On two functionals connected to the Laplacian in a class of doubly connected domains in space-forms

2005/03/05 by M H C Anisa, A R Aithal
Mathematics · #math.AP #math.DG #msc:35J25 #msc:35P15 #msc:53C21 #msc:58J32 #msc:58J50

paper · pdf

published as Proc. Indian Acad. Sci. (Math. Sci.), Vol. 115, No. 1, February 2005, pp. 93-102 · 10 pages

arxiv created 2005/03/05 · arxiv updated 2009/12/01

Abstract

Let B₁ be a ball of radius r₁ in Sⁿ(\Hyⁿ), and let B₀ be a smaller ball of radius r₀ such that B₀⊂ B₁. For Sⁿ we consider r₁< π. Let u be a solution of the problem -\La u =1 in \Om := B₁∖ B₀ vanishing on the boundary. It is shown that the associated functional J(\Om) is minimal if and only if the balls are concentric. It is also shown that the first Dirichlet eigenvalue of the Laplacian on \Om is maximal if and only if the balls are concentric.

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