For any bounded regular domain Ω of a real analytic Riemannian
manifold M, we denote by λk(Ω) the k-th eigenvalue of the
Dirichlet Laplacian of Ω. In this paper, we consider λk and as
a functional upon the set of domains of fixed volume in M. We introduce and
investigate a natural notion of critical domain for this functional. In
particular, we obtain necessary and sufficient conditions for a domain to be
critical, locally minimizing or locally maximizing for λk. These
results rely on Hadamard type variational formulae that we establish in this
general setting.Comment: To appear in Illinois J. Mat