Timeline for Minimisation in dual Sobolev space
Current License: CC BY-SA 4.0
10 events
| when toggle format | what | by | license | comment | |
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| May 21, 2021 at 10:52 | history | edited | Daniele Tampieri | CC BY-SA 4.0 |
Minor Math Jaxing (used $\|\cdot\|$ instead of $||\cdot||$)
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| Mar 17, 2021 at 15:29 | comment | added | Willie Wong | (BTW, this is true in general in Hilbert spaces. Critical points of $\| f(\lambda)\|_H$ in some real HIlbert space satisfies $\langle f(\lambda), \partial_\lambda f(\lambda)\rangle_H = 0$.) | |
| Mar 17, 2021 at 15:24 | comment | added | Willie Wong | So this can be solved explicitly as $\lambda = - \langle \Delta u, u^p \rangle_{H^{-1}} / \langle u^p, u^p\rangle_{H^{-1}}$. | |
| Mar 17, 2021 at 15:23 | comment | added | Willie Wong | In your specific case [Edit 2]: I am assuming $u$ is a fixed function? assume both $\Delta u$ and $u^p$ are in $H^{-1}$. (The former is, the latter probably only is true for some range of $p$.) You are asking to minimize the norm along a line inside a HIlbert space. This happens when with respect to the Hilbert space inner product $\Delta u + \lambda u^p \perp u^p$ (velocity is orthogonal to position). | |
| Mar 17, 2021 at 14:02 | history | edited | Student | CC BY-SA 4.0 |
deleted 151 characters in body
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| Mar 17, 2021 at 13:32 | comment | added | Willie Wong | To make Michael Renardy's comment extremely explicit. The function $f: \mathbb{R}\ni \theta \mapsto (\cos\theta,\sin\theta) \in \mathbb{R}^2$ has the property that $\|f(\lambda)\|$ is constant, but $\| \partial_\lambda f(\lambda)\|$ is $1$. You can cook up similar examples in any normed space of dimension $\geq 2$. It is not that taking the derivative of norm is not justified; it is that what you hope to prove is plainly false. | |
| Mar 17, 2021 at 13:02 | history | edited | Student | CC BY-SA 4.0 |
added 309 characters in body
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| Mar 17, 2021 at 11:42 | review | Close votes | |||
| Apr 1, 2021 at 3:05 | |||||
| Mar 17, 2021 at 11:26 | comment | added | Michael Renardy | You are asking if the derivative of the norm is equal to the norm of the derivative. This is false, even in finite dimensions. | |
| Mar 17, 2021 at 10:46 | history | asked | Student | CC BY-SA 4.0 |