Solution
= Solution
Equality in the <Cauchy-Schwarz inequality> requires $\sqrt r\,v(r)=\Omega r^{3/2}$ for one constant $\Omega$. Thus $v(r)=\Omega r$ and
$$
\boxed{\omega(r)=\Omega\text{ independent of }r}.
$$
This is <solid-body rotation>. At fixed $L$, the constant is $\Omega=2L/(Va^2)$. The value at the single axis point $r=0$ does not affect the <integrals>; if $\omega$ is continuous, it has the same constant value there as well.