Advertisement

Lipschitz continuity for solutions of the -Poisson equation

  • Xingdi ChenEmail author
Articles
  • 1 Downloads

Abstract

In this paper, we study the Lipschitz continuity for solutions of the -Poisson equation. After characterizing the boundary conditions for the Lipschitz continuity of -harmonic mappings, we present four equivalent conditions for the (K,K′)-quasiconformal solutions of the -Poisson equation with a nonhomogeneous term to be Lipschitz continuous.

Keywords

weighted Laplacian operator Lipschitz continuity Dirichlet boundary problem harmonic mapping (K,K′)-quasiconformal mapping 

MSC(2010)

30C65 31A05 30C40 35J25 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Notes

Acknowledgements

This work was supported by National Natural Science Foundation of China (Grant No. 11471128), the Natural Science Foundation of Fujian Province of China (Grant No. 2016J01020), Promotion Program for Young and Middle-aged Teacher in Science and Technology Research of Huaqiao University (Grant Nos. ZQN-YX110 and ZQN-PY402). The author is deeply indebted to Professor David Kalaj for his helpful suggestion about the study of Lipschitz continuity. The author is grateful to the anonymous referees for their careful reading of the manuscript and many helpful suggestions.

References

  1. 1.
    Ahlfors L V. Lectures on Quasiconformal Mappings. University Lecture Series, vol. 38. Providence: Amer Math Soc, 2006Google Scholar
  2. 2.
    Astala K. Iwaniec T, Martin G. Elliptic Partial Differential Equations and Quasiconformal Mappings in the Plane. Princeton: Princeton University Press, 2009zbMATHGoogle Scholar
  3. 3.
    Behm G. Solving Poisson equation for the standard weighted Laplacian in the unit disc. ArXiv:1306.2199v2, 2013Google Scholar
  4. 4.
    Chen J L, Li P J, Sahoo S K, et al. On the Lipschitz continuity of certain quasiregular mappings between smooth Jordan domains. Israel J Math, 2017, 220: 453–478MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Chen M, Chen X D. (K,K’)-quasiconformal harmonic mappings of the upper half plane onto itself. Ann Acad Sci Fenn Math, 2012, 37: 265–276MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Chen S L, Vuorinen M. Some properties of a class of elliptic partial differential operators. J Math Anal Appl, 2015, 431: 1124–1137MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Chen X D, Kalaj D. A representation theorem for standard weighted harmonic mappings with an integer exponent and its applications. J Math Anal Appl, 2016, 444: 1233–1241MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Chen X D, Kalaj D. Representation theorem of standard weighted harmonic mappings in the unit disk. https://doi.org/www.researchgate.net/publication/330243622, 2018Google Scholar
  9. 9.
    Clunie J, Theil-Small T. Harmonic univalent functions. Ann Acad Sci Fenn Math, 1984, 9: 3–25MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Duren P. Harmonic Mappings in the Plane. Cambridge: Cambridge University Press, 2004CrossRefzbMATHGoogle Scholar
  11. 11.
    Finn R, Serrin J. On the Hölder continuity of quasi-conformal and elliptic mappings. Trans Amer Math Soc, 1958, 89: 1–15MathSciNetzbMATHGoogle Scholar
  12. 12.
    Garabedian P R. A partial differential equation arising in conformal mapping. Pacific J Math, 1951, 1: 485–524MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Garnett J B. Bounded Analytic Functions, 1st ed. Graduate Texts in Mathematics. pNew York: Springer, 2007Google Scholar
  14. 14.
    Hedenmalm H, Korenblum B, Zhu K. Theory of Bergman spaces. Graduate Texts in Mathematics. New York: Springer-Verlag, 2000Google Scholar
  15. 15.
    Kalaj D. Quasiconformal mapping between Jordan domains. Math Z, 2008, 260: 237–252MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Kalaj D. On quasi-conformal harmonic maps between surfaces. Int Math Res Not IMRN, 2015, 2015: 355–380MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Kalaj D, Mateljević M. Inner estimate and quasiconformal harmonic maps between smooth domains. J Anal Math, 2006, 100: 117–132MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Kalaj D, Mateljević M. (K,K’)-quasiconformal harmonic mappings. Potential Anal, 2012, 36: 117–135MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Kalaj D, Pavlović M. On quasiconformal self-mappings of the unit disk satisfying Poisson equation. Trans Amer Math Soc, 2011, 363: 4043–4061MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Lehto O, Virtanen K I. Quasiconformal Mappings in the Plane. Berlin-Heidelberg-New York: Springer-Verlag, 1973CrossRefzbMATHGoogle Scholar
  21. 21.
    Li P J, Chen J L, Wang X T. Quasiconformal solution of Poisson equations. Bull Aust Math Soc, 2015, 92: 420–428MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Manojlović V. Bi-Lipschicity of quasiconformal harmonic mappings in the plane. Filomat, 2009, 23: 85–89MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Martio O. On harmonic quasiconformal mappings. Ann Acad Sci Fenn Ser A, 1968, 425: 3–10MathSciNetzbMATHGoogle Scholar
  24. 24.
    Mateljević M. Quasiconformality of harmonic mappings between Jordan domains. Filomat, 2012, 26: 479–510MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Morrey C B. On the solutions of quasi-linear elliptic partial differential equations. Trans Amer Math Soc, 1938, 43: 126–166MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Mu J J, Chen X D. Landau-type theorems for solutions of a quasilinear differential equation. J Math Study, 2014, 47: 295–304MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Nirenberg L. On nonlinear elliptic partial differential equations and Hölder continuity. Comm Pure Appl Math, 1953, 6: 103–156MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Olofsson A. Differential operators for a scale of Poisson type kernels in the unit disc. J Anal Math, 2014, 123: 227–249MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Olofsson A, Wittsten J. Poisson integrals for standard weighted Laplacians in the unit disc. J Math Soc Japan, 2013, 65: 447–486MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Partyka D, Sakan K. On bi-Lipschitz type inequalities for quasiconformal hamonic mappings. Ann Acad Sci Fenn Math, 2007, 32: 579–594MathSciNetzbMATHGoogle Scholar
  31. 31.
    Pavlović M. Boundary correspondence under harmonic quasiconformal homeomorphisms of the unit disk. Ann Acad Sci Fenn Math, 2002, 27: 365–372MathSciNetzbMATHGoogle Scholar
  32. 32.
    Simon L. A Hölder estimate for quasiconformal maps between surfaces in Euclidean space. Acta Math, 1977, 139: 19–51MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Science China Press and Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of MathematicsHuaqiao UniversityQuanzhouChina

Personalised recommendations