A $\mathcal{C}$-Spectral Sequence Associated with Free Boundary Variational Problems

Abstract

The $\mathcal{C}$–spectral sequence is a cohomological theory naturally associated with a space of infinite jets, which allows to write down many concepts of the variational calculus by using the same logic of the standard differential calculus. In this paper we use such a language (called Secondary Calculus by A. Vinogradov) to describe a delicate aspect of the variational calculus: the appearance of some “natural” boundary conditions in the context of variational problems with free boundary (e.g., transversality conditions). We discover that the Euler–Lagrange operator is actually a graded operator, producing simultaneously the standard Euler–Lagrange equations and these new boundary conditions as different homogeneous components of an unique object. Simple applicative examples will be presented.