Calculus of Variations with Differential Forms

TL;DR

This paper explores the calculus of variations involving differential forms, introducing new convexity notions, analyzing their relationships, and applying these concepts to a minimization problem.

Contribution

It introduces and studies new convexity concepts for integrals of differential forms, expanding the theoretical framework of the calculus of variations with forms.

Findings

Defined ext. one convexity, ext. quasiconvexity, and ext. polyconvexity.

Analyzed relationships and provided examples and counterexamples.

Applied the concepts to a minimization problem.

Abstract

We study integrals of the form $\int_{\Omega}f\left( d\omega\right)$, where $1\leq k\leq n$, $f:\Lambda^{k}\rightarrow\mathbb{R}$ 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.