Timeline for Existence of solution to linear inhomogeneous first order PDEs systems
Current License: CC BY-SA 4.0
8 events
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| Jan 4, 2023 at 21:22 | history | edited | Robert Bryant | CC BY-SA 4.0 |
Clarified the ranges of a few implied summations
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| Jan 4, 2023 at 17:26 | comment | added | A. J. Pan-Collantes | I have seen your edition. What a wonderful answer. I agree the general proof is more enlightening. Really, thank you very much for your effort. I have learned a lot reading and re-reading your answer and comments, together with your book. But I don't understand: why condition (1) implies that the rank of $\Sigma^*\omega$ is $n-r$? what do you mean by "involutive submanifold"? | |
| Jan 4, 2023 at 17:17 | vote | accept | A. J. Pan-Collantes | ||
| Jan 4, 2023 at 14:06 | history | edited | Robert Bryant | CC BY-SA 4.0 |
Added a proof of the fundamental existence result that is based on the Frobenius Theorem only.
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| Jan 2, 2023 at 13:00 | comment | added | Robert Bryant | .... The point is that involutivity implies that there are sections $\phi:V\to \Sigma\subset J^1(U,\mathbb{R})$ such that $\phi^*\omega=0$, which implies that $\phi=j^1(f)$ for some (unique) $f:V\to\mathbb{R}$. The Pfaff-Darboux theorem is being applied to the form $\Sigma^*(\omega)$, i.e., the pull-back of $\omega$ to $\Sigma$. Now, in your special case, because the equations are inhomogeneous linear, one can actually prove the result I described in the case that $D$ is involutive using only the Frobenius theorem, but the more general result is the more natural one, and it uses Pfaff-Darboux. | |
| Jan 2, 2023 at 12:53 | comment | added | Robert Bryant | @A.J.Pan-Collantes: What I had in mind is this: A system of $r$ first-order equations on a single unknown function $f:U\to\mathbb{R}$ can be thought of as defining a submanifold $\Sigma\subset J^1(U,\mathbb{R})$ of codimension $r$, and a local solution is a function $f:V\to\mathbb{R}$ for which the section $j^1(f):V\to J^1(U,\mathbb{R})$ has image in $\Sigma$. The submanifold $\Sigma$ is involutive if the Pfaff rank of the form $\Sigma^*\omega$ is $n{-}r$, where $\omega$ is the contact form on $J^1(U,\mathbb{R})$. The system $(1)$ is that your $\Sigma$ is involutive, so solutions exist.... | |
| Jan 2, 2023 at 9:13 | comment | added | A. J. Pan-Collantes | Thank you very much for your extensive answer. Unfortunately I don't understand how do you use the Pfaff-Darboux theorem. I have read about it this paper of you and in your book Exterior differential systems, but it seems it is applied to 1-forms. So in the context of my question only could be applied in case $r=n-1$. Probably I am missing something... | |
| Jan 1, 2023 at 13:26 | history | answered | Robert Bryant | CC BY-SA 4.0 |