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May/June 2018 On a generalization of the Poincaré Lemma to equations of the type $dw+a\wedge w=f$
David Strütt
Differential Integral Equations 31(5/6): 353-374 (May/June 2018).

Abstract

We study the system of linear partial differential equations given by \[ dw+a\wedge w=f, \] on open subsets of $\mathbb R^n$, together with the algebraic equation \[ da\wedge u=\beta, \] where $a$ is a given $1$-form, $f$ is a given $(k+1)$-form, $\beta$ is a given $k+2$-form, $w$ and $u$ are unknown $k$-forms. We show that if $\text{rank}[da]\geq 2(k+1)$ those equations have at most one solution, if $\text{rank}[da] \equiv 2m \geq 2(k+2)$ they are equivalent with $\beta=df+a\wedge f$ and if $\text{rank}[da]\equiv 2 m\geq2(n-k)$ the first equation always admits a solution.

Moreover, the differential equation is closely linked to the Poincaré lemma. Nevertheless, as soon as $a$ is nonexact, the addition of the term $a\wedge w$ drastically changes the problem.

Citation

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David Strütt. "On a generalization of the Poincaré Lemma to equations of the type $dw+a\wedge w=f$." Differential Integral Equations 31 (5/6) 353 - 374, May/June 2018.

Information

Published: May/June 2018
First available in Project Euclid: 23 January 2018

zbMATH: 06861583
MathSciNet: MR3749213

Subjects:
Primary: 35F35, 58A10

Rights: Copyright © 2018 Khayyam Publishing, Inc.

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Vol.31 • No. 5/6 • May/June 2018
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