Timeline for Weak (Sobolev) derivative and the Frechet derivative (chain rule) [closed]
Current License: CC BY-SA 3.0
13 events
| when toggle format | what | by | license | comment | |
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| Feb 22, 2015 at 18:32 | comment | added | Deane Yang | $w = Af(u)$ satisfies a PDE boundary value problem in terms of $f$, its derivatives, $u$, and its derivatives. So does its gradient. You have to work with that to infer whatever you need, including linearization. | |
| Feb 22, 2015 at 16:21 | comment | added | MainW | @DeaneYang I see. $A(v)$ in my case is the harmonic extension with data $v$ in some domain. Sadly the nonlinearity $f$ does not give me a good formula. | |
| Feb 22, 2015 at 10:35 | history | closed |
Deane Yang Stefan Kohl♦ Alex Degtyarev Joonas Ilmavirta Daniel Moskovich |
Needs details or clarity | |
| Feb 21, 2015 at 14:46 | review | Close votes | |||
| Feb 22, 2015 at 10:35 | |||||
| Feb 21, 2015 at 14:28 | comment | added | Deane Yang | If $A$ is the inverse to a PDO, then you can use implicit differentiation to extract some information about the gradient. | |
| Feb 21, 2015 at 14:27 | comment | added | Deane Yang | There is no simple formula. You need to use whatever you know about $A$. You can see this by defining it to be "multiply by a fixed non-constant function" or more generally a linear PDO with variable coefficients. | |
| Feb 21, 2015 at 10:36 | history | edited | MainW | CC BY-SA 3.0 |
added 12 characters in body
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| Feb 21, 2015 at 10:31 | comment | added | MainW | @AlexDegtyarev For example, let $g = A \circ f$. Then, although it is wrong, one could think of $\nabla g(u)$ as $g'(u)\nabla u$ (this obviously does not make sense since $ u \notin H^1$ but this is the kind of thing I wanted to think about. | |
| Feb 21, 2015 at 10:29 | history | edited | MainW | CC BY-SA 3.0 |
added 12 characters in body
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| Feb 21, 2015 at 10:29 | comment | added | MainW | @AlexDegtyarev I was hoping to rewrite it so as to have a gradient term which is linear in $u$, but I'd be happy for whatever the right expression should be. | |
| Feb 21, 2015 at 10:17 | comment | added | Alex Degtyarev | I don't quite understand how you are planning to interchange $A$ and $f$? This is not just the chain rule anymore. | |
| Feb 21, 2015 at 9:59 | review | First posts | |||
| Feb 21, 2015 at 10:17 | |||||
| Feb 21, 2015 at 9:58 | history | asked | MainW | CC BY-SA 3.0 |