Domain-independent upper bounds for eigenvalues of elliptic operators

Abstract:

Let $\Omega \subseteq {\mathbb {R}^m}$ be a bounded open set, $\partial \Omega$ its boundary and $\Delta$ the Laplacian on ${\mathbb {R}^m}$. Consider the elliptic differential equation: (1) \[ - \Delta u = \lambda u\quad {\text {in}}\;\Omega ;\qquad u = 0\quad {\text {on}}\;\partial \Omega .\] It is known that the eigenvalues, ${\lambda _i}$, of (1) satisfy (2) \[ \sum \limits _{i = 1}^n {\frac {{{\lambda _i}}} {{{\lambda _{n + 1}} - {\lambda _i}}}} \geqslant \frac {{mn}} {4}\] provided that ${\lambda _{n + 1}} > {\lambda _n}$. 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.