Differential and Integral Equations

On a generalization of the Poincaré Lemma to equations of the type $dw+a\wedge w=f$

David Strütt

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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.

Article information

Source
Differential Integral Equations Volume 31, Number 5/6 (2018), 353-374.

Dates
First available in Project Euclid: 23 January 2018

Permanent link to this document
https://projecteuclid.org/euclid.die/1516676430

Subjects
Primary: 35F35: Linear first-order systems 58A10: Differential forms

Citation

Strütt, David. On a generalization of the Poincaré Lemma to equations of the type $dw+a\wedge w=f$. Differential Integral Equations 31 (2018), no. 5/6, 353--374. https://projecteuclid.org/euclid.die/1516676430


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