Timeline for Weak convergence + convergence of the norm implies strong convergence in Orlicz spaces
Current License: CC BY-SA 4.0
6 events
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| Nov 26, 2023 at 4:46 | comment | added | leo monsaingeon | Merci Ayman. Unfortunately for my specific prolem I really don't see how to get started on $\int \rho\log\rho_n\to \int\rho\log\rho$. The entropy just hows up as a whoe, part of a functional that I'm minimizing, but that's it. So unless I manage to get some additional information I'm stuck. And I've tried hard, already! | |
| Nov 25, 2023 at 5:54 | comment | added | Ayman Moussa | In fact it's the increasingness of $\log$ which matters (see above the edit of my answer), however you need an extra estimate on $(\rho \log(\rho_n)_n$ to close the argument. | |
| Nov 25, 2023 at 5:50 | history | edited | Ayman Moussa | CC BY-SA 4.0 |
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| Nov 24, 2023 at 21:39 | comment | added | Ayman Moussa | Ah ! That's indeed funny ! About your problem, I was hoping for some kind of localization argument because for $\varepsilon>0$ and suitable $C_\varepsilon>0$, $z\log z + C_\varepsilon z$ is indeed increasing for $z>\varepsilon$, but it's not that clear that you can indeed localize without oscillations $\rho_n \geq \varepsilon$. I'll think about that next week and tell you if I get to anything new ! Bon week-end :) | |
| Nov 24, 2023 at 21:31 | comment | added | leo monsaingeon | Hi Ayman, funny to bump into you, I'm a friend and coauthor of Clément Cancès, and I'm currently trying to apply your "time compactness" trick with him and Boris. Small world. I'm well aware of Minty's trick, but unfortunately the entropy $z\log z$ is not monotone. And I don't think I have any other angle of attack for my speciic problem, the $L^p$ norms are unfortunately out of reach in my particular problem. Thanks antway, I appreciate you taking the time to answer and I hope we can meet in person some day. À plus! | |
| Nov 24, 2023 at 21:03 | history | answered | Ayman Moussa | CC BY-SA 4.0 |