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Calculus of Variations: A Differential Form Approach

Swarnendu Sil

TL;DR

This work develops a unified differential-form framework for calculus of variations by studying functionals $\int_{\Omega} f\big(d\omega_1, \dots, d\omega_m\big)$ with $\omega_i$ a $(k_i-1)$-form. It introduces vectorially ext. one convexity, vectorially ext. quasiconvexity, and vectorially ext. polyconvexity, proving links to weak lower semicontinuity and weak continuity via growth conditions, decomposition lemmas, and distributional wedge products. The paper provides necessary and sufficient conditions for lower semicontinuity, shows both positive results (existence theorems for quasiconvex and polyconvex energies) and fundamental limitations (gauge invariance leading to nonexistence in general when $\omega$-dependence is present), and develops a robust wedge-product calculus in the exterior-algebra setting. It thereby generalizes classical vectorial calculus of variations and single-form theories, clarifying the role of gauge invariance and offering a geometric, exterior-algebra perspective on minimization, continuity, and convergence questions in nonlinear analysis. The results have potential implications for nonlinear elasticity, compensated compactness, and gauge-field-type problems where differential forms and exterior calculus naturally arise.

Abstract

We study integrals of the form $\int_Ωf\left( dω_1 , \ldots , dω_m \right), $ where $m \geq 1$ is a given integer, $1 \leq k_{i} \leq n$ are integers and $ω_{i}$ is a $(k_{i}-1)$-form for all $1 \leq i \leq m$ and $ f:\prod_{i=1}^m Λ^{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.

Calculus of Variations: A Differential Form Approach

TL;DR

This work develops a unified differential-form framework for calculus of variations by studying functionals with a -form. It introduces vectorially ext. one convexity, vectorially ext. quasiconvexity, and vectorially ext. polyconvexity, proving links to weak lower semicontinuity and weak continuity via growth conditions, decomposition lemmas, and distributional wedge products. The paper provides necessary and sufficient conditions for lower semicontinuity, shows both positive results (existence theorems for quasiconvex and polyconvex energies) and fundamental limitations (gauge invariance leading to nonexistence in general when -dependence is present), and develops a robust wedge-product calculus in the exterior-algebra setting. It thereby generalizes classical vectorial calculus of variations and single-form theories, clarifying the role of gauge invariance and offering a geometric, exterior-algebra perspective on minimization, continuity, and convergence questions in nonlinear analysis. The results have potential implications for nonlinear elasticity, compensated compactness, and gauge-field-type problems where differential forms and exterior calculus naturally arise.

Abstract

We study integrals of the form where is a given integer, are integers and is a -form for all and 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.

Paper Structure

This paper contains 22 sections, 22 theorems, 154 equations.

Table of Contents

  1. Introduction
  2. Notations
  3. Notions of Convexity
  4. Definitions
  5. Basic Properties
  6. The quasiaffine case
  7. Weak lower semicontinuity
  8. Necessary condition
  9. Lower semicontinuity for quasiconvex functions without lower order terms
  10. Lower semicontinuity for general quasiconvex functions
  11. Failure of semicontinuity in $W^{d,p}$ for general functional
  12. Semicontinuity in $W^{1,\boldsymbol{p}}$ for general functional
  13. Weak Continuity
  14. Necessary condition
  15. Weak continuity of wedge products
  16. ...and 7 more sections

Key Result

Theorem 3.6

Let $f:\boldsymbol{\Lambda^k}\rightarrow\mathbb{R}.$ Then Moreover if $f:\boldsymbol{\Lambda^{k}}\left( \mathbb{R}^{n}\right) \rightarrow \mathbb{R}$ is vectorially ext. one convex, then $f$ is locally Lipschitz.

Theorems & Definitions (54)

  • Definition 3.1
  • Remark 3.2
  • Remark 3.3
  • Remark 3.4
  • Definition 3.5
  • Theorem 3.6
  • Proposition 3.7
  • Remark 3.8
  • Theorem 3.9
  • Remark 3.10
  • ...and 44 more