Timeline for Underdetermined system of linear PDEs
Current License: CC BY-SA 4.0
10 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Nov 11, 2021 at 15:46 | vote | accept | Matteo Raffaelli | ||
| Jan 6, 2019 at 9:45 | comment | added | Matteo Raffaelli | Dear Robert, thank you for your message. Your answer is definitely helpful, but I am particularly interested in the case where the functions $a$ and $b$ are linearly independent, and no further assumption is made. Could you add something about such case? | |
| Jan 5, 2019 at 22:47 | comment | added | Robert Bryant | @MK7: Don't worry about that; sometimes, when you are in unfamiliar territory, even familiar facts can seem mysterious. I was wondering whether you had any other questions. Since you haven't accepted my answer, I'm assuming that either there is still something more you'd like to know or that this answer is not helpful for what you were trying to do. I'm happy to think more about it, but it would be good to know what the issues are. | |
| Dec 21, 2018 at 15:23 | history | edited | Robert Bryant | CC BY-SA 4.0 |
Expanded some remarks on finding explicit solutions in the generic case.
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| Dec 15, 2018 at 21:48 | comment | added | Matteo Raffaelli | Of course, sorry for the trivial question | |
| Dec 14, 2018 at 22:37 | comment | added | Robert Bryant | @MK7: Given 5 vectors spanning a 4-dimensional vector space, there has to be a linear relation between them, and that linear relation is unique up to a nonzero multiple. (I'm not sure what you are asking if it's not this.) That's all that I'm using to get the existence of the scalar functions $p$, $q$, $r$, $f$ and $g$ on $I^2$ that do not simultaneously vanish. The functions $p$, $q$, and $r$ cannot simultaneously vanish either because, at such a point in $I^2$, we'd then have a linear relation between $a$ and $b. | |
| Dec 14, 2018 at 16:13 | comment | added | Matteo Raffaelli | Also, I am not sure I understand how equation 4 follows from the assumption that the 5 vector-valued functions span $\mathbb{R}^{4}$ everywhere. | |
| Dec 14, 2018 at 16:11 | comment | added | Matteo Raffaelli | Thanks a lot for the help. I have now included in the question the assumption that $a$ and $b$ are always linearly independent. Could you add something about that specific case in the answer? | |
| Dec 13, 2018 at 11:06 | history | edited | Robert Bryant | CC BY-SA 4.0 |
Added some remarks about the 'auxilliary conditions'.
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| Dec 13, 2018 at 8:43 | history | answered | Robert Bryant | CC BY-SA 4.0 |