On Two Functionals Connected to the Laplacian in a Class of Doubly Connected Domains in Space-Forms

Proceedings Mathematical Sciences

A R Aithal¹  M H C Anisa² 

[1] $$;Department of Mathematics, University of Mumbai, Mumbai 00 0, India$$

关键词: Eigenvalue problem;    Laplacian;    maximum principles.;   

DOI  :  

学科分类：数学（综合）

来源: Indian Academy of Sciences

【 摘 要 】

Let $B_1$ be a ball of radius $r_1$ in $S^n(mathbb{H}^n)$, and let $B_0$ be a smaller ball of radius $r_0$ such that $overline{B_0} subset B_1$. For $S^n$ we consider $r_1 < 𝜋$. Let 𝑢 be a solution of the problem $-𝛥 u = 1$ in $𝛺: = B_1 overline{B_0}$ vanishing on the boundary. It is shown that the associated functional 𝐽(𝛺) is minimal if and only if the balls are concentric. It is also shown that the first Dirichlet eigenvalue of the Laplacian on 𝛺 is maximal if and only if the balls are concentric.

【 授权许可】

Unknown