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.
Dipartimento di Matematica e Applicazioni, Università Federico II di Napoli, Via Cintia, Monte S. Angelo 80126 Napoli, Italy
Bibliografia
[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.
[2] L. Ambrosio, B. Kircheim, Rectifiable sets in metric and Banach spaces, Math. Ann. 318 (3) (2000) 527–555.
[3] L. Ambrosio, B. Kircheim, Metric currents, Acta Mathematica 185 no. 1(2000), 1-80.
[4] L. Ambrosio, N. Fusco, D. Pallara, Function of Bounded Variations and Free Discontinbuity Problems,Oxford University Press, New York, 2000.
[5] L. Ambrosio, P. Tilli, Topics on Analysis in Metric Spaces, Oxford University Press, 2004.
[6] Erik Barvínek, Ivan Daler and Jan Franku, Convergence of sequences of inverse functions, Archivium Mathematicum, Vol. 27 3-4 (1991), 201–204.
[7] Y. Benyamini, J. Lindenstrauss, Geometric Nonlinear Functional Analysis, Vol.1, AMS, 2000.
[8] Chung-Wu Ho, A note on proper maps, Proceedings AMS, Vol. 51 1 (1975), 237-241.
[9] P.G. Ciarlet, J. Necas, Injectivity and self-Contact in nonlinear elasticity, Arch. Rational Mech. Anal. 97 (1987), no. 3, 171-188.
[10] B. Dacorogna, Direct Methods in the Calculus of Variations, Springer 2008.
[11] G. De Marco, G. Gorni, G. Zampieri, Global inversion of functions: an introduction, NODEA 1 (1994), 229-248.
[12] E. Durand-Cartagena, J.A. Jaramillo, Pointwise Lipschitz functions on metric spaces, J. Math. Anal. Appl. 363 (2010), 525– 548.
[13] I. Fonseca, W. Gangbo, Degree Theory in Analysis and Applications, Clarendom Press Oxford, 1995.
[14] W. Fulton, Algebraic Topology, Springer, 1995.
[15] J. Gevirtz, Metric conditions that imply local invertibility, Communications in Pure and AppliedMathematics 23 (1969), 243– 264.
[16] J. Gevirtz, Injectvity in Banach spaces and the Mazur-Ulam Theorem on Isometries, Transactions AMS 274, no. 1 (1982), 307–318.
[17] J. Gevirtz, A sharp condition for univalence in Euclidean spaces, Proceedings AMS 57, no. 2 (1976), 261–265.
[18] L. Granieri, Metric currents and geometry of Wasserstein spaces, Rend. Semin. Mat. Univ. Padova 124 (2010), 91-125.
[19] L. Granieri, F. Maddalena, A metric approach to elastic reformations, Acta Mathematica Applicandae, published online December 2013.
[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.
[22] F. John, On Quasi-Isometric Mappings I, Communications in Pure and Applied Mathematics 21 (1968), 77–110.
[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.
[25] L. V. Kovalev, Jani Onninen, On invertibility of Sobolev mappings, Journal für die reine und angewandte Mathematik, 656, (2011), 1–16.
[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.
[27] A. Lytchak, Open map Theorem for metric spaces, St. Petersburg Math. J. 17 (2006), No. 3, 477–491.
[28] V. Magnani, An Area formula in metric spaces, Colloq. Math. 124 (2011), no. 2, 275-283.
[29] J. S. Raymond, Local inversion for differentiable functions and Darboux property, Mathematika 49 (2002), no. 1-2, 141-158. [Crossref]
[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.
[32] M. Seeotharama Gowda, R. Sznajder, Weak connectedness of inverse images of continuus functions, Mathematics of Operations Reaserch 24 (1999), No. 1, 255-261.
[33] R. Van Der Putten, A note on the local invertibility of Sobolev functions, Math. Scand. 83 (1998), 255–264.