Timeline for Existence and uniqueness for $\Delta f + \lambda f = g$ on $S^2$ for $\lambda>0$ [closed]
Current License: CC BY-SA 4.0
10 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jan 16, 2022 at 1:22 | history | closed |
LeechLattice Eric Peterson Carl-Fredrik Nyberg Brodda Dima Pasechnik Alec Rhea |
Not suitable for this site | |
| Dec 11, 2021 at 2:01 | review | Close votes | |||
| Jan 16, 2022 at 1:22 | |||||
| Dec 11, 2021 at 0:56 | comment | added | Laithy | Thank you for your replies! :) | |
| Dec 11, 2021 at 0:03 | comment | added | Jochen Glueck | @Laithy: Yes, $H^2(S)$ is the right domain. The estimate you mention is also true. if $-\lambda$ is not in the spectrum, then $(\Delta + \lambda)^{-1}$ is a bounded operator from $L^2$ to $H^2$. | |
| Dec 10, 2021 at 21:05 | comment | added | Laithy | Oh thank you. I forgot the definition of the spectrum. Domain of $\Delta$ can be chosen to be $H^2(S^2)$, correct? Also, do we get the estimate $||f||_{H^2} \leq C ||g||_{L^2}$? | |
| Dec 10, 2021 at 20:55 | comment | added | Jochen Glueck | @Laithy: The operator $\Delta + \lambda: \operatorname{dom}(\Delta) \to L^2$ is bijective if and only if $-\lambda$ is not in the spectrum of $\Delta$. This is immediate from the definition of the spectrum. (I think this question would have been a better fit for Mathematics StackExchange.) | |
| Dec 10, 2021 at 19:30 | comment | added | Laithy | So what does that mean in terms of the existence and uniqueness of that PDE? | |
| Dec 10, 2021 at 19:29 | comment | added | paul garrett | As @JochenGlueck says... and the same applies to any compact Riemannian manifold, for the same reason. Of course, in general, it's less explicit. | |
| Dec 10, 2021 at 19:15 | comment | added | Jochen Glueck | Since $\Delta$ has compact resolvents, the only spectral values of $\Delta$ are eigenvalues. | |
| Dec 10, 2021 at 19:02 | history | asked | Laithy | CC BY-SA 4.0 |