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On the rigidity and boundary regularity for Bakry-Emery-Kohn harmonic functions in Bergman metric on the unit ball in $ C^n$


Author: QiQi Zhang
Journal: Proc. Amer. Math. Soc. 145 (2017), 2971-2979
MSC (2010): Primary 32A50
DOI: https://doi.org/10.1090/proc/13501
Published electronically: February 22, 2017
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Abstract: In this paper we study the rigidity theorem for smooth Bakry-Emery-Kohn harmonic function $ u$ in the unit ball $ B_n$ in $ \textit {C}^n$, which satisfies

$\displaystyle \Box ^{\psi } u=\sum _{i, j=1}^n (\delta _{ij}-z_i \overline {z}_... ...t^2) \sum _{j=1}^n \overline {z}_j {\partial u\over \partial \overline {z}_j}=0$    

with some restriction of the coefficients of Taylor expansion for $ \psi $ at $ 1$. We prove that any smooth B-E-K harmonic function on $ \overline {B}_n$ must be holomorphic in $ B_n$. We study the regularity problem for the solution of the Dirichlet boundary value problem:

\begin{displaymath}\begin {cases}\Box ^\psi u=0, \hbox { if } z\in B_n,\cr \quad \ u=f, \hbox { if } z\in \partial B_n.\cr \end{cases}\end{displaymath}    


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Additional Information

QiQi Zhang
Affiliation: School of Mathematics and Computer Science, FuJian Normal University, Fuzhou 350007, People’s Republic of China
Email: zqq123_{good}@163.com

DOI: https://doi.org/10.1090/proc/13501
Keywords: Regularity, rigidity, Bakry-Emery-Kohn harmonic functions
Received by editor(s): August 8, 2016
Published electronically: February 22, 2017
Communicated by: Lei Ni
Article copyright: © Copyright 2017 American Mathematical Society