Timeline for Dirichlet to Neumann operator for a nonlocal ODE
Current License: CC BY-SA 4.0
7 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Dec 13, 2021 at 20:22 | vote | accept | Laithy | ||
| Dec 13, 2021 at 20:17 | comment | added | Laithy | The goal hasn't changed. I was asking for a different approach for the $\textbf{same}$ thing. Anyhow, thank you for your answer. | |
| Dec 13, 2021 at 20:04 | comment | added | Willie Wong | on the interval $[0,\infty)$ with boundary conditions at $0$, and require $g(r) \sim 1/ r^2$ as $r\nearrow \infty$, you will find that the Dirichlet-Newmann map will have norm independent of $\ell$ (in fact, equal 1). So there is a possibility that the norm you are looking for can be sensitive to the precise terms that show up. | |
| Dec 13, 2021 at 19:59 | comment | added | Willie Wong | (a) I don't like to deal with moving goal posts. (b) It is easy enough to write down a homogeneous ODE of the form $p(r) f''(r) + 2 q(r) f'(r) + h(\ell,r) f(r) = 0$, depending on the parameter $\ell$, whose solutions look like $\frac{1}{(r+1)(1 + r^2)^{\ell/2}}$. We can even guarantee that $p(r) \sim r^2$ and $q(r) \sim r$ as $r\nearrow \infty$, and that the leading order term of $h$ behaves like $-\ell(\ell+1)$. So something qualitatively similar to what you have. If you try to solve the equation $$p(r) f'' + 2q(r) f' + h(\ell,r) f = (f(0) + f'(0)) g(r) $$ ... | |
| Dec 13, 2021 at 19:23 | comment | added | Laithy | The ODE that I am actually working with is more complicated (though still linear with explicit coefficients), and it will be very difficult to apply this approach. Is there a way to prove this $O(\ell)$ result without using the simplicity of the ODE? | |
| Dec 13, 2021 at 18:52 | history | edited | Willie Wong | CC BY-SA 4.0 |
added 500 characters in body
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| Dec 13, 2021 at 18:34 | history | answered | Willie Wong | CC BY-SA 4.0 |