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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Hegehr, H.G.: Complex Analytic Methods for Partial Differential Equations. World Scientific, Singapore (1994)
Begehr, H., Vu, T.N.H., Zhang, Z.-X.: Polyharmonic dirichlet problems. Proc. Steklov Inst. Math. 255, 13–34 (2006)
Borichev, A., Hedenmalm, H.: Weighted integrability of polyharmonic functions. Adv. Math. 264, 464–505 (2014)
Bonk, M.: On Bloch’s constant. Proc. Am. Math. Soc. 110(3), 889–894 (1990)
Burns, D.M., Krantz, S.G.: Rigidity of holomorphic map**s and a new Schwarz lemma at the boundary. J. Am. Math. Soc. 7, 661–676 (1994)
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
Chen, Sh., Ponnusamy, S.: Schwarz lemmas for map**s satisfying Poisson’s equation, ar**s of the unit ball in \({\mathbb{C}}^n\). Pure Appl. Math. Q. 11(1), 115–130 (2015)
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)
Kalaj, D.: Heinz-Schwarz inequalities for harmonic map**s in the unit ball. Ann. Acad. Sci. Fenn. Math. 41, 457–464 (2016)
Krantz, S.G.: The Schwarz lemma at the boundary. Complex Var. Elliptic Equa. 56, 455–468 (2011)
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)
Mayboroda, S., Maz’ya, V.: Regularity of solutions to the polyharmonic equation in general domains. Invent. Math. 196, 1–68 (2014)
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)
Osserman, R.: A sharp Schwarz inequality on the boundary. Proc. Amer. Math. Soc. 128(12), 3513–3517 (2000)
Pavlović, M.: Introduction to function spaces on the disk, Matematic̆ki institut SANU, Belgrade, (2004)
Wang, X., Zhu, J.-F.: Boundary Schwarz lemma for solutions to Poisson’s equation. J. Math. Anal. Appl. 463, 623–633 (2018)
Zhu, J.-F.: Schwarz lemma and boundary Schwarz lemma for pluriharmonic map**s. Filomat 32(15), 5385–5402 (2018)
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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Communicated by Adrian Constantin.
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Mohapatra, M.R. Schwarz-type lemma at the boundary for map**s satisfying non-homogeneous polyharmonic equations. Monatsh Math 192, 409–418 (2020). https://doi.org/10.1007/s00605-020-01402-x
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DOI: https://doi.org/10.1007/s00605-020-01402-x