Calculus of Variations: A Differential Form Approach

TL;DR

This paper develops a differential form approach to the calculus of variations, introducing new convexity notions, proving lower semicontinuity and continuity theorems, and applying these to minimization problems in a generalized setting.

Contribution

It introduces vectorial exterior convexity concepts and extends classical calculus of variations results to differential forms.

Findings

Established weak lower semicontinuity theorems for form integrals.

Proved weak continuity theorems in the context of differential forms.

Generalized classical variational results to a broader differential form framework.

Abstract

We study integrals of the form $\int_{\Omega}f\left( d\omega_1 , \ldots , d\omega_m \right), $ where $m \geq 1$ is a given integer, $1 \leq k_{i} \leq n$ are integers and $\omega_{i}$ is a $(k_{i}-1)$-form for all $1 \leq i \leq m$ and $ f:\prod_{i=1}^m \Lambda^{k_i}\left( \mathbb{R}^{n}\right) \rightarrow\mathbb{R}$ is a continuous function. We introduce the appropriate notions of convexity, namely vectorial ext. one convexity, vectorial ext. quasiconvexity and vectorial ext. polyconvexity. We prove weak lower semicontinuity theorems and weak continuity theorems and conclude with applications to minimization problems. These results generalize the corresponding results in both classical vectorial calculus of variations and the calculus of variations for a single differential form.