Asymptotic Formulas with Remainder Estimates for Eingevalues of Elliptic Operators

Abstract

We propose to discuss in this lecture a number of results related to the problem of eigenvalue distribution of elliptic operators. We start with some classical results. Let Δ be the Laplacian in Rⁿ and consider the eigenvalue problem:

$$\begin{array}{*{20}c}{ - \Delta = \lambda {\text{u}}} & {{\text{in}}\,\,\,\Omega \,,}\\{{\text{u}} = 0} & {{\text{on}}\,\,\partial \Omega \,,}\\\end{array}$$

((1))

where Ω is a bounded open set in Rⁿ. Let {λ_j} be the sequence of eigenvalues of (1), each repeated according to its multiplicity and set

$$N\left( t \right) = \sum\limits_{\lambda _j < t} 1.$$

((2))