Inequalities for the minimal eigenvalue of the Laplacian in an annulus

Abstract

We discuss the behavior of the minimal eigenvalue λ of the Dirichlet Laplacian in the domain D1\D2 := D (an annulus) where D1 is a circular disc and D2 ⊂ D1 is a smaller circular disc. It is conjectured that the minimal eigenvalue λ has a maximum value when D2 is a concentric disc. If h is a displacement of the center of the disc D2 and λ(h) is the corresponding minimal eigenvalue, then dλ(h)/dh < 0 so that λ(h) is minimal when ∂D2 touches ∂D1, where ∂D is the boundary of D. Numerical results are given to back the conjecture. Upper and lower bounds are given for λ(h).