Timeline for A continuous bi-Lipschitz shrinking of a domain into a compact subset
Current License: CC BY-SA 4.0
19 events
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| May 16, 2019 at 8:26 | answer | added | Mateusz Kwaśnicki | timeline score: 2 | |
| May 16, 2019 at 1:40 | comment | added | Ben Ciotti | @MateuszKwaśnicki Thank you, your comment has been very helpful (indeed, I may cite it at some point). It also directed me towards a section (14.6) in Gilbarg and Trudinger that had some useful lemmas regarding existence, regularity, and bounds on the distance function. They showed that it is $C^k$ in a neighborhood of a $C^k$ boundary. I would be interested if there was a way to get $G$ to be $C^{1,1}$ on a (merely) $C^{0,\alpha}$ domain as well. | |
| May 10, 2019 at 17:26 | comment | added | Mateusz Kwaśnicki | If $\Omega$ is $C^{1,1}$, then also the signed distance to the boundary is $C^{1,1}$ (near the boundary), and, appropriately regularised away from the boundary, can serve as the function $G$: the gradient flow does the job in this case. However, my feeling is that $C^{1,1}$ is way too much, the same should be true for $C^{0,\alpha}$ domains for any $\alpha > 0$. If I find time (and if this is what you are looking for), I will try to sketch the argument. | |
| May 10, 2019 at 16:53 | history | edited | Ben Ciotti | CC BY-SA 4.0 |
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| May 10, 2019 at 16:50 | comment | added | Ben Ciotti | @NateEldredge Another excellent point, you may have saved me from possibly making a fool of myself. | |
| May 10, 2019 at 16:06 | comment | added | Nate Eldredge | In particular, this suggests that your approach with $G$ has a gap. If $\Omega$ is a punctured disk, then $\partial \Omega$ is the zero set of the $C^\infty$ function $G(x) = |x|^2(1-|x|^2)$. But $\nabla G$ vanishes at the origin, and so your flows will just fix 0 instead of mapping it inside $\Omega$. You'd really need a $G$ whose gradient doesn't vanish on $\partial \Omega$, and the implicit function theorem says you can't have that at the puncture. | |
| May 10, 2019 at 15:54 | comment | added | Nate Eldredge | I'm wondering about topological obstructions. Say $\Omega$ is a punctured disk in $\mathbb{R}^2$. Then $\overline{\Omega}$ is a closed disk, and it's contractible, so your map $F_n$ can't just "enlarge the hole". You have to map the whole thing to a contractible blob that doesn't surround the missing point, and I'm having a hard time visualizing how those could converge to the identity. | |
| May 10, 2019 at 15:30 | history | edited | Ben Ciotti | CC BY-SA 4.0 |
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| May 10, 2019 at 10:42 | comment | added | Mateusz Kwaśnicki | For the existence of $G$, regularised distance might be a good search term; see, for example, here for construction and basic properties, and here for further developments. | |
| S May 10, 2019 at 10:00 | history | suggested | user64494 | CC BY-SA 4.0 |
The title is improved.
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| May 10, 2019 at 7:41 | review | Suggested edits | |||
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| May 10, 2019 at 6:38 | history | edited | Ben Ciotti | CC BY-SA 4.0 |
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| May 10, 2019 at 6:32 | history | edited | Ben Ciotti | CC BY-SA 4.0 |
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| May 10, 2019 at 6:27 | history | edited | Ben Ciotti | CC BY-SA 4.0 |
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| May 10, 2019 at 6:15 | comment | added | Ben Ciotti | @NateEldredge Excellent point! However it made me realize that I must add some more conditions in order to suit my purpose (which concerns the Coulomb energy) - I shall edit the question now to be in terms of bilipschitz maps. | |
| May 10, 2019 at 6:05 | history | edited | Ben Ciotti | CC BY-SA 4.0 |
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| May 10, 2019 at 5:40 | comment | added | Nate Eldredge | I don't think any assumptions on $\Omega$ should be needed. I think, for instance, that you could choose $\mu_n$ to be supported on a finite set of points in $\Omega$. Partition $\overline{\Omega}$ into a finite number of disjoint Borel sets $A_i$, $i=1,\dots, k$ of diameter less than $1/n$, each containing at least one point $x_i$ of $\Omega$. Then let $\mu_n$ put mass $\mu(A_i)$ at the point $x_i$. I think such $\mu_n$ should converge vaguely to $\mu$. Note that your test functions $\varphi$ are uniformly continuous. | |
| May 10, 2019 at 4:54 | history | edited | Ben Ciotti | CC BY-SA 4.0 |
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| May 10, 2019 at 4:48 | history | asked | Ben Ciotti | CC BY-SA 4.0 |