This chapter concerns various geometric applications of Fourier series that either do not have higher dimensional analogues or serve as good illustrations for the methods used in the more complicated d-dimensional case. For a survey of the results discussed here see Groemer (1993c, chapter 2). Although most of these results are relatively old, some of the proofs have been modified to avoid smoothness assumptions quite often present (explicitly or implicitly) in the original literature.

A Proof of Hurwitz of the Isoperimetric Inequality

The aim of this section is to present a proof of the isoperimetric inequality (in E2) based on the ideas of the classical paper of Hurwitz (1901). It is remarkable that this proof can be arranged in such a way that no smoothness assumption and not even convexity are required.

We first discuss a few concepts and known results regarding curves in E2 that will be used here. A curve is defined as a continuous mapping of a closed interval [α, β] into E2 that is not constant on any subinterval of [α, β]. In this connection intervals are always assumed to have positive length. Any two such curves, say Γ1 and Γ2, are considered to be the same if Γ2 is obtained from Γ1 by an admissible change of parameter.

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).