Timeline for boundary integral estimates for elliptic pde
Current License: CC BY-SA 4.0
8 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jun 19, 2018 at 18:33 | vote | accept | Math604 | ||
| Jun 19, 2018 at 12:27 | answer | added | fedja | timeline score: 2 | |
| Jun 19, 2018 at 3:44 | comment | added | Math604 | i pretty much talk nonsense continuously these days... If you add a little detail to your above comments I can attempt to see if i understand (sorry) | |
| Jun 19, 2018 at 1:55 | comment | added | fedja | Yes, that is what you obtain from integrating against the first eigenfunction of the Laplacian. However Sobolev tells you that you can estimate some $L^q$-norm with fixed $q>2$ by some close to $1$ power of $L^{p+1}$-norm and, thereby, by Holder by some power of $L^{p/3}$-norm (if $p$ is close enough to $1$), but the latter is controlled by your expression for domains with decent boundary. Am I talking nonsense? (this happens sometimes :lol:) | |
| Jun 18, 2018 at 22:22 | comment | added | Math604 | then in the case of the above equation one can use some weighted estimates of Souplet, Quittner to start a bootstrap (it is presicely this part is causing problems with the extra advection terms that i didn't add) | |
| Jun 18, 2018 at 22:19 | comment | added | Math604 | yes i am assuming $u>0$. Maybe i missing something but I thought this doesn't give me anything. So i was under the impression the best initial estimate i can get is something like $ \int_\Omega u_m(x)^p \delta(x) dx \le C $ ($C$ independent of $m$) where $ \delta$ is the distance to the boundary. | |
| Jun 18, 2018 at 21:31 | comment | added | fedja | Do you assume that $u>0$ inside $\Omega$? Also why don't you just integrate against $u_m$ and apply Sobolev inequality (should work if $p$ is only slightly above $1$) | |
| Jun 18, 2018 at 20:27 | history | asked | Math604 | CC BY-SA 4.0 |