Timeline for Two doubts in the paper of Brezis Merle in blow up analysis of the equation $-\Delta u=Ve^u$
Current License: CC BY-SA 4.0
14 events
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| Jul 14 at 18:34 | comment | added | Elio Li | For question 2, it can be proved by rearrangement inequality, see math.stackexchange.com/questions/4756719/…. | |
| Apr 3, 2023 at 19:26 | vote | accept | User1723 | ||
| Apr 3, 2023 at 1:30 | answer | added | Marc | timeline score: 5 | |
| Mar 12, 2023 at 14:32 | comment | added | Marc | @GiorgioMetafune Ok thank you very much for these references! Yeah it was a little difficult for me to find resources doing L^p theory when p=1. | |
| Mar 12, 2023 at 13:48 | comment | added | Giorgio Metafune | Then you can localize or perturb these estimates. (I wrote this in an old papewr with A. Lunardi, in a more general setting) | |
| Mar 12, 2023 at 13:47 | comment | added | Giorgio Metafune | I have notes in Italian by L. Boccardo, dealing with elliptic equations with data in $L^1$ or even measures and these estimates are there (they go back to Stampacchia). With these names, I hope, you should find appropriate references; if not I can try again. There is also a parabolic approach: in the whole space solve $u_t-\Delta u=f$ through the heat kernel and write the solution of $\lambda v-\Delta v=f$ as the laplace transform of the semigroup. Then you differentiate under the integral and the $L^q$ norm of the gradient is finite exactly for $q<N/(N-1)$. | |
| Mar 12, 2023 at 12:40 | comment | added | Marc | @GiorgioMetafune Yea I certainly agree with the $L^1$ estimate you stated. In my proof I need the $ \|v_{n}\|_{1,q} \leq C $ bound you stated. Do you have a reference for this particular estimate? | |
| Mar 12, 2023 at 10:51 | comment | added | Giorgio Metafune | I admit that everything is vague. Concerning the $L^1$ estimate it follows from the Green function (or either if $u-\Delta u=f$ by integrating by parts against smooth aprroximation of the signum of $u$). Since elliptic regularity does not hold in $L^1$ one cannot get $W^{2,1}$ estimates. However the estimate for gradient is true (if needed) or also, having $f_n \in L^p$ one can use interior $W^{2,p}$ estimates to get bounds for the secpond derivatives. | |
| Mar 11, 2023 at 21:46 | comment | added | Marc | @GiorgioMetafune I see how these estimates imply the result however can you give a reference for these? I believe $ \|v_{n}\|_{1} \leq C$ follows from Green's identity but I can't quite see how to prove the $ W^{1,q} $ bound you state. | |
| Dec 26, 2022 at 23:46 | comment | added | Giorgio Metafune | For 1., a possibility is the following. Having $\|f_n\|_1 \leq C$ first infer that $\|v_n\|_1 \leq C_1$ and also $\|v_n\|_{1,q} \leq C_2$ if $q<n/(n-1)=2$. These estimates are not obvious but well-known. Then use interior estimates for the Laplacian (since $f_n$ are bounded in $L^p_{loc}$) to deduce compactness, hence convergence, in $W^{2,p}_{loc}$. Since $p>n/2=1$ you have uniform convergence, by Sobolev embedding. | |
| Dec 26, 2022 at 11:18 | comment | added | Giorgio Metafune | Concerning 2., they use the fact that the integral of a radial decreasing function on a set $\Omega$ is maximized when $\Omega$ is a ball centered at 0 having the same volume. However, in the computation for $B_R$ I also find a different constant. | |
| Dec 26, 2022 at 10:21 | history | edited | Daniele Tampieri | CC BY-SA 4.0 |
Fixed typos.
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| Dec 26, 2022 at 9:42 | history | edited | Daniele Tampieri | CC BY-SA 4.0 |
Minor Math Jaxing and formatting
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| Dec 26, 2022 at 8:48 | history | asked | User1723 | CC BY-SA 4.0 |