Timeline for Reference for Rellich Kondrachov theorem on bounded domains and spaces with finite measure
Current License: CC BY-SA 4.0
11 events
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| Jun 11, 2020 at 16:23 | comment | added | Hannes | @George $\mu$ is a weighted Lebesgue measure in the paper, with the weight $\rho$ being $L^\infty$ and locally bounded away from zero on $\overline\Gamma$, so for compact $\overline\Gamma$, bounded away from zero. Thus $\mu$ is equivalent to the Lebesgue measure in the sense of $\rho_\bullet |A| \leq \mu(A) \leq \rho^\bullet |A|$ for every measureable set $A$, where $0<\rho_\bullet\leq\rho^\bullet$ are constants. (This is also mentioned and used e.g. in the proof of Prop. 2.2 in your paper.) The definition of the $\rho$-weighted Sobolev spaces then shows that these are just the usual ones. | |
| Jun 10, 2020 at 14:09 | comment | added | George | Adams considers the Lebesgue measure, but I have a doubt now: in what sense and how the measure defined in the paper is equivalent to the Lebesgue measure? Do you have a reference for this statement? | |
| Jun 10, 2020 at 14:07 | comment | added | George | The immersion compact is guaranteed by the theorem is $W^{m,p}(\Omega) \longrightarrow W^{m,p}(\Omega_0^k)$ , where $\Omega_0$ is a subdomain of $\Omega$ and $\Omega_0^k$ is the intersection of a $k$-plane in $\mathbb{R}^n$ with $\Omega_0$. The immersion holds for $0 < n - mp < k \leq n$ and $1 \leq q < \frac{kp}{n - mp}$ or $n = mp$, $1 \leq k \leq n$ and $1 \leq q < \infty$, but think better I think is not a problem because you can take $\Omega_0 = \Omega$ in the case of $\Omega$ be bounded and $k = n$ to obtain $\Omega_0^k = \Omega$. | |
| Jun 10, 2020 at 6:35 | comment | added | Hannes | @George I did mean the R-K Theorem in Adams' book, why can it not be used? (Is it the measure? That should be equivalent to the Lebesgue measure here and then the involved spaces are the same, no?) Regarding the domain assumption, personally I personally would understand that the assumption on $\Gamma$ at the beginning of Section 2 holds for the rest of the paper and the remark in question only considers the additional property that $\Gamma$ is in fact bounded. | |
| Jun 9, 2020 at 19:09 | comment | added | George | @Hannes, can you cite the corresponding theorem in Adams' book that you commented please? If you are talk about the Rellich-Kondrachov theorem in Adams' book, then I can not see how the theorem in the book can be used to prove the remark that I want a proof. Other thing, the author of the paper comments that $\Gamma \subset \mathbb{R}^n$ is a unbounded domain with the cone and segment property at the beginning of the section $2$, but he did not comment this in the remark for bounded domains. Can I assume these properties hold for bounded domains? | |
| Jun 8, 2020 at 11:26 | comment | added | Jochen Glueck | @Hannes: That's a great argument, thank you! Maybe I should spend more time reading Daniel Daners' papers instead of writing new ones with him ;-). ("Everybody writes, nobody reads"...) | |
| Jun 8, 2020 at 11:06 | comment | added | Hannes | @JochenGlueck By considering $W^{1,2}_0$ in the interior and using the gap $r>2$ for a small "boundary layer": let $\tau$ be smooth cutoff function associated to $\Gamma_0$ with $|\Gamma\setminus\Gamma_0| = \varepsilon$ and with support strictly contained in $\Gamma$. If $u_n \to 0$ weakly in $W^{1,2}(\Gamma)$, then $(\tau u_n) \subset W^{1,2}_0(\Gamma)$ has a subsequence going to zero strongly in $L^2(\Gamma)$. For $((1-\tau)u_n)$, we use the Hölder inequality: $\|(1-\tau)u_n\|_2 \leq \|1-\tau\|_{2r/r-2} \|u_n\|_r \lesssim \varepsilon^{(r-2)/2r}$. I learned this from a paper of Daners :-) | |
| Jun 8, 2020 at 9:26 | comment | added | Jochen Glueck | @Hannes: Why would it be enough to have an embedding $W^{1,2}(\Gamma) \hookrightarrow L^r(\Gamma)$ for some $r > 2$? | |
| Jun 8, 2020 at 8:40 | comment | added | Hannes | I think the author assumes that a cone property holds at the beginning of Section 2. The corresponding result can then be found in the book of Adams. (Reference [2] in the paper, which is also mentioned there in relation to the cone property.) The mentioned embedding will not in general not be compact for an arbitrary domain, but the conditions are rather mild I would say. (For example it would be enough to have an embedding $W^{1,2}(\Gamma) \hookrightarrow L^r(\Gamma)$ with an $r>2$, and this allows for outward cusps in $\Gamma$.) | |
| Jun 8, 2020 at 7:34 | history | edited | leo monsaingeon |
added the [sobolev-spaces] tag
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| Jun 7, 2020 at 22:33 | history | asked | George | CC BY-SA 4.0 |