Calculus of variations with differential forms

Abstract

We study integrals of the form ${\int }_{\Omega }f\left(d\omega \right)$, where $1\le k\le n$, $f:{\Lambda }^k\to \mathbb{R}$ is continuous and $\omega$ is a $\left(k-1\right)$-form. We introduce the appropriate notions of convexity, namely ext. one convexity, ext. quasiconvexity and ext. polyconvexity. We study their relations, give several examples and counterexamples. We finally conclude with an application to a minimization problem.