Timeline for Critical elliptic equation; kernel of linearized operator
Current License: CC BY-SA 3.0
6 events
| when toggle format | what | by | license | comment | |
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| Jul 19, 2013 at 3:24 | history | edited | Craig | CC BY-SA 3.0 |
added 3331 characters in body
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| Jul 18, 2013 at 21:36 | comment | added | Craig | Igor, I am not sure of the correct way of saying this in words... $w_\lambda(x)$ is a dilated version of $w(x)$ and one can check that $-\Delta w_\lambda(x)=w_\lambda(x)^p$. Also $ w_1(x)=w(x)$ and hence taking derivative in $ \lambda$ and setting $ \lambda=1$ shows that $ \partial_\lambda w_\lambda(x)|_{\lambda=1}$ is in the kernel of $L$. I was confused as to whether it was independent of the other terms. | |
| Jul 18, 2013 at 21:31 | comment | added | Craig | Michael. Thanks, that is a nice way to see it. | |
| Jul 18, 2013 at 20:22 | comment | added | Igor Khavkine | Honestly, I'm having trouble following your definition of $\phi_{N+1}$. However, it sounds like you mean the result of the dilation operator acting on the solution $w$. If that's the case, then all of these $\phi_i$ should be in the kernel of $L$. This should follow from the general fact that symmetry generators acting on solutions create linearized solutions, which is fairly straight forward to work out for yourself. | |
| Jul 18, 2013 at 19:33 | comment | added | Michael Renardy | It is easy to see that $\phi_{N+1}$ cannot be a linear combination of the others. Note that $\phi_{N+1}$ is radially symmetric, and all the others have zero average over the sphere. | |
| Jul 18, 2013 at 19:01 | history | asked | Craig | CC BY-SA 3.0 |