5 - Elliptic Operators

Summary

Introduction

This chapter deals with the distribution of eigenvalues of degenerate elliptic operators in domains and on Rⁿ. It is based on the results of the previous chapters and demonstrates the symbiotic relationship between the diverse ingredients treated so far:

1. (i) spectral theory in quasi-Banach spaces, especially the connection between entropy numbers and eigenvalues obtained in 1.3.4;

2. (ii) some new results in the theory of function spaces, especially the assertions about Hölder inequalities in 2.4;

3. (iii) sharp estimates of the behaviour of entropy numbers of compact embeddings between function spaces on bounded domains obtained in Chapter 3;

4. (iv) corresponding assertions for weighted spaces on Rⁿ described in Chapter 4.

The combination of these ingredients is the basis for the study of the distribution of eigenvalues of degenerate elliptic operators. In 5.2 we concentrate on elliptic operators in bounded smooth domains in nonlimiting situations. As a by-product we obtain some results, based on the Birman–Schwinger principle, about the problem of the “negative spectrum” of self-adjoint operators. But we shall be very brief here and defer a detailed study of this topic until 5.4, when we deal with corresponding problems on Rⁿ, which are more natural for problems of the “negative spectrum”. In 5.3 we complement the results of 5.2 by the study of limiting situations, again on bounded smooth domains. Finally, 5.4 deals with corresponding problems on Rⁿ, including a more detailed study of the “negative spectrum” of some self-adjoint elliptic operators in L₂(Rⁿ).