Formal Divergence Equations

Abstract

In this chapter, we introduce formal divergence equations on Euclidean spaces and study their basic properties. These partial differential equations naturally appear in the variational geometry on fibered manifolds, but also have a broader meaning related to differential equations, conservation laws, and integration of forms on manifolds with boundary. A formal divergence equation is not always integrable; we show that the obstructions are connected with the Euler–Lagrange expressions known from the higher-order variational theory of multiple integrals. If a solution exists, then it defines a solution of the associated “ordinary” divergence equation along any section of the underlying fibered manifold. The notable fact is that the solutions of formal divergence equations of order r are in one–one correspondence with a class of differential forms on the $\boldsymbol{(r - 1)}$-st jet prolongation of the underlying fibered manifold, defined by the exterior derivative operator.