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. 1. 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 Delta u = 0 in B-1 \ (B) over bar (0), u = 0 on partial derivative B-0, partial derivative u/partial derivative v = tau u on partial derivative B-1, attains its maximum if and only if B-0 and B-1 are concentric. Here nu is the outward unit normal on. partial derivative B-1 and tau is a real number. 2. Let B-0 subset of M be a geodesic ball of radius r centered at a point p is an element of M, where M denote either a non-compact rank-1 symmetric space (M, ds(2)) with curvature -4 <= K-M <= -1 or M = R-m. Let D subset of M be a domain of fixed volume which is geodesically symmetric with respect to the point p is an element of M such that (B-0) over bar. D. Then the first non-zero eigenvalue of Delta u = mu u in D \ (B) over bar (0)), partial derivative u/partial derivative v = 0 on partial derivative (D \ (N) over bar (0))(,) attains its maximum if and only if D is a geodesic ball centered at p. Here nu represents the outward unit normal on partial derivative(D \ (B) over bar (0)) and mu is a real number.