Exterior convexity and classical calculus of variations
Saugata Bandyopadhyay, Swarnendu Sil
TL;DR
The article establishes a precise bridge between exterior convexity notions and classical calculus of variations by introducing a projection map $\pi$ from certain matrix spaces to exterior forms. The main result proves that ext. one convex, ext. quasiconvex, and ext. polyconvex properties of a function $f:\Lambda^{k}\to\mathbb{R}$ are equivalent to rank-one convexity, quasiconvexity, and polyconvexity of the composed function $f\circ\pi$, respectively. A key technical component is the construction of higher projections $\pi_s$ and an adjugate-based algebra that underpins a crucial lemma, enabling representation of exterior convexity in terms of classical convexity and vice versa. The work also provides simple corollaries and new proofs for results in BDS1, clarifying the algebraic structure linking exterior forms with matrix-based variational frameworks. Overall, the paper unifies exterior and classical calculi of variations, opening a two-way translational path between exterior forms and standard vectorial variational techniques, with implications for lower semicontinuity and quasiaffine characterizations in exterior spaces.
Abstract
We study the relation between various notions of exterior convexity introduced in Bandyopadhyay-Dacorogna-Sil \cite{BDS1} with the classical notions of rank one convexity, quasiconvexity and polyconvexity. To this end, we introduce a projection map, which generalizes the alternating projection for two-tensors in a new way and study the algebraic properties of this map. We conclude with a few simple consequences of this relation which yields new proofs for some of the results discussed in Bandyopadhyay-Dacorogna-Sil \cite{BDS1}.