Timeline for are these norms equivalent?
Current License: CC BY-SA 3.0
7 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Sep 28, 2016 at 9:28 | comment | added | Fedor Petrov | At first, you miss $\int$ in your formula for the norm. At second, your norm is Hilbert norm for a positive-definite inner product $\langle u,v\rangle=\int \sum_{i,j=1}^{n}a_{ij}\frac{\partial u}{\partial x_i}\frac{\partial v}{\partial x_j}$. This guarantees the triangle inequality. | |
| Sep 28, 2016 at 8:44 | comment | added | Alexander | I just noticed that the triangle inequality for the 'norm' on $W_0^{1,2}(\Omega)$ defined by $||u||_{1,2}=(\sum_{i,j=1}^{n}a_{ij}\frac{\partial u}{\partial x_i}\frac{\partial u}{\partial x_j})^{1/2}$ is the difficult part. There are two extra terms which creeps in and that is what is the root cause of the difficulty. Can you please at least give me a hint so that I can go ahead?. | |
| Sep 28, 2016 at 8:30 | comment | added | Igor Khavkine | Ah, yes, Fedor is right. Since the inequality you gave actually bounds $a_{ij}$ from below, the operator norm of $(a^{-1})_{ij}$ (the components of the inverse matrix, not the inverses of the matrix elements) should be bounded by $1/\alpha^2$ (and maybe a constant depending on $n$). | |
| Sep 28, 2016 at 8:12 | comment | added | Fedor Petrov | Yes, since also $\sum_{i,j=1}^{n}a_{ij}\xi_i\xi_j\leq \beta^2|\xi|^2$ for large enough $\beta$ depending on $n$ and suprema of $a_{ij}$'s. | |
| Sep 28, 2016 at 7:32 | comment | added | Alexander | No information is available on the $1/a_{ij}$'s though. | |
| Sep 28, 2016 at 7:10 | comment | added | Igor Khavkine | Is $(a^{-1})_{ij}$ also bounded and measurable? | |
| Sep 28, 2016 at 6:46 | history | asked | Alexander | CC BY-SA 3.0 |