Timeline for boundary integral estimates for elliptic pde
Current License: CC BY-SA 4.0
15 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jun 28, 2018 at 19:57 | comment | added | Math604 | Let me ask you another question (well its the same question as the original) but now my interest is in what range of $p$. For ultra weak solutions we know they are bounded for $p<\frac{N+1}{N-1}$ and if we assume no boundary singularlarities then its bounded for $p<\frac{N}{N-2}$. Now suppose $u_m$ smooth positive solutions. Without using moving plane near the boundary (or a blow up argument) is the best one can show that the solutions are bounded for $p<\frac{N+1}{N-1}$ ? I just figured you might know the answer to this without doing any thinking. Your help on this is great | |
| Jun 28, 2018 at 14:09 | comment | added | Math604 | Regarding extending the range; I am able to do this with a weighted space approach and duality (different than the approach from Quittner / Souplet book); but i run into problems with the advection term. Regarding your comment. So the approach I am using i don't use the first eigenfunction. I am using a positive `first integral of a(x)' and I assume its roughly linear near the boundary. I am still a bit confused about what exact estimates regarding the 1D case we are talking about; so by all means you can type up some details if its easy. | |
| Jun 28, 2018 at 9:36 | comment | added | fedja | @Math604 Yeah, you need to work more to extend the range. BTW, I checked my computations and stay by my words: you are in trouble for a constant field on $[0,1]$ and I do not see why you should be any better off in higher dimensions. Do you want me to post the details? | |
| Jun 26, 2018 at 21:32 | comment | added | Math604 | Let me ask a related question (since maybe you know the answer off the top of your head). Consider the original setting. Now without using a blow up argument (or say moving plane on a convex domain) i would like to know when we can obtain that $u_m$ is uniformly bounded independently of $m$. Using the weighted stuff (which I cannot generalize to the real setting i care about) it seems one obtains the result for $ 1<p<\frac{N+1}{N-1}$. Using a very slight generalization of your approach (generalize the 3) I think i only get $p<\frac{N}{N-1}$. Is this what you suspect? | |
| Jun 20, 2018 at 3:59 | comment | added | fedja | @Math604 I'll check my computation again then (though it looks correct). There may be some chance for large $p$ that you can control some $L^q$-norm with $q\le p-1$ but I do not see of what use such low norms may be and you declared that your $p$ is close to $1$ anyway | |
| Jun 20, 2018 at 3:50 | comment | added | Math604 | i was convinced there was some results regarding the case of $a(x)$ divergence free...specially if there as a first integral of $a(x)$. For instance i think there might be some results in (but i can't currently see it) Communications in Mathematical Physics February 2005, Volume 253, Issue 2, pp 451–480 | Cite as Elliptic Eigenvalue Problems with Large Drift and Applications to Nonlinear Propagation Phenomena | |
| Jun 20, 2018 at 3:33 | comment | added | fedja | @Math604 It looks like you are out of luck (in the sense that you cannot control even the $L^1$ norm) in 1D with constant vector field already. Have you checked this case before? ($\Omega=[0,1], a(x)=1$) | |
| Jun 20, 2018 at 1:43 | comment | added | Math604 | actually i haven't thought on whether i really need this eigenfunction bound...was just a thought that if i had it; then probably everything ok... but it might be okay without it. I can think on it first and if i have questions i can post more here...in either case thanks for all the help | |
| Jun 19, 2018 at 23:52 | comment | added | fedja | @Math604 OK, let me think a bit then. If I find some way around this, I'll let you know :-) | |
| Jun 19, 2018 at 22:47 | comment | added | Math604 | so this was exactly the type of thing i was looking for... now whether it works in my situation might be a problem. I think unless i can get a $C^{0,1}$ estimate on the first eigenfunction (indepenent of the parameter) this might be a problem. But i can make some other assumptions (maybe) to get around this. So the actually equation i was working with is $ -\Delta u_m(x) + \lambda_m a(x) \cdot \nabla u_m(x) = u_m(x)^p$ where $a$ is smooth and divergence free but where $ \lambda_m \rightarrow \infty$. | |
| Jun 19, 2018 at 21:20 | comment | added | fedja | @Math604 You are welcome. Does it work in your setting or you need something even more robust? In the latter case, just post the actual setup. | |
| Jun 19, 2018 at 18:33 | vote | accept | Math604 | ||
| Jun 19, 2018 at 18:33 | comment | added | Math604 | I think i understand the proof. Thank you very much. | |
| Jun 19, 2018 at 16:17 | comment | added | Math604 | thanks you very much for the detailed answer... let me attempt to digest it and see if i understand. | |
| Jun 19, 2018 at 12:27 | history | answered | fedja | CC BY-SA 4.0 |