Timeline for Regularity on Neumann problem on polygonal domain
Current License: CC BY-SA 3.0
10 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Dec 7, 2015 at 6:24 | comment | added | Math604 | here are what appear to be some nice notes related to the above questions. hal.archives-ouvertes.fr/hal-00453934/document | |
| Nov 27, 2015 at 2:26 | comment | added | Math604 | ya, i played around with that a bit and saw there were restrictions on $p$ but I think i figured that must be a general theorem and the case of a cube one can do much better (at least that is what i was hoping). So by playing around with even extension i am half convinced it is true for all $1<p< \infty$... but i could be compeltely out to lunch also... I suspect on cubes these answers must be well known (just not by me). | |
| Nov 26, 2015 at 23:51 | comment | added | Delio Mugnolo | Thanks, I didn't know that paper by Dauge. I must admit I don't understand it quite precisely, but still: If one tries to apply her Theorem 1.1, don't her remarks right after the theorem yield the regularity you are looking for if and only if $p<6/(5-\sqrt{5})\simeq 2,17$ (the first condition being automatically satisfied in the case of a cube, where $\omega=\pi/2$ and $k=0$)? | |
| Nov 26, 2015 at 11:56 | comment | added | Math604 | Take a look at this paper. I think they have some stuff regarding this $p$ dependence... perso.univ-rennes1.fr/monique.dauge/publis/Da_mixed.pdf | |
| Nov 26, 2015 at 9:23 | comment | added | Delio Mugnolo | In the case of domains smooth enough I have never seen a dependence on $p$ - even the spectrum and the eigenfunctions of all these operators is identical! But I do know instances of $p$-dependence in the case of certain nonlinear operators, like the $p$-Laplacian (but here $p$ has nothing to do with "your $p$"). So I would be interested in the kind of results you are mentioning. | |
| Nov 25, 2015 at 18:49 | comment | added | Math604 | Thanks for the comment. I saw some papers online and it appears the value of $p$ actually plays a role. They have results that say provided $p$ is less than some geometrical quantity then one has the desired $W^{2,p}$ estimate (this is for both the Dirichlet and Neumann problem). In the case of the Neumann problem it appears that one is not really able to compute this geometrical quantity. Putting this aside it seems that since I am on a cube one can do even reflections to obtain the desired regularity from interior estimates (but maybe i am completely missing something). | |
| Nov 25, 2015 at 18:25 | comment | added | Delio Mugnolo | The philosophy in that book is that if anything can go wrong in a domain with a nonsmooth domain, it is because of sharp edges/corners, which means that for convex domains the situation will generally be benign. That said, the result you are looking for is Thm. 3.2.1.3 in the case $p=2$, but I had no time to look for the general case. | |
| Nov 25, 2015 at 16:20 | comment | added | Math604 | They seem to cover very general cases and since I have the very special case I thought someone probably just knows the answer off the top of their head. Honestly I attempted to decipher the results but gave up before I could come to a conclusion... | |
| Nov 25, 2015 at 8:00 | comment | added | Delio Mugnolo | Can you please explain why the results by Grisvard are not sufficient? He also considers spaces of Hölder continuous functions btw. | |
| Nov 24, 2015 at 23:33 | history | asked | Math604 | CC BY-SA 3.0 |