Timeline for Inverse trace theorem for partial trace
Current License: CC BY-SA 3.0
12 events
| when toggle format | what | by | license | comment | |
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| May 2, 2016 at 11:01 | comment | added | Peter | @WillieWong thank you very much! Your advice has helped me a lot! | |
| May 1, 2016 at 22:45 | comment | added | Willie Wong | Define $f(\vec{x},y) = g(\vec{x}) \phi(y)$ where $g$ is the function on the submanifold, $y$ is the transversal coordinate in the tubular, and $\phi$ is a smooth cut-off function. $f$ is as smooth and as integrable as $g$ is. | |
| May 1, 2016 at 20:40 | comment | added | Peter | @WillieWong and why is it trivial for a submanifold? You used a tubular, i think this is easy to imagine in two dimensional case, but for instance, how do you connect the boundary for a surface of a ball in three dimension by using a tubular? Perhaps this is a stupid question, but i'm not very familiar with such problems in differential geometry. | |
| May 1, 2016 at 20:29 | history | edited | Peter | CC BY-SA 3.0 |
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| May 1, 2016 at 20:22 | history | edited | Peter | CC BY-SA 3.0 |
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| May 1, 2016 at 20:15 | comment | added | Peter | @WillieWong thanks for your suggestion. I will make some comments in the question. | |
| May 1, 2016 at 19:58 | comment | added | Willie Wong | What I don't understand is your notation: if $\partial\Omega'$ is an arbitrary subset, for example, a discrete subset of $\partial\Omega$, how do you intend to define $W^{1-1/p,p}(\partial\Omega')$? Similarly if $\partial\Omega$ is a submanifold of positive codimension? Your question may admit a good technical answer, but first you have to specify what you mean. | |
| May 1, 2016 at 19:56 | comment | added | Willie Wong | In terms of simply extension theorems for Sobolev spaces, the recent works of Fefferman, Israel, and Luli (in some combination) come to mind. If your $\partial\Omega'$ is sufficiently nice you can extend to $\partial\Omega$ and apply the standard results. One of the problems however is with the definition of $W^{1 - 1/p,p}(\partial\Omega')$. If $\partial\Omega'$ is a submanifold of $\partial\Omega$ and the Sobolev space is the intrinsic one, then by taking a tubular nbhd the extension is trivial. | |
| May 1, 2016 at 17:30 | comment | added | Peter | @WillieWong what I mean "subsets" is that we can deal with the sets directly, for instance ths sobolev spaces, without using a definition of charts. And I think there should exist some similar extension results for manifolds like the ones for Sobolev spaces of usual domains. | |
| May 1, 2016 at 17:27 | comment | added | Peter | @WillieWong Nope. But can u tell me in which references I can find relevant contents? I think $\partial\Omega'$ should some how regular, but I dont know how to define the regularity for a manifold, since Im mostly deal with subsets in $R^n$. I think the regularity should some how relate to the charts or not? | |
| May 1, 2016 at 16:25 | comment | added | Willie Wong | This will be true as soon you have an extension operator $W^{1-1/p,p}(\partial\Omega') \to W^{1-1/p,p}(\partial\Omega)$. Do you assume more on $\partial\Omega'$ than just "arbitrary subset"? | |
| May 1, 2016 at 15:36 | history | asked | Peter | CC BY-SA 3.0 |