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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles 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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A remark on a result of Ding-Jost-Li-Wang
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by Yunyan Yang and Xiaobao Zhu PDF
Proc. Amer. Math. Soc. 145 (2017), 3953-3959 Request permission

Abstract:

Let $(M,g)$ be a compact Riemannian surface without boundary, $W^{1,2}(M)$ be the usual Sobolev space, and $J: W^{1,2}(M)\rightarrow \mathbb {R}$ be the functional defined by \[ J(u)=\frac {1}{2}\int _M|\nabla u|^2dv_g+8\pi \int _M udv_g-8\pi \log \int _Mhe^udv_g,\] where $h$ is a positive smooth function on $M$. In an inspiring work (Asian J. Math., vol. 1, pp. 230-248, 1997), Ding, Jost, Li and Wang obtained a sufficient condition under which $J$ achieves its minimum. In this note, we prove that if the smooth function $h$ satisfies $h\geq 0$ and $h\not \equiv 0$, then the above result still holds. Our method is to exclude blow-up points on the zero set of $h$.
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Additional Information
  • Yunyan Yang
  • Affiliation: Department of Mathematics, Renmin University of China, Beijing 100872, People’s Republic of China
  • Email: yunyanyang@ruc.edu.cn
  • Xiaobao Zhu
  • Affiliation: Department of Mathematics, Renmin University of China, Beijing 100872, People’s Republic of China
  • MR Author ID: 923728
  • Email: zhuxiaobao@ruc.edu.cn
  • Received by editor(s): January 17, 2016
  • Received by editor(s) in revised form: September 30, 2016
  • Published electronically: March 27, 2017
  • Communicated by: Nimish Shah
  • © Copyright 2017 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 145 (2017), 3953-3959
  • MSC (2010): Primary 46E35
  • DOI: https://doi.org/10.1090/proc/13515
  • MathSciNet review: 3665046