Hostname: page-component-745bb68f8f-cphqk Total loading time: 0 Render date: 2025-01-08T23:44:14.906Z Has data issue: false hasContentIssue false
  • English
  • Français

BOUNDARY SCHWARZ LEMMA FOR SOLUTIONS TO NONHOMOGENEOUS BIHARMONIC EQUATIONS

Part of: Geometric function theory Two-dimensional theory

Published online by Cambridge University Press:  09 September 2019

MANAS RANJAN MOHAPATRA Open the ORCID record for MANAS RANJAN MOHAPATRA [Opens in a new window] ,
XIANTAO WANG Open the ORCID record for XIANTAO WANG [Opens in a new window]  and
JIAN-FENG ZHU Open the ORCID record for JIAN-FENG ZHU [Opens in a new window]
MANAS RANJAN MOHAPATRA
Affiliation:
Department of Mathematics, Shantou University, Shantou, 515063, PR China email [email protected]
XIANTAO WANG*
Affiliation:
MOE-LCSM and School of Mathematics and Statistics, Hunan Normal University, Changsha, Hunan, 410081, PR China Department of Mathematics, Shantou University, Shantou, Guangdong, 515063, PR China email [email protected]
JIAN-FENG ZHU
Affiliation:
Department of Mathematics, Shantou University, Shantou, 515063, PR China School of Mathematical Sciences, Huaqiao University, Quanzhou, 362021, PR China email [email protected]
*
✉[email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We establish a boundary Schwarz lemma for solutions to nonhomogeneous biharmonic equations.

Keywords

boundary Schwarz lemmasolutionnonhomogeneous biharmonic equation

MSC classification

Primary: 30C80: Maximum principle; Schwarz's lemma, Lindelöf principle, analogues and generalizations; subordination
Secondary: 31A30: Biharmonic, polyharmonic functions and equations, Poisson's equation
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 100 , Issue 3 , December 2019 , pp. 470 - 478
Copyright
© 2019 Australian Mathematical Publishing Association Inc. 

Footnotes

The research was partly supported by NSF of China (Nos. 11571216, 11671127 and 11720101003) and STU SRFT. The third author was supported by NSF of Fujian Province (No. 2016J01020) and the Promotion Program for Young and Middle-aged Teachers in Science and Technology Research of Huaqiao University (ZQN-PY402).

References

Axler, S., Bourdon, P. and Ramey, W., Harmonic Function Theory, 2nd edn (Springer, New York, Berlin, Heidelberg, 2004).Google Scholar
Begehr, H., ‘Dirichlet problems for the biharmonic equation’, Gen. Math. 13 (2005), 65–72.Google Scholar
Bonk, M., ‘On Bloch’s constant’, Proc. Amer. Math. Soc. 110 (1990), 889–894.Google Scholar
Burns, D. M. and Krantz, S. G., ‘Rigidity of holomorphic mappings and a new Schwarz lemma at the boundary’, J. Amer. Math. Soc. 7 (1994), 661–676.Google Scholar
Chang, S.-Y. A., Wang, L. and Yang, P., ‘A regularity theory of biharmonic maps’, Comm. Pure Appl. Math. 52 (1999), 1113–1137.Google Scholar
Chen, Sh. and Kalaj, D., ‘The Schwarz type lemmas and the Landau type theorem of mappings satisfying Poisson’s equations’, Complex Anal. Oper. Theory 13 (2019), 2049–2068.Google Scholar
Chen, Sh., Li, P. and Wang, X., ‘Schwarz-type lemma, Landau-type theorem, and Lipschitz-type space of solutions to inhomogeneous biharmonic equations’, J. Geom. Anal. 29 (2019), 2469–2491.Google Scholar
Garnett, J., Bounded Analytic Functions (Academic Press, New York, 1981).Google Scholar
Krantz, S. G., ‘The Schwarz lemma at the boundary’, Complex Var. Elliptic Equ. 56 (2011), 455–468.Google Scholar
Liu, T. and Tang, X., ‘A new boundary rigidity theorem for holomorphic self-mappings of the unit ball in ℂ n ’, Pure Appl. Math. Q. 11 (2015), 115–130.Google Scholar
Liu, T. and Tang, X., ‘Schwarz lemma at the boundary of strongly pseudoconvex domain in ℂ n ’, Math. Ann. 366 (2016), 655–666.Google Scholar
Liu, T., Wang, J. and Tang, X., ‘Schwarz lemma at the boundary of the unit ball in ℂ n and its applications’, J. Geom. Anal. 25 (2015), 1890–1914.Google Scholar
Mayboroda, S. and Maz’ya, V., ‘Boundedness of gradient of a solution and Wiener test of order one for biharmonic equation’, Invent. Math. 175 (2009), 287–334.Google Scholar
Osserman, R., ‘A sharp Schwarz inequality on the boundary’, Proc. Amer. Math. Soc. 128 (2000), 3513–3517.Google Scholar
Strzelechi, P., ‘On biharmonic maps and their generalizations’, Calc. Var. Partial Differ. Equ. 18 (2003), 401–432.Google Scholar
Wang, X. and Zhu, J.-F., ‘Boundary Schwarz lemma for solutions to Poisson’s equation’, J. Math. Anal. Appl. 463 (2018), 623–633.Google Scholar
Zhu, J.-F., ‘Schwarz lemma and boundary Schwarz lemma for pluriharmonic mappings’, Filomat 32 (2018), 5385–5402.Google Scholar