Harmonic mappings and distance function
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 10 (2011) no. 3, pp. 669-681.

We prove the following theorem: every quasiconformal harmonic mapping between two plane domains with C 1,α (α<1) and, respectively, C 1,1 compact boundary is bi-Lipschitz. This theorem extends a similar result of the author [10] for Jordan domains, where stronger boundary conditions for the image domain were needed. The proof uses distance function from the boundary of the image domain.

Publié le :
Classification : 58E20, 30C62
Kalaj, David 1

1 University of Montenegro Faculty of Natural Sciences and Mathematics Cetinjski put b.b. 81000 Podgorica, Montenegro
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Kalaj, David. Harmonic mappings and distance function. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 10 (2011) no. 3, pp. 669-681. http://www.numdam.org/item/ASNSP_2011_5_10_3_669_0/

[1] L. Ahlfors, Lectures on Quasiconformal mappings, Van Nostrand Mathematical Studies, Vol. 10, D. Van Nostrand 1966. | MR | Zbl

[2] G. Alessandrini and V. Nesi, Invertible harmonic mappings, beyond Kneser, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (5) 8 (2009), 451–468. | Numdam | MR | Zbl

[3] M. Arsenovic, V. Kojic, and M. Mateljevic, On lipschitz continuity of harmonic quasiregular maps on the unit ball in n , Ann. Acad. Sci. Fenn. Math. 33 (2008), 315–318. | EuDML | MR | Zbl

[4] P. Li, and L. Tam, Uniqueness and regularity of proper harmonic maps, Ann. of Math. (2) 137 (1993), 167–201. | MR | Zbl

[5] P. Li, and L. Tam, Uniqueness and regularity of proper harmonic maps. II, Indiana Univ. Math. J. 42 (1993), 591–635. | MR | Zbl

[6] G. M. Goluzin, Geometric function theory, Nauka Moskva, Russian, 1966. | Zbl

[7] D. Gilbarg and N. Trudinger, “Elliptic Partial Differential Equations of Second Order”, Vol. 224, Second Edition, Springer 1977, 1983. | MR | Zbl

[8] W. Hengartner and G. Schober, Harmonic mappings with given dilatation, J. London Math. Soc. 33 (1986), 473–483. | MR | Zbl

[9] E. Hopf, A remark on linear elliptic differential equations of second order, Proc. Amer. Math. Soc. 3 (1952), 791–793. | MR | Zbl

[10] D. Kalaj, Lipschitz spaces and harmonic mappings, Ann. Acad. Sci. Fenn. Math. 34 (2009), 475–485. | MR | Zbl

[11] D. Kalaj, Quasiconformal harmonic functions between convex domains, Publ. Inst. Math. 76 (2004), 3–20. | EuDML | MR | Zbl

[12] D. Kalaj, On harmonic quasiconformal self-mappings of the unit ball, Ann. Acad. Sci. Fenn. Math. 33 (2008), 1-11. | MR

[13] D. Kalaj, Quasiconformal harmonic mapping between Jordan domains, Math. Z. 260 (2008), 237–252. | MR | Zbl

[14] D. Kalaj, On harmonic diffeomorphisms of the unit disc onto a convex domain, Complex Var. Theory Appl. 48 (2003), 175–187. | MR | Zbl

[15] D. Kalaj, On quasiregular mappings between smooth Jordan domains, J. Math. Anal. Appl. 362 (2010), 58–63. | MR | Zbl

[16] D. Kalaj and M. Mateljević, Inner estimate and quasiconformal harmonic maps between smooth domains, J. Anal. Math. 100 (2006), 117–132. | MR | Zbl

[17] D. Kalaj and M. Mateljević, On certain nonlinear elliptic PDE and quasiconfomal maps between Euclidean surfaces, Potential Anal. 34 (2010), 13–22. | MR | Zbl

[18] D. Kalaj and M. Mateljević, On quasiconformal harmonic surfaces with rectifiable boundary, Complex Anal. Oper. Theory, to appear. doi: 10.1007/s11785-010-0062-9. | MR | Zbl

[19] D. Kalaj and M. Pavlović, Boundary correspondence under harmonic quasiconformal homeomorfisms of a half-plane, Ann. Acad. Sci. Fenn. Math. 30 (2005), 159–165. | EuDML | MR | Zbl

[20] D. Kalaj and M. Pavlović, On quasiconformal self-mappings of the unit disk satisfying the Poisson equation, Trans. Amer. Math. Soc. 363 (2011), 4043–4061. | MR | Zbl

[21] M. Knezevic and M. Mateljevic, On the quasi-isometries of harmonic quasiconformal mappings, J. Math. Anal. Appl. 334 (2007), 404–413. | MR | Zbl

[22] O. Lehto and K. I. Virtanen, “Quasiconformal Mapping”, Springer-Verlag, Berlin and New York, 1973.

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

[24] V. Manojlović, Bi-lipshicity of quasiconformal harmonic mappings in the plane, Filomat 23 (2009), 85–89. | MR | Zbl

[25] O. Martio, On harmonic quasiconformal mappings, Ann. Acad. Sci. Fenn. Ser. A I 425 (1968), 3–10. | MR | Zbl

[26] M. Mateljevic and M. Vuorinen, On harmonic quasiconformal quasi-isometries, J. Inequalities Appl. 2010 (2010), Article ID 178732, 19 p. | EuDML | MR | Zbl

[27] R. Näkki and B. Palka, Boundary regularity and the uniform convergence of quasiconformal mappings, Comment. Math. Helv. 54 (1979), 458–476. | EuDML | MR | Zbl

[28] Partyka D.; Sakan, K. On bi-Lipschitz type inequalities for quasiconformal harmonic mappings, Ann. Acad. Sci. Fenn. Math. 32 (2007), 579–594. | MR | Zbl

[29] M. Pavlović, Boundary correspondence under harmonic quasiconformal homeomorfisms of the unit disc, Ann. Acad. Sci. Fenn. Math. 27 (2002), 365–372. | EuDML | MR | Zbl

[30] C. Pommerenke, “Boundary Behaviour of Conformal Maps”, Springer-Verlag, New York, 1992. | MR | Zbl

[31] C. Wang, A sharp form of Mori’s theorem on Q-mappings, Kexue Jilu 4 (1960), 334–337. | MR

[32] T. Wan, Constant mean curvature surface, harmonic maps, and universal Teichmüller space, J. Differential Geom. 35 (1992), 643–657. | MR | Zbl

[33] S. E. Warschawski, On differentiability at the boundary in conformal mapping, Proc. Amer. Math. Soc. 12 (1961), 614–620. | MR | Zbl

[34] S. E. Warschawski, On the higher derivatives at the boundary in conformal mapping, Trans. Amer. Math. Soc. 38 (1935), 310–340. | JFM | MR