Timeline for Regularity of a shrunken domain
Current License: CC BY-SA 4.0
12 events
| when toggle format | what | by | license | comment | |
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| Jun 8, 2019 at 8:44 | comment | added | Mateusz Kwaśnicki | @GuyFsone: If $d(x) = \operatorname{dist}(x,\partial \Omega)$ is, say, $C^k$, then the level sets $\{d(x) = \delta\} = \partial \Omega_\delta$ are $C^k$, too (because $\nabla d \ne 0$). | |
| Jun 7, 2019 at 23:41 | comment | added | Guy Fsone | @MateuszKwaśnicki The subject treated in Gilbard and Trudinger is a different one. It mostly talks about the smoothness of the distance function on a dilated version of $\Omega$. | |
| Jun 7, 2019 at 23:27 | comment | added | Math604 | okay; i;m an idiot... i thought it was $ \delta(x)=dist(x,\partial \Omega) <\epsilon$ not greater than... omit my comments | |
| Jun 7, 2019 at 23:24 | comment | added | Math604 | i quicky googled and saw this (but i know there is tons of notes about this stuff) link.springer.com/content/pdf/bbm%3A978-3-642-96379-7%2F1.pdf | |
| Jun 7, 2019 at 23:20 | comment | added | Math604 | you might want to google 'tubular domains'... i thinkthat is the name of this type of domain (and as mentioned earlier... some pde people look at these things). Also lots of people play arouind with the distannce function... i believe its as smooth as the boundary accept at the ''ridge'' (ie. the ''center'' of the set)... also i recall the level sets of $ \delta$ satisfy something like $ \Delta \delta = K(x)$ where $K$ is some sort of curvature associated to the level set (maybe the mean curvature)... | |
| Jun 7, 2019 at 21:35 | comment | added | Mateusz Kwaśnicki | @GuyFsone: Section 14.6 in Gilbarg and Trudinger has what you need. | |
| Jun 7, 2019 at 21:24 | comment | added | Kostya_I | In the example of Mateusz Kwasnicki, the "highly irregular" means "having an outward cusp", and such things do not satisfy VDC (while $C^1$ domains to, of course). | |
| Jun 7, 2019 at 21:08 | comment | added | Guy Fsone | Of course, large $\delta $ are not interesting at all. | |
| Jun 7, 2019 at 21:06 | comment | added | Guy Fsone | @MateuszKwaśnicki Please I would like to see does reference. Do you have an idea about what could happen with VDC condition? I technically like that condition very much. | |
| Jun 7, 2019 at 21:02 | history | edited | Guy Fsone | CC BY-SA 4.0 |
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| Jun 7, 2019 at 21:02 | comment | added | Mateusz Kwaśnicki | Question 1 has a clear answer: yes if $k \geqslant 2$ and $\delta > 0$ is small enough. This is given in most textbooks on PDE's, I suppose, I can search for an exact reference if you like. However, if $\delta > 0$ is too large, then smoothness of $\Omega_\delta$ may deteriorate: consider, say, a unit ball and $\delta = 1$. Also, if $k = 1$, then $\Omega_\delta$ may become highly irregular even when $\delta > 0$ is small. An example: $\Omega$ a region above the graph of $y = |x| / \log(1/|x|)$ near $x = 0$. | |
| Jun 7, 2019 at 20:52 | history | asked | Guy Fsone | CC BY-SA 4.0 |