Timeline for Estimating the size of $\Omega_r=\{x\in\Omega: \text{dist}(x,\partial\Omega)<r \}$
Current License: CC BY-SA 4.0
6 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jan 5, 2021 at 3:43 | comment | added | sharpe | Oh, I see. It certainly looks like no good. | |
| Jan 5, 2021 at 2:17 | comment | added | BigbearZzz | @sharpe unfortunately, for any continuous approximation of $f\in W^{1,p}(\Omega)$, the factor $1/r$ would blow up the error term. I haven't found a way to make it works yet. | |
| Jan 5, 2021 at 2:08 | comment | added | sharpe | I don't know well. At least if $p=2$, it follows that $C(\overline{D}) \cap W^{1,2}(D)$ is a dense subspace of $W^{1,2}(D)$. Wouldn't it work if we used this fact? | |
| Jan 4, 2021 at 18:56 | comment | added | BigbearZzz | @sharpe very interesting, thanks! This piqued my interest quite a bit. Do you know if a similar result holds when $f$ is a Sobolev function instead? | |
| Jan 4, 2021 at 11:21 | comment | added | sharpe | If $\Omega$ is a bounded Lipschitz domain, we have $\lim_{r \to 0}(1/r)\int_{\Omega_r}f(y)\,dy=\int_{\partial \Omega}f(y)\,d\mathcal{H}^{n-1}(y)$ for any bounded continuous function $f \colon \overline{\Omega} \to \mathbb{R}$. See Lemma 7.1 in this article projecteuclid.org/download/pdfview_1/euclid.aop/1485421330. | |
| Jan 4, 2021 at 10:41 | history | asked | BigbearZzz | CC BY-SA 4.0 |