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 …).