DISTORTION OF QUASICONFORMAL HARMONIC FUNCTIONS AND HARMONIC MAPPINGS M. Mateljevic Univ. of Belgrade, Belgrade, Serbia;
[email protected]
- Miodrag Mateljevi´ c-
In a series of lectures, during Iwhmhm 09, we have discussed the following subjects: (1) basic properties of harmonic and harmonic quasiconformal mappings in plane and space (2) In the first part of the communication, we plan to discuss the distortion of harmonic functions in plane. In particular we show that quasiconformal harmonic type mappings are the quasi-isometries with respect to quasihyperbolic geometry, cf. [MaK], [Man] and [Ma2]; and in connection with this we consider versions of the Kobe theorem for quasiregular harmonic mappings, cf. [Ma3]. (3) Harmonic quasiconformal mappings were first studied by O. Martio. We plan to present some recent results on the topic of quasiconformal harmonic maps. We estimate the modulus of derivatives of mappings which satisfy a certain estimate concerning laplacian and gradient and under certain conditions we show that harmonic quasiconformal maps are bi-Lipschitz in the plane, cf. [KaM, MP]. Characterizations of harmonic quasiconformal maps by the boundary mappings are given in some settings. In [Ka4], it is proved that harmonic quasiconformal mappings have a property which is typical for conformal mappings: every quasiconformal harmonic mapping w of C 1,µ Jordan domain Ω1 onto C 1,µ Jordan domain Ω is Lipschitz continuous. In addition, if Ω has C 2,µ boundary, using a version of the Hopf lemma, which seems to be a new idea in the subject, it is proved that w is bi-Lipschitz continuous, cf. [Ka5]. These results have been extended by the author and his colaborators in various ways. For example, in [BM] we generalize the result in [Ka5] and prove that any quasi-conformal harmonic mapping between two Jordan domains, with C 1,α , boundaries is bi-Lipschitz. Our proof is is based on new ideas and in particular we use GehringOsgood theorem, cf. [GOs]; note that we do not use any version of the Hopf lemma. 1
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(4) In [MAK] and [AKM], we obtain the sharp estimate of the derivatives of harmonic quasiconformal extension u = P [φ] of a Lipschitz map φ : Sn−1 → Rn . We also consider additional conditions which provide that u is Lipschitz on the unit ball; in particular, we give characterizations of Lipschitz continuity of u in the planar case and in the upper half space setting. We also answer a question posed by O. Martio in [OM] and extend this to the case of several variables. (5) The purpose of this lecture is to explore conditions which guarantee Lipschitz-continuity of harmonic maps w.r.t. quasihyperbolic metrics. For instance, we prove that harmonic quasiconformal maps are Lipschitz w.r.t. quasihyperbolic metrics in space cf. [MaVu], and byLipschitz in plane, cf. [Man] and [Ma2]. (6) We stady mappings in space which satisfy (0.1)
|∆u| ≤ a|∇u|2 + b . Suppose that domains D and Ω are bounded domains in Rn and its boundaries belong to class C k,α , 0 ≤ α ≤ 1, k ≥ 2. Suppose further that g andg 0 are C 1 metric on D and Ω respectively. Usung inner estimate (cf. Theorem 6.14 [GT]) we prove Theorem 0.1. If u : D → Ω a qc (g, g 0 )-harmonic map (or satisfies (0.1)), then u is Lipschitz on D.
We also point some differences between the theory in the plane and the space. In [KaM]it is proved the following theorem: a K quasiconformal harmonic mapping of the unit ball B n (n > 2) onto itself is Euclidean bi-lipschitz, providing that u(0) = 0 and that K < 2n−1 , where n is the dimension of the space. It is an extension of a similar result for hyperbolic harmonic mappings with respect to hyperbolic metric (see Tam and Wan, (1998)). The proof makes use of M¨obius transformations in the space, and of a recent result which states that, harmonic quasiconformal self-mappings of the unit ball are Lipschitz continuous. References [Ah] L. Ahlfors: Lectures on quasiconformal mappings, Van Nostrand Princeton, N.J., 1966. [ABR] S. Axler, P. Bourdon and W. Ramey: Harmonic function theory, Springer-Verlag, New York 1992. c, V. Koji´ c, M. Mateljevi´ c, On Lipschitz continuity of harmonic quasiregural [AKM] M. Arsenovi´ mappings on the unit ball in Rn , Ann. Acad. Sci. Fenn. Math. Vol. 33, (2008), 315-318, Zbl 1140.31003. zin, M. Mateljevi´ c: Quasiconformal harmonic mapping between Jordan domains C 1,α , [BM] V. Boˇ unpublished manuscript [GOs] F.W. Gehring and B.G. Osgood: Uniform domains and the quasi-hyperbolic metric, J. Anal. Math. 36(1979), 50–74. [GT] Gilbarg, D., Trudinger, N., Elliptic partial Differential Equation of Second Order, Second Edition, 1983.
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[Ka] D. Kalaj: Univalent harmonic mappings between Jordan domains. Publ. Inst. Math., Nouv. Ser. 69(83), 108-112 (2001). [Ka1] D. Kalaj, Harmonic and quasiconformal functions between convex domains, Doctoral Thesis, Beograd, submited 2000, (2002). [Ka2] D. Kalaj: On harmonic diffeomorphisms of the unit disc onto a convex domain. Complex Variables, Theory Appl. 48, No.2, 175-187 (2003). [KaP] D. Kalaj; M. Pavlovi´ c: Boundary correspondence under harmonic quasiconformal homeomorfisms of a half-plane Ann. Acad. Sci. Fenn., Math. 30, No.1, (2005) 159-165. [Ka3] D. Kalaj: Quasiconformal harmonic functions between convex domains, Publ. Inst. Math., Nouv. Ser. 76(90), 3-20 (2004). [Ka4] D. Kalaj: Quasiconformal harmonic mapping between Jordan domains Math. Z. V 260, Number 2, 237-252, 2008. [Ka5] D. Kalaj: Harmonic quasiconformal mappings and Lipschitz spaces, Ann. Acad. Sci. Fenn. Math. Math. 34:2 (2009), 475-485. [KaM] D. Kalaj, M. Mateljevi´ c: Inner estimate and quasiconformal harmonic maps between smooth domains, Journal d’Analise Math. 100. 117-132, (2006). [KaM] David Kalaj, Miodrag Mateljevi´ c, Harmonic quasiconformal self-mappings and M¨ obius transformations of the unit ball, to appear in Pacific J. Math. [Ko] O. Kellogg: On the derivatives of harmonic functions on the boundary, Trans. Amer. Math. Soc. 33 (1931), 689-692. [MaK] M. Kneˇ zevi´ c, M. Mateljevi´ c, On the quasi-isometries of harmonic quasi-conformal mappings J Math Anal Appl, 2007, 334(1), 404–413. [L] H. Lewy: On the non-vanishing of the Jacobian in certain in one-to-one mappings, Bull. Amer. Math. Soc. 42. (1936), 689-692. [OM] O. Martio: On harmonic quasiconformal mappings, Ann. Acad. Sci. Fenn., Ser. A I 425 (1968), 3-10, Zbl 0162.37902. [Ma] M. Mateljevic, Ahlfors-Schwarz lemma and curvature, Kragujevac Journal of Mathematics (Zbornik radova PMF), Vol 25, 2003, 155–164. ´, Estimates for the modulus of the derivatives of harmonic univalent map[Ma1] M. Mateljevic pings, Proceedings of International Conference on Complex Analysis and Related Topics (IXth Romanian-Finnish Seminar, 2001), Rev Roum Math Pures Appliq (Romanian Journal of Pure and Applied mathematics) 47 (2002) 5-6, 709 -711. [Ma2] M. Mateljevi´ c, Distortion of harmonic functions and harmonic quasiconformal quasiisometry, Revue Roum. Math. Pures Appl. Vol.51,(2006)), 5–6, 711–722 [Ma3] M. Mateljevi´ c, Quasiconformal and Quasiregular harmonic analogues of Koebe’s Theorem and Applications, Ann. Acad. Sci. Fenn. -M vol. 32, 2007, 301-315 [Ma4] M. Mateljevi´ c: On bi-Lipschitz continuity of quasiconformal ρ-harmonic with positive Gauss curvature, preprint. [MP] M. Pavlovi´ c: Boundary correspondence under harmonic quasiconformal homeomorfisms of the unit disc, Ann. Acad. Sci. Fenn., Vol 27, 2002, 365–372. [MAK] Miodrag Mateljevi´ c, Miloˇs Arsenovi´ c and Vesna Manojlovi´ c, Lipschitz-type spaces and harmonic mappings in the space, to appear in Ann. Acad. Sci. Fenn. Math. [Ma5] M. Mateljevic, DISTORTION OF QUASICONFORMAL HARMONIC FUNCTIONS AND HARMONIC MAPPINGS, Abstracts, International Conference on Complex Analysis and the 12th Romanian-Finnish Seminar, Turku,2009 [MaVu] M. Mateljevic, M. Vuorinen: On harmonic quasiconformal quasi-isometries, arXiv: 0709.4546v1. [KaM] David Kalaj, Miodrag Mateljevi´ c, QUASICONFORMAL AND HARMONIC MAPPINGS BETWEEN SMOOTH JORDAN DOMAINS, Novi Sad J. Math. Vol. 38, No. 3, 2008, 147-156 [Man] V. Manojlovic, Bi-Lipschicity of quasiconformal harmonic mappings in the plane, FILOMAT Volume 23, Number 1, February 2009, 85-89 [Iwhm09] https://sites.google.com/site/iwhmhm09/course-materials,Workshop on Harmonic Mappings and Hyperbolic Metrics, Chennai, India, Dec. 10-19, 2009
