Timeline for separable BV space for PDE's, Whats stopping us? [closed]
Current License: CC BY-SA 3.0
19 events
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| Dec 1, 2015 at 20:09 | comment | added | Pietro Majer | @RajeshDachiraju: right, I misread for $TV(u-v)$, sorry. So in this convergence $u_n$ converges to $u$ iff $u_n$ converges to $u$ in $L^1(\Omega)$ and $TV(u_n)$ converges to $TV(u)$. This convergence should be equivalent to w* convergence of $Du_n$ to $Du$ as measures on the compact set clos($\Omega$): no mass escapes to $\partial\Omega$. Have a look to Tigth Convergence of Measures. | |
| Dec 1, 2015 at 19:00 | comment | added | Fan Zheng | @RajeshDachiraju That's probably "putting the cart before the horse", again quoting Wille Wong, but good luck anyway! | |
| Dec 1, 2015 at 18:56 | comment | added | Rajesh D | Ok, but i want to fibd a pde for which this space is suitable! | |
| Dec 1, 2015 at 18:54 | comment | added | Fan Zheng | @RajeshDachiraju If that is your question, then I think you should read the first comment of Willie Wong, in particular this line: "in solving PDEs, one does not choose a function space first. Instead, one chooses a method and finds a function space in which the method can be applied." If your metric doesn't show up much in the literature, that is probably just a sign that it isn't a good one for expressing "closeness" in the BV sense. | |
| Dec 1, 2015 at 18:53 | comment | added | Rajesh D | @Fan : yes i know. Thats why i am nterested in it. I want to study this metric space. I know its not having anything to do with banach space. | |
| Dec 1, 2015 at 18:51 | comment | added | Fan Zheng | @RajeshDachiraju This metric actually seems weird to me: it is not equivalent to the usual metric $d(u,v)=TV(u-v)$ on BV spaces. To wit, let $u$ and $v$ be bounded by $\epsilon$ and have the same total variation, but $u$ oscillates wildly on $[0,1/3]$ and $v$ oscillates like crazy on $[2/3,1]$. Then $u$ and $v$ will be $2\epsilon$ close in your metric, but very far away in the standard BV metric. | |
| Dec 1, 2015 at 18:49 | comment | added | Rajesh D | @PietroMajer : plz take a close look at second term on the right of equation. This is not coming from a norm. | |
| Dec 1, 2015 at 16:17 | history | closed |
Willie Wong Jan-Christoph Schlage-Puchta Wolfgang YCor Pietro Majer |
Opinion-based | |
| Dec 1, 2015 at 16:17 | comment | added | Pietro Majer | To be precise: 1) This distance comes from a Banach space norm. 2) It Is NOT compact as claimed, and not even locally compact (for TVS, LC=finite dimension. 3) It is a dual space, so that its closed unit ball is compact in the w* topology by Banach-Alaoglu, and it is also metrizable due to separability (the distance being strictly weaker than the one you wrote). 4) However on the whole space, the w* topology is not metrizable. | |
| Dec 1, 2015 at 16:03 | answer | added | Bazin | timeline score: 1 | |
| Dec 1, 2015 at 15:11 | comment | added | Delio Mugnolo | I don't exactly get your point either, @RajeshDachiraju. $BV$ is usually a proxy for the space $W^{1,1}$, so it's a space where energy functionals (like the TV-functional) are defined and minimised, rather than taken as an ambient space. But if this instances are ok for you, then there is a lot you can discover in the neighbourhood of the calculus of variations: e.g., P.L. Lions has worked on the mathematical theory of image processing in that space and Kawohl has proved a Cheeger inequality working on $BV$. | |
| Dec 1, 2015 at 14:54 | comment | added | Willie Wong | books.google.com/books/about/… onlinelibrary.wiley.com/doi/10.1002/cpa.3160180408/abstract for your last comment. To your original question: what makes you think that your metric space is good/not good for PDEs? | |
| Dec 1, 2015 at 14:30 | review | Close votes | |||
| Dec 1, 2015 at 16:18 | |||||
| Dec 1, 2015 at 14:28 | comment | added | Rajesh D | @Willie : Also, could you please give a quick reference to the works on 1+1 conservation laws. | |
| Dec 1, 2015 at 14:26 | comment | added | Rajesh D | If you look at the reference I gave, there is a concept of vector metric space, where continuous linear functionals need not be uniformly continuous. | |
| Dec 1, 2015 at 14:25 | comment | added | Rajesh D | @WillieWong : I am interested in works on using the exact metric space mentioned, and not just BV as a Banach space! | |
| Dec 1, 2015 at 14:23 | comment | added | Willie Wong | So the "obvious thing" would be that the solution schemes used by people studying DE and PDE require certain properties that is not satisfied by the space listed. That said, bounded variation function spaces (in general) are used to study 1+1 dimensional conservation laws. A large part of the theory of shocks that originated with Lax and Glimm is built upon BV spaces. So I don't really get your question. | |
| Dec 1, 2015 at 14:15 | comment | added | Willie Wong | I quote my answer from here: "in solving PDEs, one does not choose a function space first. Instead, one chooses a method and finds a function space in which the method can be applied. In fact, a large and necessary part of modern analysis of partial differential equations consists of clever choices of function spaces in order to implement certain solution schemes." | |
| Dec 1, 2015 at 12:46 | history | asked | Rajesh D | CC BY-SA 3.0 |