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Notions of affinity in calculus of variations with differential forms

Saugata Bandyopadhyay, Swarnendu Sil

Abstract

Ext-int.\ one affine functions are functions affine in the direction of one-divisible exterior forms, with respect to exterior product in one variable and with respect to interior product in the other. The purpose of this article is to prove a characterization theorem for this class of functions, which plays an important role in the calculus of variations for differential forms.

Notions of affinity in calculus of variations with differential forms

Abstract

Ext-int.\ one affine functions are functions affine in the direction of one-divisible exterior forms, with respect to exterior product in one variable and with respect to interior product in the other. The purpose of this article is to prove a characterization theorem for this class of functions, which plays an important role in the calculus of variations for differential forms.

Paper Structure

This paper contains 5 sections, 11 theorems, 70 equations.

Table of Contents

  1. Introduction
  2. Notations
  3. Notions of exterior convexity
  4. Some algebraic lemmas
  5. Characterization of ext-int. one affine functions

Key Result

Theorem 1

Let $1\leqslant k\leqslant n-1$. Then, $f:\Lambda^{k+1}\times \Lambda^{k-1}\rightarrow\mathbb{R}$ is ext-int. one affine if and only if there exist $c_{s}\in\Lambda^{(k+1)s}$ and $d_{r}\in\Lambda^{(n-k+1)r},$ for all $0\leqslant s\leqslant\left[ \frac{n}{k+1}\right]$, $0 \leqslant r\leqslant\left[ \

Theorems & Definitions (22)

  • Theorem 1
  • Proposition 2
  • Definition 3
  • Definition 4
  • Definition 5: Hodge transform
  • Remark 6
  • Lemma 7
  • Remark 8
  • Lemma 9: Decomposition lemma
  • Remark 10
  • ...and 12 more