Timeline for Spectrum of a linear elliptic operator
Current License: CC BY-SA 4.0
6 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| May 29, 2019 at 22:32 | comment | added | asv | @WillieWong : Many thanks. Seems to work. | |
| May 29, 2019 at 21:18 | comment | added | Willie Wong | Gagliardo-Nirenberg-Sobolev on $\mathbb{R}^4$ states that $\| u\|_{L^4} \leq C \|\partial u\|_{L^2}$ (among other things). en.wikipedia.org/wiki/… | |
| May 29, 2019 at 17:59 | comment | added | asv | @WillieWong: I am not sure how you use Sobolev theorem and get rid of the term $\alpha$ without derivatives. Are you using also some version of Poincaré inequality in $\mathbb{R}^n$? | |
| May 29, 2019 at 2:19 | comment | added | Willie Wong | Consider the corresponding quadratic form $\int |\nabla \alpha|^2 + A_\mu \partial_\mu \alpha \cdot \alpha$. In $\mathbb{R}^4$ by Sobolev $W^{1,2} \to L^4$ you can bound $\int A_\mu \partial_\mu \alpha \cdot \alpha$ by $C \|A_\mu\|_{L^4} \|\partial \alpha\|_{L^2}^2$. So provided that $\|A_\mu\|_{L^4} < C^{-1}$ you have that the quadratic form is positive semidefinite. | |
| May 28, 2019 at 22:02 | history | edited | asv | CC BY-SA 4.0 |
edited title
|
| May 28, 2019 at 21:44 | history | asked | asv | CC BY-SA 4.0 |