Abstract
We first investigate the Lipschitz continuity of (K,K’)-quasiregular C 2 mappings between two Jordan domains with smooth boundaries, satisfying certain partial differential inequalities concerning Laplacian. Then two applications of the obtained result are given: As a direct consequence, we get the Lipschitz continuity of ρ-harmonic (K,K’)-quasiregular mappings, and as the other application, we study the Lipschitz continuity of (K,K’)- quasiconformal self-mappings of the unit disk, which are the solutions of the Poisson equation Δw = g. These results generalize and extend several recently obtained results by Kalaj, Mateljević and Pavlović.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
M. Chen and X. Chen, (K,K’)-quasiconformal harmonic mappings of the upper half plane onto itself, Annales Academiæ Scientiarum Fennicæ. Mathematica 37 (2012), 265–276.
P. Duren, Harmonic Mappings in the Plane, Cambridge Tracts in Mathematics, Vol. 156, Cambridge University Press, Cambridge, 2004.
R. Finn and J. Serrin, On the Hölder continuity of quasiconformal and elliptic mappings, Transactions of the American Mathematical Society 89 (1958), 1–15.
G. M. Goluzin, Geometric Theory of Functions of a Complex Variable, Translations of Mathematical Monographs, Vol. 26, American Mathematical Society, Providence, RI, 1969
E. Heinz, On certain nonlinear elliptic differential equations and univalent mappings, Journal d’Analyse Mathématique 5 (1956/57), 197–272.
E. Heinz, On one-to-one harmonic mappings, Pacific Journal of Mathematics 9 (1959), 101–105.
L. Hörmander, Notions of Convexity, Progress in Mathematics, Vol. 127, Birkhäuser Boston, Boston, MA, 1994.
D. Kalaj, Quasiconformal and harmonic mappings between Jordan domains, Matheamtische Zeitschrift 260 (2008), 237–252.
D. Kalaj, On harmonic quasiconformal self-mappings of the unit ball, Annales Academiæ Scietiarum Fennicæ. Mathematica 33 (2008), 261–271.
D. Kalaj, On quasiregular mappings between smooth Jordan domains, Journal of Mathematical Analysis and Applications 362 (2010), 58–63.
D. Kalaj, On the quasiconformal self-mappings of the unit ball satisfying the Poisson differential equation, Annales Academiæ Scientiarum Fennicæ. Mathematica 36 (2011), 177–194.
D. Kalaj and M. Mateljević, Inner estimate and quasiconformal harmonic maps between smooth domains, Journal d’Analyse Mathématique 100 (2006), 117–132.
D. Kalaj and M. Mateljević, On certain nonlinear elliptic PDE and quasiconfomal maps between Euclidean surfaces, Potential Analysis 34 (2011), 13–22.
D. Kalaj and M. Mateljević, (K,K’)-quasiconformal harmonic mappings, Potential Analysis 36 (2012), 117–135.
D. Kalaj and M. Pavlović, Boundary correspondence under harmonic quasiconformal diffeomorphisms of a half-plane, Annales Academiæ Scientiarum Fennicæ. Mathematica 30 (2005), 159–165.
D. Kalaj and M. Pavlović, On quasiconformal self-mappings of the unit disk satisfying Poisson equation, Transactions of the American Mathematical Society 363 (2011), 4043–4061.
M. Knežević and M. Mateljević, On the quasi-isometries of harmonic quasiconformal mappings, Journal of Mathematical Analysis and Applications 334 (2007), 404–413.
P. Li, J. Chen and X. Wang, Quasiconformal solutions of Poisson equations, Bulletin of the Australian Mathematical Society 92 (2015), 420–428.
O. Martio, On harmonic quasiconformal mappings, Annales Academiæ Scientiarum Fennicæ. Mathematica 425 (1968), 3–10.
M. Mateljević and M. Vuorinen, On harmonic quasiconformal quasi-isometries, Journal of Inequalities and Applications (2010), Art. ID 178732, 19.
L. Nirenberg, On nonlinear elliptic partial differential equations and Hölder continuity, Communications on Pure and Applied Mathematics 6 (1953), 103–156.
M. Pavlović, Boundary correspondence under harmonic quasiconformal homeomorphisms of the unit disk, Annales Academiæ Scientiarum Fennicæ. Mathematica 27 (2002), 365–372.
Ch. Pommerenke, Boundary Behavour of Conformal Maps, Grundlehren der Mathematischen Wissenschaften, Vol. 299, Springer-Verlag, Berlin, 1992.
L. F. Tam and T. Y. H. Wan, Harmonic diffeomorphisms into Cartan-Hadamard surfaces with prescribed Hopf differentials, Communications in Analysis and Geometry 2 (1994), 593–625.
L. F. Tam and T. Y. H. Wan, Quasi-conformal harmonic diffeomorphism and the universal Teichmüller space, Journal of Differential Geometry 42 (1995), 368–410.
L. F. Tam and T. Y. H. Wan, On harmonic quasiconformal maps, Pacific Journal of Mathematics 182 (1998), 359–383.
T. Y. H. Wan, Constant mean curvature surface, harmonic maps and universal Teichm üller space, Journal of Differential Geometry 35 (1992), 643–657.
S. E. Warschawski, On the higher derivatives at the boundary in conformal mapping, Transactions of the American Mathematical Society 38 (1935), 310–340.
S. E. Warschawski, On differentiability at the boundary in conformal mapping, Proceedings of the American Mathematical Society 12 (1961), 614–620.
H. Whitney, Analytic extensions of differentiable functions defined in closed sets, Transactions of the American Mathematical Society 36 (1934), 63–89.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Chen, J., Li, P., Sahoo, S.K. et al. On the Lipschitz continuity of certain quasiregular mappings between smooth Jordan domains. Isr. J. Math. 220, 453–478 (2017). https://doi.org/10.1007/s11856-017-1522-y
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11856-017-1522-y