Timeline for A Question about compactness of an embedding into $L^p$ spaces
Current License: CC BY-SA 3.0
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| Jun 15, 2020 at 7:27 | history | edited | CommunityBot |
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| Oct 9, 2015 at 20:01 | comment | added | Math604 | Maybe you can argue directly. Let $ u_m$ be bounded by say $1$ in $H$ and use the improved imbedding to see that $u_m$ is bounded in $L^2(\Omega)$. Then directly from the inequality again we see that $ u_m$ should be bounded in $H^1_0_{loc}( overline{\Omega} \backslash \{0\} )$ (trying to say its bounded in $H^1$ except near origin). By diagonal argument there should be some $ u in H^1_{loc} (remove near origin) such that $ u_m \rightharpoonup u$ in $H^1_{loc}(away from origin). Now try and show that $ u_m$ converges in $L^1(\Omega)$ (so in $L^1$ it can't concentrate at origin). Then... | |
| Oct 9, 2015 at 18:52 | comment | added | Hheepp | Thanks. I must give a seminar in the classroom in this regard. I want to minimize a function in this space and I need an compact embedding. | |
| Oct 9, 2015 at 18:47 | comment | added | Math604 | there should be a bunch of "improved Hardy-Rellich papers". Try googling that phrase (also Amir Moradifam wrote some papers on this in the radial case). Regarding the space $H$ I don't really know of many references.... Do you need to understand $H$ for a particular reason or ?? | |
| Oct 9, 2015 at 18:31 | comment | added | Hheepp | @Math604: thanks. Did you know any other article about improved case of rellich inequality and similar space H as completion with respect to norm $$ \bigg( \int_{\Omega} \Big((\Delta u)^2 - \dfrac{N^2(N-4)^2}{16} \dfrac{u^2}{|x|^4}\Big) \,\mathrm{d}x \bigg)^{\frac{1}{2}} $$ | |
| Oct 8, 2015 at 18:54 | comment | added | Math604 | in the `improve Hardy inequality' you need to renormalize the $L^p$ term; ie. it should be $ \| u \|_{L^p}^2$. Take a look at this paper regarding $H(\Omega)$ , I think it says that some of the earlier papers regarding $H$ may have had some flaw??? arxiv.org/pdf/1102.5661v1.pdf Further here is a paper regarding improved $L^2$ Hardy inequaliteis : arxiv.org/abs/0805.0610 | |
| Oct 8, 2015 at 17:58 | history | edited | Hheepp | CC BY-SA 3.0 |
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| Oct 8, 2015 at 16:58 | history | asked | Hheepp | CC BY-SA 3.0 |