Timeline for When is Sobolev space a subset of the continuous functions?
Current License: CC BY-SA 4.0
5 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Feb 25, 2022 at 15:18 | history | edited | coudy | CC BY-SA 4.0 |
correct spelling
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| May 29, 2012 at 20:06 | history | edited | Pietro Majer | CC BY-SA 3.0 |
edited body
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| Sep 6, 2010 at 19:46 | comment | added | Dorian | Also, for the $d=2$ case, consider the function $u(x) = \log |x|$. Then $|Du(x)| = \frac{1}{|x|}$ and so $\int |Du(x)|^2 = 2\pi \int_0^r r^2/r$ which is finite obviously. So $\log|x|$ is in $H_0^1(\Omega)$ but not continuous. A similar example can be made for $d=3$ with $1/|x|^{\alpha}$ for an appropriate choice of $\alpha > 0$. | |
| Sep 6, 2010 at 19:19 | comment | added | alext87 | Cheers! :) The first one $d=1$ is simple doing it this way. Now off to find out about the co-area formula... | |
| Sep 5, 2010 at 2:48 | history | answered | Dorian | CC BY-SA 2.5 |