Timeline for how to use the sobolev inequality to obtain the embedding theorem
Current License: CC BY-SA 3.0
28 events
| when toggle format | what | by | license | comment | |
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| S Apr 11, 2017 at 12:49 | history | bounty ended | CommunityBot | ||
| S Apr 11, 2017 at 12:49 | history | notice removed | CommunityBot | ||
| Apr 11, 2017 at 4:16 | vote | accept | pxchg1200 | ||
| Apr 10, 2017 at 1:49 | comment | added | pxchg1200 | Since $S_{0}^{1,2}(U)$ can be compact embedding in to $L^2(U)$, I think we can prove the following Poincare inequality for $p=2$, $$ \lambda_{1}\int_{U}|u(x)|^2dx\leq \int_{U}|D_{L}u(x)|^2dx $$ after variational calculus. the $\lambda_{1}$ means the first Dirichlet eigenvalues of $L$ which is strict positive. | |
| Apr 8, 2017 at 13:52 | answer | added | Henry.L | timeline score: 2 | |
| Apr 6, 2017 at 3:05 | comment | added | Deane Yang | I have to concede that I appear to be wrong about this. I thought I knew how to do this, but I currently don't see how. If there is a global Poincaré inequality ($\|f\|_p \le C\|D_Lf\|_p$), then you would get the inequality. | |
| Apr 3, 2017 at 15:09 | comment | added | pxchg1200 | It seems we can only obtain $$\|f\|_{L^{q}(U)}\leq C( \|D_{L}f\|_{L^{p}(U)}+\|f\|_{L^{p}(U)} )$$ will the sharp estimate | |
| Apr 3, 2017 at 14:31 | comment | added | pxchg1200 | Do you mean use $\|D_{L}(\phi_{i}f) \|_{L^{p}(B_{i})}=\|(D_{L}\phi_{i})f+\phi_{i}D_{L}f\|_{L^{p}(B_{i}}$ ? then how can I combine it to $\|D_{L}f\|_{L^{p}(U)}$ ? @DeaneYang | |
| Apr 3, 2017 at 14:25 | comment | added | Deane Yang | Use product rule for differentiation and Holder's inequality. | |
| Apr 3, 2017 at 14:21 | history | edited | pxchg1200 | CC BY-SA 3.0 |
added 219 characters in body
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| Apr 3, 2017 at 14:01 | history | edited | pxchg1200 | CC BY-SA 3.0 |
added 847 characters in body
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| S Apr 3, 2017 at 10:50 | history | bounty started | pxchg1200 | ||
| S Apr 3, 2017 at 10:50 | history | notice added | pxchg1200 | Authoritative reference needed | |
| S Oct 27, 2016 at 5:34 | history | bounty ended | CommunityBot | ||
| S Oct 27, 2016 at 5:34 | history | notice removed | CommunityBot | ||
| S Oct 19, 2016 at 4:16 | history | bounty started | pxchg1200 | ||
| S Oct 19, 2016 at 4:16 | history | notice added | pxchg1200 | Canonical answer required | |
| Oct 18, 2016 at 3:15 | comment | added | Deane Yang | The set $B_R$ is still an open set, so if you cover $U$ with such "balls", you can still construct a partition of unity subordinate to that cover. | |
| Oct 17, 2016 at 12:23 | comment | added | Nate Eldredge | Good point, I did not see the $c$ subscript. | |
| Oct 17, 2016 at 5:25 | comment | added | pxchg1200 | @NateEldredge how does closed graph theorem works here? first you need $S_{c}^{1,p}(U)$ to be a Banach space in norm $\|\cdot\|=\left(\int_{U}|D_{L}u|^{p}dx\right)^{1/p}$ | |
| Oct 17, 2016 at 4:08 | comment | added | pxchg1200 | I wonder the subunit ball can cover $U$ or not,because the subunit ball is not like the open ball we used before. so I don't know how to use partition of unity, can you write the detail proof? @DeaneYang thank you very much! | |
| Oct 17, 2016 at 3:45 | comment | added | Deane Yang | Nate, thanks. Didn't know I had such powers. | |
| Oct 17, 2016 at 3:02 | review | Close votes | |||
| Oct 17, 2016 at 17:26 | |||||
| Oct 17, 2016 at 2:38 | comment | added | Nate Eldredge | @DeaneYang: You have enough reputation to vote to migrate... | |
| S Oct 17, 2016 at 1:29 | history | suggested | T. Amdeberhan | CC BY-SA 3.0 |
Language errata corrected and clarifications added.
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| Oct 17, 2016 at 0:40 | review | Suggested edits | |||
| S Oct 17, 2016 at 1:29 | |||||
| Oct 17, 2016 at 0:32 | comment | added | Deane Yang | This should be migrated to math.stackexchange.com. But the idea is to use a partition of unity subordinate to a locally finite covering of $U$ by open balls to write $u$ as a sum of functions, each compactly supported on one of the balls, and apply Theorem 2.3 to each of these functions. | |
| Oct 17, 2016 at 0:26 | history | asked | pxchg1200 | CC BY-SA 3.0 |