# Calculus of variations with differential forms

### Saugata Bandyopadhyay

IISER Kolkata, Mohanpur, India### Bernard Dacorogna

Ecole Polytechnique Fédérale de Lausanne, Switzerland### Swarnendu Sil

Ecole Polytechnique Fédérale de Lausanne, Switzerland

## 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.

## Cite this article

Saugata Bandyopadhyay, Bernard Dacorogna, Swarnendu Sil, Calculus of variations with differential forms. J. Eur. Math. Soc. 17 (2015), no. 4, pp. 1009–1039

DOI 10.4171/JEMS/525