Timeline for Sobolev imbedding result; proof
Current License: CC BY-SA 4.0
9 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jun 18, 2021 at 16:46 | comment | added | Math604 | okay. thanks for your response. | |
| Jun 18, 2021 at 13:26 | comment | added | Willie Wong | In your case $\nabla_\theta$ is not the unit angular derivative, it is the angular derivatived weighted by distance. Note that $$ \int_{B_1} |\nabla u|^2 ~dx = \int_0^1\int_0^{2\pi} (|\partial_r u|^2 + \frac{1}{r^2} |\partial_\theta u|^2 ) r ~d\theta ~dr \neq \int_0^1 \int_0^{2\pi} (|\partial_r u|^2 + |\partial_\theta u|^2) r ~d\theta ~dr $$ | |
| Jun 17, 2021 at 21:23 | comment | added | Math604 | i ran out of room. So we gained a dimension and hence have worse imbedding... i know i must be being stupid here...but... | |
| Jun 17, 2021 at 21:22 | comment | added | Math604 | Hi Willie. I have a question related to your answer that seems to be bothering me. Its this idea of viewing stuff using a Sobolev imbedding on the product space; i seem to (or actually gain) some dimensions when i change to polar coordinates (i fully realize this isn't what happened in what you did though). So for a trivial case lets assume $H_0^1(B_1)$ imbedding in $L^p(B_1)$ where $B_1$ is the unit ball in $ R^2$. Once we change to polar coordinates we can view it as a product with measure $r^{2-1} \theta^{1-1} dr d \theta$ and then count the dimensions as $ 2+1=3$ which clearly we | |
| May 12, 2021 at 6:12 | vote | accept | Math604 | ||
| May 12, 2021 at 6:12 | comment | added | Math604 | Sorry for the delay. I think i understand your proof and also your scaling (I think my problem with the scaling was that I only concentrated in $ t $ but didn't even touch $s$ which clearly lost me some). Your compactness proof i mostly can follow. Thanks a bunch for your answer; it helps me a lot. | |
| May 7, 2021 at 17:26 | comment | added | Math604 | thank you very much for the comments and answers. It will take me a while to digest (I had also scaled in $ t $ like you did and I thought i was getting the $\frac{2n}{n-2}$ but I guess i made a mistake). I will look at both the scaling and compactness proof and get back to you. thanks | |
| May 7, 2021 at 13:53 | history | edited | Willie Wong | CC BY-SA 4.0 |
added 1774 characters in body
|
| May 7, 2021 at 13:26 | history | answered | Willie Wong | CC BY-SA 4.0 |