2
$\begingroup$

I read at the top of the page $580$ of this paper that the imbedding $W^{1,2}(\Gamma,d\mu) \hookrightarrow L^2(\Gamma,d\mu)$ is compact for a bounded domain $\Gamma \subset \mathbb{R}^n$ and a measure $\mu$ as defined in the second paragraph of the section $2$. I would like a reference for this result.

Thanks in advance!

Improve this question
$\endgroup$
9
  • $\begingroup$ 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 $\Gamma$.) $\endgroup$
    – Hannes
    Commented Jun 8, 2020 at 8:40
  • 1
    $\begingroup$ @Hannes: Why would it be enough to have an embedding $W^{1,2}(\Gamma) \hookrightarrow L^r(\Gamma)$ for some $r > 2$? $\endgroup$
    – Jochen Glueck
    Commented Jun 8, 2020 at 9:26
  • 1
    $\begingroup$ @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 :-) $\endgroup$
    – Hannes
    Commented Jun 8, 2020 at 11:06
  • 1
    $\begingroup$ @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"...) $\endgroup$
    – Jochen Glueck
    Commented Jun 8, 2020 at 11:26
  • 1
    $\begingroup$ @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. $\endgroup$
    – Hannes
    Commented Jun 11, 2020 at 16:23

0

Reset to default

You must log in to answer this question.