Pizzadililehz

## Answered question

2022-03-27

How do I get an estimate for this nonlocal ODE?
Consider the following nonlocal ODE on $\left[1,\mathrm{\infty }\right)$:

$f\left(1\right)=\alpha$
$\underset{r\to \mathrm{\infty }}{lim}f\left(r\right)=0$
where l is a positive integer and $\alpha$ is a real number.
Define the following norm $‖·‖$
$‖f{‖}^{2}:={\int }_{1}^{\infty }{r}^{2}f\text{'}\left(r{\right)}^{2}dr+l\left(l+1\right){\int }_{1}^{\infty }f\left(r{\right)}^{2}dr$
I want to prove the estimate:
$‖f‖\le C\sqrt{l\left(l+1\right)}|\alpha |$
for some constant C independent of $\alpha$, l and f. But I am stuck.
Here is what I tried. Multiply both sides by f and integrate by parts to get:

where I used Cauchy-Schwartz in the before last line. I am not sure how to continue and how to get rid of the f'(1) term.
Any help is appreciated.

### Answer & Explanation

Jonathon Hanson

Beginner2022-03-28Added 9 answers

Step 1
Here is how we deal with the f'(1) term:

$\begin{array}{rl}{f}^{\prime }\left(1\right)& =-{\int }_{1}^{\mathrm{\infty }}{f}^{″}\left(r\right)dr\\ & =-{\int }_{1}^{\mathrm{\infty }}\left[-\frac{f\left(1\right)+{f}^{\prime }\left(1\right)}{{r}^{4}}+\frac{l\left(l+1\right)}{{r}^{2}}f\left(r\right)-\frac{1}{r}{f}^{\prime }\left(r\right)\right]dr\end{array}$

Step 2
Since ${\int }_{1}^{\mathrm{\infty }}\frac{1}{{r}^{4}}dr<1$, we can just solve for f'(1) and use Cauchy Schwartz to estimate it by |f(1)| and ∥f∥. Then I can easily get the estimate I wanted.

Do you have a similar question?

Recalculate according to your conditions!

Ask your question.
Get an expert answer.

Let our experts help you. Answer in as fast as 15 minutes.

Didn't find what you were looking for?