Invertible harmonic mappings, beyond Kneser

Alessandrini, Giovanni; Nesi, Vincenzo

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 8 (2009) no. 3, pp. 451-468.

We prove necessary and sufficient criteria of invertibility for planar harmonic mappings which generalize a classical result of H. Kneser, also known as the Radó–Kneser–Choquet theorem.

[1] G. Alessandrini and A. Diaz Valenzuela, Unique determination of multiple cracks by two measurements, SIAM J. Control Optim. 34 (1996), 913–921. | MR | Zbl

[2] G. Alessandrini and V. Nesi, Univalent $\sigma$-harmonic mappings, Arch. Ration. Mech. Anal. 158 (2001), 155–171. | MR | Zbl

[3] G. Alessandrini and V. Nesi, Univalent $\sigma$-harmonic mappings: applications to composites, ESAIM Control Optim. Calc. Var. 7 (2002), 379–406 (electronic). | EuDML | Numdam | MR | Zbl

[4] K. Astala, T. Iwaniec, G. J. Martin and J. Onninen, Extremal mappings of finite distortion, Proc. London Math. Soc. (3) 91 (2005), 655–702. | MR | Zbl

[5] P. Bauman, A. Marini and V. Nesi, Univalent solutions of an elliptic system of partial differential equations arising in homogenization, Indiana Univ. Math. J. 50 (2001), 747–757. | MR | Zbl

[6] D. Bshouty and W. Hengartner, Univalent harmonic mappings in the plane, In: “Handbook of Complex Analysis: Geometric Function Theory”, Vol. 2, Elsevier, Amsterdam, 2005, 479–506. | MR | Zbl

[7] G. Choquet, Sur un type de transformation analytique généralisant la représentation conforme et définie au moyen de fonctions harmoniques, Bull. Sci. Math. (2) 69 (1945), 156–165. | MR | Zbl

[8] J. Clunie and T. Sheil-Small, Harmonic univalent functions, Ann. Acad. Sci. Fenn. Ser. A I Math. 9 (1984), 3–25. | MR | Zbl

[9] P. Duren, “Harmonic Mappings in the Plane”, Cambridge Tracts in Mathematics, Vol. 156, Cambridge University Press, Cambridge, 2004. | MR | Zbl

[10] A. Friedman and M. Vogelius, Determining cracks by boundary measurements, Indiana Univ. Math. J. 38 (1989), 527–556. | MR | Zbl

[11] H. Kim and J. K. Seo, Unique determination of a collection of a finite number of cracks from two boundary measurements, SIAM J. Math. Anal. 27 (1996), 1336–1340. | MR | Zbl

[12] H. Kneser, Lösung der Aufgabe 41, Jber. Deutsch. Math.-Verein. 35 (1926), 123–124.

[13] H. Lewy, On the non-vanishing of the Jacobian in certain one-to-one mappings, Bull. Amer. Math. Soc. 42 (1936), 689–692. | JFM | MR

[14] G. H. Meisters and C. Olech, Locally one-to-one mappings and a classical theorem on schlicht functions, Duke Math. J. 30 (1963), 63–80. | MR | Zbl

[15] B. P. Palka, “An Introduction to Complex Function Theory”, Undergraduate Texts in Mathematics, Springer-Verlag, New York, 1991. | MR | Zbl

[16] T. Radó, Aufgabe 41, Jber. Deutsch. Math.-Verein. 35 (1926), 49. | JFM

[17] T. Radó, “On the problem of Plateau, Subharmonic Functions”, Reprint, Springer-Verlag, New York, 1971. | MR | Zbl

[18] F. Sauvigny, Uniqueness of Plateau’s problem for certain contours with a one-to-one, nonconvex projection onto a plane, In: “Geometric Analysis and the Calculus of Variations”, J. Jost (ed.), Int. Press, Cambridge, MA, 1996, 297–314. | Zbl

[19] V. I. Smirnov, “A Course of Higher Mathematics”, Vol. III, Part two, Complex variables. Special functions, Translated by D. E. Brown. Translation edited by I. N. Sneddon, Pergamon Press, Oxford, 1964. | MR

[20] S. Stoïlow, “Leçons sur les Principes Topologiques de la Théorie des Fonctions Analytiques”, deuxième édition, augmentée de notes sur les fonctions analytiques et leurs surfaces de Riemann, Gauthier-Villars, Paris, 1956. | MR | Zbl

[21] B. Von Kerékjártó, “Vorlesungen über Topologie, I.: Flächentopologie”, Die Grundlehrender mathematischen Wissenschaften. Bd. 8 , J. Springer, Berlin, 1923. | JFM

[22] A. Weinstein, A global invertibility theorem for manifolds with boundary, Proc. Roy. Soc. Edinburgh Sect. A 99 (1985), 283–284. | MR | Zbl

[23] J. C. Wood, Singularities of harmonic maps and applications of the Gauss-Bonnet formula, Amer. J. Math. 99 (1977), 1329–1344. | MR | Zbl