Linear Differential Operators

Abstract

In this chapter, we consider only linear systems of partial differential equations, and use the notation and terminology introduced in Chapter IX. In general, if D: e → ℱ is a linear differential operator, where E, F are vector bundles over the manifold X, and if f is a section of F, the inhomogeneous equation $$ Du = f $$ is not solvable for a section u of E unless f satisfies a requisite compatibility condition. Indeed, certain conditions must be imposed on the formal power series expansion j∞(f)(x) of f at x ∈ X in order that it may be written as j∞(Du)(x), for some section u of E. Under certain regularity assumptions on D, they can be expressed in terms of a differential operator P: ℱ → B of finite order, where B is a vector bundle over X. This operator is called the compatibility condition for D and is obtained by repeatedly differentiating the equation. We then obtain a complex of differential operators $$ \varepsilon \mathop{ \to }\limits^D F\mathop{ \to }\limits^P B $$ which is exact at the formal power series level: the formal power series expansion j∞(f)(x) of f at x can be written in the form j ∞ (Du)(x), for some section u of E, if and only if Pf vanishes to infinite order at x. For example, the inhomogeneous equation du — f, where u is a real-valued function and f is a 1-form on X, is not solvable for u unless df = 0.