Timeline for About a completion of a Sobolev space
Current License: CC BY-SA 3.0
6 events
| when toggle format | what | by | license | comment | |
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| Jan 30, 2015 at 12:54 | comment | added | Jean Van Schaftingen | @jamesC If you complete a space with respect to the norm (2.15), then the restriction of the norm on your completed to your original space is (2.15). For other points in your completed space, the norm can be computed by taking the limit of (2.15) on a Cauchy sequence. | |
| Jan 29, 2015 at 12:27 | comment | added | riem | @JeanVanSchaftingen is it really the case that the norm in the completion space is given by the natural extension of expression (2.15) in the op? | |
| Jan 22, 2015 at 14:41 | comment | added | jamesC | Hmm, I think you're right, we have $\int_{\mathcal{C}}|\nabla u_n|^2 = |\Omega|\frac{1}{n}$. | |
| Jan 22, 2015 at 14:16 | comment | added | Jean Van Schaftingen | I think that your sequence is Cauchy: all the traces are equal and $\int_{\mathcal{C}} \vert \nabla u_n \vert^2 = O (n)$ as $n \to \infty$. | |
| Jan 22, 2015 at 13:07 | comment | added | jamesC | Thanks for the answer. I agree with your first paragraph. If I understand you, you propose: 1) Define $H^\epsilon$ as the completion given in the OP. 2) Extend $tr:H^1 \to L^2$ to an operator $T:H^{\epsilon} \to L^2$ (done via limits of Cauchy sequences and density). 3) Find a sequence $u_n$ which is Cauchy in $H^\epsilon$-norm with $tr(u_n) = c$ and which converges in some sense to the constant function $c$. Unfortunately the sequence I listed in the OP is not Cauchy, but presumably another one can be constructed... | |
| Jan 22, 2015 at 9:29 | history | answered | Jean Van Schaftingen | CC BY-SA 3.0 |