Calculus of Variations with Differential Forms
Saugata Bandyopadhyay, Bernard Dacorogna, Swarnendu Sil
TL;DR
The paper extends the calculus of variations to integrals of the form $\int_{\Omega} f(d\omega)$ for differential forms, introducing exterior analogs of convexity—$ext.$-one convexity, $ext.$-quasiconvexity, and $ext.$-polyconvexity—and detailing their interrelations. It develops the quasiaffine case, proving equivalences among ext. polyaffine, ext. quasiaffine, and ext. one affine, with a canonical representation $f(\xi)=\sum_{s=0}^{\left[ n/k\right]} \langle c_s; \xi^{s}\rangle$; special simplifications occur when $k$ is odd or $2k>n$. The quadratic-case analysis yields precise criteria for ext. polyconvexity, ext. quasiconvexity, and ext. one convexity, including Sverak-type counterexamples that separate these notions in various regimes. An application to minimization shows existence of minimizers under ext. quasiconvexity with $p$-growth, and illustrates how non-quasiconvexity affects solvability, situating the results within compensated compactness. Overall, the work provides a coordinate-free, algebraic framework for variational problems involving differential forms and connects to differential inclusions and wave-cone techniques.
Abstract
We study integrals of the form $\int_Ωf\left( dω\right)$, where $1\leq k\leq n$, $f:Λ^{k}\rightarrow\mathbb{R}$ is continuous and $ω$ 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.