Domain-independent upper bounds for eigenvalues of elliptic operators

Abstract

Let Ω ⊆ R m \Omega \subseteq {\mathbb {R}^m} be a bounded open set, ∂ Ω \partial \Omega its boundary and Δ \Delta the Laplacian on R m {\mathbb {R}^m} . Consider the elliptic differential equation: (1) \[ − Δ u = λ u in Ω ; u = 0 on ∂ Ω . - \Delta u = \lambda u\quad {\text {in}}\;\Omega ;\qquad u = 0\quad {\text {on}}\;\partial \Omega . \] It is known that the eigenvalues, λ i {\lambda _i} , of (1) satisfy (2) \[ ∑ i = 1 n λ i λ n + 1 − λ i ⩾ m n 4 \sum \limits _{i = 1}^n {\frac {{{\lambda _i}}} {{{\lambda _{n + 1}} - {\lambda _i}}}} \geqslant \frac {{mn}} {4} \] provided that λ n + 1 > λ n {\lambda _{n + 1}} > {\lambda _n} . In this paper we abstract the method used by Hile and Protter [2] to establish (2) and apply the method to a variety of second-order elliptic problems, in particular, to all constant coefficient problems. We then consider a variety of higher-order problems and establish an extension of (2) for problem (1) where the Laplacian is replaced by a more general operator in a Hilbert space.