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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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On the rigidity and boundary regularity for Bakry-Emery-Kohn harmonic functions in Bergman metric on the unit ball in $\textit {C}^n$
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by QiQi Zhang PDF
Proc. Amer. Math. Soc. 145 (2017), 2971-2979 Request permission

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 \begin{equation*} \Box ^{\psi } u=\sum _{i, j=1}^n (\delta _{ij}-z_i \overline {z}_j){\partial ^2 u \over \partial z_i \partial \overline {z}_j}+(1-|z|^2) \psi ’(|z|^2) \sum _{j=1}^n \overline {z}_j {\partial u\over \partial \overline {z}_j}=0 \end{equation*} 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{equation*} \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{equation*}
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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
  • Received by editor(s): August 8, 2016
  • Published electronically: February 22, 2017
  • Communicated by: Lei Ni
  • © Copyright 2017 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 145 (2017), 2971-2979
  • MSC (2010): Primary 32A50
  • DOI: https://doi.org/10.1090/proc/13501
  • MathSciNet review: 3637945