Linear Differential Equations

Abstract

Linear differential equations are simpler in many respects. The truth of this statement is already obvious from the fact that their solution spaces possess the structure of a vector space. Thus it is not surprising that some of our previous results may be improved in this special case. In the first section we study how the linearity can be expressed within our geometric framework for differential equations. This topic includes in particular a geometric formulation of the linearisation of an arbitrary equation along one of its solutions.

In the discussion of the Cartan–Kähler Theorem in Section 9.4 we emphasised that the uniqueness statement holds only within the category of analytic functions; it is possible that further solutions with lower regularity exist. In the case of linear systems stronger statements hold. In Section 10.2 we will use our proof of the Cartan–Kähler Theorem to extend the classical Holmgren Theorem on the uniqueness of $\mathcal{C}^1$ solutions from normal equations to arbitrary involutive ones.

A fundamental topic in the theory of partial differential equations is the classification into elliptic and hyperbolic equations. We will study the notion of ellipticity in Section 10.3 for arbitrary involutive equations. One of our main results will be that the approach to ellipticity via weights usually found in the literature is not only insufficient but also unnecessary, if one restricts to involutive equations: if a system is elliptic with weights, then its involutive completion is also elliptic without weights; the converse is not true.