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Schwarz-type lemma at the boundary for mappings satisfying non-homogeneous polyharmonic equations

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Abstract

In this paper, we establish the boundary Schwarz lemma for solutions to non-homogeneous polyharmonic equations defined on the unit disk.

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References

  1. Hegehr, H.G.: Complex Analytic Methods for Partial Differential Equations. World Scientific, Singapore (1994)

    Google Scholar 

  2. Begehr, H., Vu, T.N.H., Zhang, Z.-X.: Polyharmonic dirichlet problems. Proc. Steklov Inst. Math. 255, 13–34 (2006)

    Article  MathSciNet  Google Scholar 

  3. Borichev, A., Hedenmalm, H.: Weighted integrability of polyharmonic functions. Adv. Math. 264, 464–505 (2014)

    Article  MathSciNet  Google Scholar 

  4. Bonk, M.: On Bloch’s constant. Proc. Am. Math. Soc. 110(3), 889–894 (1990)

    MathSciNet  MATH  Google Scholar 

  5. Burns, D.M., Krantz, S.G.: Rigidity of holomorphic mappings and a new Schwarz lemma at the boundary. J. Am. Math. Soc. 7, 661–676 (1994)

    Article  MathSciNet  Google Scholar 

  6. Chen, Sh, Li, P., Wang, X.: Schwarz-type lemma, Landau-type theorem, and Lipschitz-type space of solutions to inhomogenous biharmonic equations. J. Geom. Anal. (2019). https://doi.org/10.1007/s12220-018-0083-6

    Article  MATH  Google Scholar 

  7. Chen, Sh., Ponnusamy, S.: Schwarz lemmas for mappings satisfying Poisson’s equation, arXiv:1708.00715v1 [math.CV] 2 Aug (2017)

  8. Liu, T., Tang, X.: A new boundary rigidity theorem for holomorphic self-mappings of the unit ball in \({\mathbb{C}}^n\). Pure Appl. Math. Q. 11(1), 115–130 (2015)

    Article  MathSciNet  Google Scholar 

  9. Liu, T.S., Tang, X.M.: Schwarz lemma at the boundary of strongly pseudoconvex domain in \({\mathbb{C}}^{n}\). Math. Ann. 366, 655–666 (2016)

    Article  MathSciNet  Google Scholar 

  10. Kalaj, D.: Heinz-Schwarz inequalities for harmonic mappings in the unit ball. Ann. Acad. Sci. Fenn. Math. 41, 457–464 (2016)

    Article  MathSciNet  Google Scholar 

  11. Krantz, S.G.: The Schwarz lemma at the boundary. Complex Var. Elliptic Equa. 56, 455–468 (2011)

    Article  MathSciNet  Google Scholar 

  12. Liu, T., Wang, J., Tang, X.: Schwarz lemma at the boundary of the unit ball in \({\mathbb{C}}^n\) and its applications. J. Geom. Anal. 25, 1890–1914 (2015)

    Article  MathSciNet  Google Scholar 

  13. Mayboroda, S., Maz’ya, V.: Regularity of solutions to the polyharmonic equation in general domains. Invent. Math. 196, 1–68 (2014)

    Article  MathSciNet  Google Scholar 

  14. Mohapatra, M.R., Wang, X., Zhu, J.-F.: Boundary Schwarz lemma for solutions to nonhomogeneous biharmonic equations. Bull. Aust. Math. Soc. 100(3), 470–478 (2019)

    Article  MathSciNet  Google Scholar 

  15. Osserman, R.: A sharp Schwarz inequality on the boundary. Proc. Amer. Math. Soc. 128(12), 3513–3517 (2000)

    Article  MathSciNet  Google Scholar 

  16. Pavlović, M.: Introduction to function spaces on the disk, Matematic̆ki institut SANU, Belgrade, (2004)

  17. Wang, X., Zhu, J.-F.: Boundary Schwarz lemma for solutions to Poisson’s equation. J. Math. Anal. Appl. 463, 623–633 (2018)

    Article  MathSciNet  Google Scholar 

  18. Zhu, J.-F.: Schwarz lemma and boundary Schwarz lemma for pluriharmonic mappings. Filomat 32(15), 5385–5402 (2018)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

The author would like to thank the referee of this paper for his/her careful reading and constructive comments. The research was partly supported by NSFs of China (Nos. 11571216, 11671127 and 11720101003) and STU SRFT. The author would like to thank Dr. Jian-Feng Zhu for useful discussions on this topic.

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Correspondence to Manas Ranjan Mohapatra.

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Communicated by Adrian Constantin.

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Mohapatra, M.R. Schwarz-type lemma at the boundary for mappings satisfying non-homogeneous polyharmonic equations. Monatsh Math 192, 409–418 (2020). https://doi.org/10.1007/s00605-020-01402-x

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