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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Partially overdetermined problem in some integral equations


Authors: Boqiang Lv, Fengquan Li and Weilin Zou
Journal: Proc. Amer. Math. Soc. 141 (2013), 3073-3081
MSC (2010): Primary 45K05, 45P05; Secondary 35J67
Published electronically: June 7, 2013
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Abstract: In this paper, we consider the partially overdetermined problem in integral equations as follows:

\begin{align*}\begin {cases}u(x)=A \int _{\Omega }\frac {1}{\vert x-y\vert^{n-\a... ...mega , \\ u=C,~~&x\in \Gamma \subseteq \partial \Omega , \end{cases}\end{align*}    

where $ 0<\alpha <n, ~p>\frac {n}{n-\alpha }, ~A,~ B,~ C $ are positive constants, $ \Omega \subset R^n\ (n\geq 2)$ is a bounded domain with $ \partial \Omega \in C^1$, and $ \Gamma $ is a proper open set of $ \partial \Omega $. Under some assumptions on the geometry of $ \Gamma $, we prove that $ \Omega $ must be a ball and $ u$ is radially symmetric and monotone decreasing with respect to the radius.

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Additional Information

Boqiang Lv
Affiliation: College of Mathematics and Information Science, Nanchang Hangkong University, Nanchang 330063, People’s Republic of China
Email: lbq86@yahoo.com.cn

Fengquan Li
Affiliation: School of Mathematical Sciences, Dalian University of Technology, Dalian 116024, People’s Republic of China
Email: fqli@dlut.edu.cn

Weilin Zou
Affiliation: College of Mathematics and Information Science, Nanchang Hangkong University, Nanchang 330063, People’s Republic of China
Email: zwl267@yahoo.com.cn

DOI: http://dx.doi.org/10.1090/S0002-9939-2013-12192-1
PII: S 0002-9939(2013)12192-1
Keywords: Integral equation, moving planes in integral forms, symmetry of domain and solution, partially overdetermined problem
Received by editor(s): August 29, 2011
Published electronically: June 7, 2013
Communicated by: James E. Colliander
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.