On eigenvalue problems related to the laplacian in a class of doubly connected domains

We study eigenvalue problems in some specific class of doubly connected domains. In particular, we prove the following. Let B 1 be an open ball in R n , n > 2 and B 0 be an open ball contained in B 1 . Then the first eigenvalue of the problem Δ u = 0 in B 1 \ B ¯ 0 , u = 0 on ∂ B 0 , ∂ u ∂ ν = τ u on ∂ B 1 , attains its maximum if and only if B 0 and B 1 are concentric. Here ν is the outward unit normal on ∂ B 1 and τ is a real number. Let B 0 ⊂ M be a geodesic ball of radius r centered at a point p ∈ M , where M denote either a non-compact rank-1 symmetric space ( M , d s 2 ) with curvature - 4 ≤ K M ≤ - 1 or M = R m . Let D ⊂ M be a domain of fixed volume which is geodesically symmetric with respect to the point p ∈ M such that B 0 ¯ ⊂ D . Then the first non-zero eigenvalue of Δ u = μ u in D \ B ¯ 0 , ∂ u ∂ ν = 0 on ∂ ( D \ B ¯ 0 ) , attains its maximum if and only if D is a geodesic ball centered at p . Here ν represents the outward unit normal on ∂ ( D \ B ¯ 0 ) and μ is a real number.