Introduction

In Chapter 3, we developed the spectral theory of elliptic operators L on smooth bounded domains D, concentrating in particular on the principal eigenvalue λ0. This theory hinged on the fact that L possessed a compact resolvent, since only then could the Fredholm theory be applied. In the case of an arbitrary domain D, L does not possess a compact resolvent in general, and the above spectral theory breaks down. Indeed, it is no longer even clear on what space to define L or on what space to define the corresponding semigroup Tt. In this chapter, we develop a generalized spectral theory for elliptic operators L on arbitrary domains. Specifically, we will extend the definition of the principal eigenvalue λ0 via the existence or non-existence of positive harmonic functions for L – λ in D, that is, functions u satisfying (L – λ)u = 0 and u > 0 in D.

In this chapter, we will assume that L satisfies Assumption locally:

Assumptionis defined on a domain D ⊆ Rdand satisfies Assumption (defined in Chapter 3, Section 7) on every subdomain D' ⊂⊂ D.

(In Exercise 4.16, the reader is asked to check that all the theorems in this chapter which involve only L and not hold if L satisfies Assumption H (defined in Chapter 3, Section 2) locally.)

An indispensable tool in this chapter is Harnack's inequality, which we state here for operators in non-divergence form since the formulation is simpler in this case. See the notes at the end of the chapter for references.