Timeline for Some questions about convergence
Current License: CC BY-SA 4.0
15 events
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| Jun 15, 2020 at 7:27 | history | edited | CommunityBot |
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| May 15, 2020 at 21:49 | comment | added | leo monsaingeon | you can use the Poincaré inequality with $p=2$ instead of $p=1$! (here this is OK because the sequence of gradients is bounded in $L^2$ and the boundary datum is fixed) | |
| S Sep 5, 2019 at 23:01 | history | bounty ended | CommunityBot | ||
| S Sep 5, 2019 at 23:01 | history | notice removed | CommunityBot | ||
| S Aug 28, 2019 at 21:57 | history | bounty started | yoshi | ||
| S Aug 28, 2019 at 21:57 | history | notice added | yoshi | Authoritative reference needed | |
| Aug 28, 2019 at 21:40 | history | edited | yoshi | CC BY-SA 4.0 |
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| Aug 28, 2019 at 18:15 | comment | added | yoshi | okay I'm almost there: for 1) Applying the poincare inequality gives: $$\|u_k - u^0\|_{L^1_{loc}} \leq \|Du_k - Du^0\|_{L^1_{loc}}$$ How do I get to $u_k-u^0$ are bounded in $L^2_{loc}$, I checked the relevant embedding theorem but it seems to be going the wrong way (Folland, Real Analysis 6.12) | |
| Aug 28, 2019 at 16:44 | history | edited | yoshi | CC BY-SA 4.0 |
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| Aug 26, 2019 at 15:28 | comment | added | Hannes | Yes; I'd say 2) is a consequence of 1) plus some tedious stuff on how to transfer the subsequences converging pw. a.e. on $B_R \cap \Omega$ to the whole $\Omega$. But maybe there is a direct argument which I am not seeing. | |
| Aug 26, 2019 at 15:23 | comment | added | vidyarthi | and again, 2) is the application of poincare inequality | |
| Aug 26, 2019 at 13:37 | comment | added | Hannes | 1) is the Poincare inequality: $u_k - u^0$ is zero on $S$, and $S \subseteq \partial(B_R \cap \Omega)$ for $R$ large enough, so there you have the Poincare inequality for $u_k - u^0$ on $B_R \cap \Omega$. 3) Is the Banach-Alaoglu theorem in $L^\infty(\Omega)$ since the characteristic functions are uniformly bounded in $L^\infty$ norm by $1$. | |
| Aug 26, 2019 at 13:24 | comment | added | yoshi | H. W. Alt & L. A. Caffarelli, Existence and regularity for a minimum problem with free boundary, J. Reine Angew. Math. 325 (1981), 105–144 | |
| Aug 26, 2019 at 13:09 | comment | added | vidyarthi | could you give a reference to the paper? | |
| Aug 26, 2019 at 12:37 | history | asked | yoshi | CC BY-SA 4.0 |