How do I get an estimate for this



Answered question


How do I get an estimate for this nonlocal ODE?
Consider the following nonlocal ODE on [1,):
r2f (r)+2rf(r)l(l+1)f(r)=(f(1)+f(1))r2
where l is a positive integer and α is a real number.
Define the following norm ·
I want to prove the estimate:
for some constant C independent of α, l and f. But I am stuck.
Here is what I tried. Multiply both sides by f and integrate by parts to get:
f2=1r2f2+1l(l+1)f2 =f(1)f(1)+(f(1)+f(1))1fr2  f(1)α+(f(1)+α)31f2  f(1)α+(f(1)+α)3f
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

Jonathon Hanson

Beginner2022-03-28Added 9 answers

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

f ( 1 ) = 1 f ( r ) d r = 1 [ f ( 1 ) + f ( 1 ) r 4 + l ( l + 1 ) r 2 f ( r ) 1 r f ( r ) ] d r

Step 2
Since 11r4dr<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.

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