An overdetermined problem for a weighted Poisson's equation. (1st May 2018)

• Record Type: Journal Article
• Authors: Ruan, Qi-hua; Wang, Weihua; Huang, Qin
• Is Part Of: Computers & mathematics with applications. Volume 75:issue 9(2018)
• Page Start: 3139
• Page End: 3146
• Publication Date: 2018-05-01
• Subjects: Overdetermined problem -- Weighted Laplacian -- Ball -- Mean value properties; Electronic data processing -- Periodicals; Mathematics -- Data processing -- Periodicals; 510.28541
• DOI: 10.1016/j.camwa.2018.01.036
• Languages: English
• ISSNs: 0898-1221

Abstract

Abstract: In this paper, we discuss an overdetermined problem for a weighted Poisson's equation. We prove that if there exists a solution of the weighted Poisson's equation △ u + ∇ log w ⋅ ∇ u = − 1 on a smooth bounded domain Ω with both Dirichlet and Neumann constant boundary condition, and the weight function w satisfies some conditions in Ω, then Ω is a ball. We also study some applications of the overdetermined problems and some overdetermined problems with nonconstant Neumann boundary condition.