Introduction

The theory of pseudo-differential operators (abbreviated ψDOp) shows the essential behavior of many operators appears most clearly when applied to rapidly oscillating functions, studying the behavior as the frequency tends to infinity. This provides a more flexible and transparent approach to our “finite-rank” problems than the method of integral operators of Chapter 7. It is also often simpler, for simple problems — but for this reason, we are encouraged to attack harder problems. (If you are not familiar with ψDOp, don't worry; our argument is direct. Here, the intention is only to show why we don't cite results from this theory.)

We have a ψDOp which, by hypothesis — a hypothesis we hope to contradict — has finite rank. A ψDOp of finite rank is trivial: its symbol is identically zero. So all we have to do is compute the symbol. But it must be computed in some detail. Often the (apparent) principal and subprincipal symbols both vanish identically and uselessly — we must go to the third stage, sometimes further. This is not surprising since we start with second order operators, so the coefficient of the zero-order part only begins to play a role at the third stage. We also deal with quite degenerate problems; after all, we expect to find a contradiction. The theory of ψDOp serves only as inspiration, since it ordinarily computes explicitly only the principal symbol.

One of the compensations for the difficulty of PDEs, compared to ordinary differential equations, is that the regions in which we work can have almost any shape. This may seem only an added difficulty, but it turns into an advantage if we restrict attention to properties which are generic with respect to perturbation of the boundary. This is quite a strong condition, as the class of perturbations is infinite-dimensional (but it is finite-dimensional for ODEs). In many problems it is also reasonable to require “genericity” with respect to certain coefficients, etc. (See Uhlenbeck [42] and Saut and Teman [34] for some results of this kind.) But some problems are very rigid in this regard (e.g., the Navier-Stokes equation). Perturbation of the boundary is almost always reasonable, if we allow for occasional symmetry constraints as in Example 6.2 below. We will in any event perturb only the boundary — as a way of testing the strength of our tools, as a first step in more general “genericity” arguments, and for the intrinsic interest of the problems.

In this chapter, “generic” properties are those that hold for most domains Ω, in the sense of Baire category, or for most domains in a certain stated class C (Cm-regular, bounded, perhaps connected, perhaps contained in a certain open set or the boundary is contained in a certain open set, or the boundary meets a certain open set, or …).

The transversality theorem (or transversal density theorem) of Thom and Abraham is a tool useful in many branches of geometry and analysis. The usual form [1, 32, 35, 42] makes unnecessarily strong hypotheses for the case of negative Fredholm index. We will prove some generalizations, aimed at infinite-dimensional applications, with special attention to the case of negative index. (We will see many examples with index —∞ in Ch. 6).

Our terminology is that of Lang [17] and Abraham and Robbin [1] except for “codimension,” defined in 5.15, and “σ-proper,” defined in 5.4(3). See 5.13–14 for semi-Fredholm operators.

This chapter is logically self-contained except for Sard's theorem. We will use only Theorem 5.4 in the sequel, but other variations on the theme of transversality seemed of sufficient interest to be worth recording.

Theorem 5.1. Sard's Theorem (Brown-Morse-Sard). Let A ⊂ ℝnbe an open set and f : A → ℝma Ckmap, where k is a positive integer and k > n — m. Then the set of critical values of f has measure zero in ℝmand is meager (= first Baire category).

Proof. See [1] or [37].

Recall that x is a regular point of ƒ if the derivative ƒ′(x) is surjective; otherwise x is a critical point.