geometry in Metric Spaces},
volume = {2},
year = {2014},
zbl = {1309.26015},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_2478_agms-2014-0008}
}
Luca Granieri. Inverse Function Theorems and Jacobians over Metric Spaces. Analysis and Geometry in Metric Spaces, Tome 2 (2014) . http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_2478_agms-2014-0008/
[1] E. Acerbi, G. Buttazzo, N. Fusco, Semicontinuity and Relaxation for Integrals Depending on Vector Valued Functions, J.Math. Pure et Appl. 62 (1983), 371-387. | Zbl 0481.49013
[2] L. Ambrosio, B. Kircheim, Rectifiable sets in metric and Banach spaces, Math. Ann. 318 (3) (2000) 527–555. | Zbl 0966.28002
[3] L. Ambrosio, B. Kircheim, Metric currents, Acta Mathematica 185 no. 1(2000), 1-80. | Zbl 0984.49025
[4] L. Ambrosio, N. Fusco, D. Pallara, Function of Bounded Variations and Free Discontinbuity Problems,Oxford University Press, New York, 2000. | Zbl 0957.49001
[5] L. Ambrosio, P. Tilli, Topics on Analysis in Metric Spaces, Oxford University Press, 2004. | Zbl 1080.28001
[6] Erik Barvínek, Ivan Daler and Jan Franku, Convergence of sequences of inverse functions, Archivium Mathematicum, Vol. 27 3-4 (1991), 201–204. | Zbl 0752.40003
[7] Y. Benyamini, J. Lindenstrauss, Geometric Nonlinear Functional Analysis, Vol.1, AMS, 2000. | Zbl 0946.46002
[8] Chung-Wu Ho, A note on proper maps, Proceedings AMS, Vol. 51 1 (1975), 237-241. | Zbl 0273.54007
[9] P.G. Ciarlet, J. Necas, Injectivity and self-Contact in nonlinear elasticity, Arch. Rational Mech. Anal. 97 (1987), no. 3, 171-188. | Zbl 0628.73043
[10] B. Dacorogna, Direct Methods in the Calculus of Variations, Springer 2008. | Zbl 1140.49001
[11] G. De Marco, G. Gorni, G. Zampieri, Global inversion of functions: an introduction, NODEA 1 (1994), 229-248. | Zbl 0820.58008
[12] E. Durand-Cartagena, J.A. Jaramillo, Pointwise Lipschitz functions on metric spaces, J. Math. Anal. Appl. 363 (2010), 525– 548. | Zbl 1200.46030
[13] I. Fonseca, W. Gangbo, Degree Theory in Analysis and Applications, Clarendom Press Oxford, 1995. | Zbl 0852.47030
[14] W. Fulton, Algebraic Topology, Springer, 1995. | Zbl 0852.55001
[15] J. Gevirtz, Metric conditions that imply local invertibility, Communications in Pure and AppliedMathematics 23 (1969), 243– 264. | Zbl 0172.33102
[16] J. Gevirtz, Injectvity in Banach spaces and the Mazur-Ulam Theorem on Isometries, Transactions AMS 274, no. 1 (1982), 307–318. | Zbl 0497.46011
[17] J. Gevirtz, A sharp condition for univalence in Euclidean spaces, Proceedings AMS 57, no. 2 (1976), 261–265. | Zbl 0344.26006
[18] L. Granieri, Metric currents and geometry of Wasserstein spaces, Rend. Semin. Mat. Univ. Padova 124 (2010), 91-125. | Zbl 1210.35076
[19] L. Granieri, F. Maddalena, A metric approach to elastic reformations, Acta Mathematica Applicandae, published online December 2013. | Zbl 1316.49051
[20] L. Granieri, G. Mola, Sequences of inverse and implicit functions, in preparation.
[21] C. Gutierrez, C. Biasi, Finite Branched Coverings in a Generalized Inverse Mapping Theorem, Int. Journal of Math. Analysis 2 no. 4 (2008), 169–179. | Zbl 1179.26043
[22] F. John, On Quasi-Isometric Mappings I, Communications in Pure and Applied Mathematics 21 (1968), 77–110. | Zbl 0157.45803
[23] M. B. Karmanova, Area and Coarea Formulas for the mappings of Sobolev classes with values in a metric space, Siberian Mathematical Journal, Vol. 48, No. 4 (2007), 621–628. Proceedings of the AmericanMathematical Society, 121, No. 1 (1994), 113–123. [WoS]
[24] B. Kirchheim, Rectifiable Metric Spaces: Local Structure and Regularity of the Hausdorff Measure, Proceedings of the American Mathematical Society, 121, No. 1 (1994), 113–123. | Zbl 0806.28004
[25] L. V. Kovalev, Jani Onninen, On invertibility of Sobolev mappings, Journal für die reine und angewandte Mathematik, 656, (2011), 1–16. | Zbl 1236.30021
[26] L. V. Kovalev, Jani Onninen, Kai Rajala, Invertibility of Sobolev mappings under minimal hypotheses, Ann. Inst. H. Poincaré Anal. Non Linéaire 27 (2010), no. 2, 517–528. | Zbl 1190.30019
[27] A. Lytchak, Open map Theorem for metric spaces, St. Petersburg Math. J. 17 (2006), No. 3, 477–491. | Zbl 1152.53033
[28] V. Magnani, An Area formula in metric spaces, Colloq. Math. 124 (2011), no. 2, 275-283. | Zbl 1231.28007
[29] J. S. Raymond, Local inversion for differentiable functions and Darboux property, Mathematika 49 (2002), no. 1-2, 141-158. [Crossref] | Zbl 1051.26009
[30] Yu. G. Reshetnyak, Space Mappings with Bounded Distortion, AMS, 1989.
[31] O. Martio, S. Rickman and J. Vaisala, Topological and Metric Properties of Quasiregular Mappings, Annales Academie Scientiarum Fennicae Mathematica, 488 (1971), 1–31. | Zbl 0223.30018
[32] M. Seeotharama Gowda, R. Sznajder, Weak connectedness of inverse images of continuus functions, Mathematics of Operations Reaserch 24 (1999), No. 1, 255-261. | Zbl 0977.90060
[33] R. Van Der Putten, A note on the local invertibility of Sobolev functions, Math. Scand. 83 (1998), 255–264. | Zbl 0921.46031