Timeline for Question on Sobolev spaces in domains with boundary
Current License: CC BY-SA 4.0
7 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| May 29, 2019 at 17:13 | vote | accept | asv | ||
| May 29, 2019 at 9:55 | answer | added | Hannes | timeline score: 4 | |
| May 29, 2019 at 8:27 | answer | added | Mateusz Kwaśnicki | timeline score: 2 | |
| May 29, 2019 at 5:28 | comment | added | Jochen Glueck | @MikeMiller: I'm not sure I follow your argument. The closure of the compactly supported smooth functions within the space $W^{1,2}(\Omega) \cap C^0(\overline{\Omega})$ might be smaller than $W^{1,2}_0(\Omega)$. | |
| May 29, 2019 at 0:06 | comment | added | mme | Sure, the trace (restriction to boundary) map $r: C^\infty(\overline \Omega) \to C^\infty(\partial \Omega)$ is continuous when you equip the domain with the $W^{1,2}$ norm and the codomain with the $L^2$ norm, or when you equip both with the $C^0$ norm, and hence extends to a continuous map $(W^{1,2} \cap C^0)(\overline \Omega) \to (L^2 \cap C^0)(\partial \Omega)$. Because the map $r$ vanishes on the set of compactly supported smooth functions in $\Omega$, it also vanishes on its closure. The continuity in $C^0$ is obvious, while the continuity in $W^{1,2} \to L^2$ is in many PDE books. | |
| May 29, 2019 at 0:04 | history | edited | asv | CC BY-SA 4.0 |
added 5 characters in body; added 12 characters in body
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| May 28, 2019 at 22:00 | history | asked | asv | CC BY-SA 4.0 |