Timeline for Given an eigenvalue equation (elliptic PDE) in a ball $B_R$, prove the convergence of the first nonzero $\lambda_R$ and its eigenfunction $\phi_R$
Current License: CC BY-SA 4.0
6 events
| when toggle format | what | by | license | comment | |
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| Sep 16, 2023 at 13:21 | comment | added | Otis Chodosh | I think you're missing a diagonal argument. | |
| Sep 16, 2023 at 7:23 | comment | added | Elio Li | I have a naive question. We can find a $\phi_{\infty}$, $\phi_R$ is compactly convergent to $\phi_{\infty}$, which means on every compact set the limit functions are the same, how to prove that on every compact set $K$, the limit function $\Phi_K$ is the same, it seems that I misunderstood some essential conceptions, but I can't figure out, could you help me find where I misunderstood? | |
| Sep 5, 2023 at 16:51 | comment | added | Otis Chodosh | Your modified equation is the Euler Lagrange equation of $\int |\nabla u|^2 + h(x) W(u)$ where $W'=F$. So probably most of the basic ideas go through. You should be able to find some papers about this equation. I don't know of any intrinsic reason to consider this modification but there may be physical models where it's natural. | |
| Sep 5, 2023 at 14:44 | comment | added | Elio Li | Really thanks for your answer! I'm reading the papers about the problems related to the 'stable outside a compact set‘ solution, and I notice that the paper in this area always talks about the $\Delta u = F(u)$ type equation, but I hardly see $\Delta u = h(x)F(u)$ type, is it much more difficult to deal with (maybe the second variation of the energy functional is much more complex)? Or study the 'stable outside a compact set‘ solution of this type is less meaningful. | |
| Sep 2, 2023 at 5:38 | vote | accept | Elio Li | ||
| Sep 1, 2023 at 15:51 | history | answered | Otis Chodosh | CC BY-SA 4.0 |