Timeline for Dense set in Sobolev space ${H^1}\left( {0,1} \right)$
Current License: CC BY-SA 4.0
10 events
| when toggle format | what | by | license | comment | |
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| Sep 13, 2019 at 23:41 | answer | added | Pietro Majer | timeline score: 3 | |
| Sep 13, 2019 at 23:37 | vote | accept | Trần Quang Minh | ||
| Sep 13, 2019 at 23:34 | answer | added | Jochen Glueck | timeline score: 6 | |
| Sep 13, 2019 at 19:50 | history | edited | Jochen Glueck | CC BY-SA 4.0 |
Corrected a few typos.
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| Sep 13, 2019 at 19:40 | comment | added | Jochen Glueck | Concerning your question itself: Yes, $V$ is dense in $H^1(0,1)$, but the proof that I just came up with is quite weird (the Laplace operator with the same boundary conditions can be define via form methods with form domain $H^1(0,1)$ and domain $V$; then one uses the spectral theorem for self-adjoint operators to see that the domain is dense in the form domain). I'll try to find a simpler proof (I strongly suspect that there is one); if I don't succeed, I'll post the details of the proof that I just sketched. | |
| Sep 13, 2019 at 19:35 | comment | added | Jochen Glueck | Thanks for your response! I slightly changed the wording of your post and replaced the imperative with a question. | |
| Sep 13, 2019 at 19:33 | history | edited | Jochen Glueck | CC BY-SA 4.0 |
Removed imperative mood and added PDE tag.
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| Sep 13, 2019 at 14:52 | comment | added | Trần Quang Minh | It's not homework problem. I consider the nonlinear wave equation with boundary conditions $${u_x}\left( {0,t} \right) - u\left( {0,t} \right) = {u_x}\left( {1,t} \right) + u\left( {1,t} \right) = 0,$$ and inital conditions $$u\left( {x,0} \right) = {u_0}\left( x \right),\,{u_t}\left( {x,0} \right) = {u_1}\left( x \right).$$ I proved the existence of weak solution with ${u_0} \in V$. Now i want to prove the existence of weak solution with ${u_0} \in {H^1}$ by by a density argument. | |
| Sep 13, 2019 at 10:29 | comment | added | Jochen Glueck | Welcome to MathOverflow! May I ask what the background of this question is? I'm asking because the imperative mood in which the question is phrased suggests that it might be a homework problem. | |
| Sep 13, 2019 at 7:02 | history | asked | Trần Quang Minh | CC BY-SA 4.0 |