Solution
= Solution
For any zero-endpoint admissible $u$, use $|\sin u|\leq|u|$ and the sharp <Poincaré inequality> on $(0,1)$:
$$
I(u)\geq\frac12\int_0^1(u')^2-\int_0^1u^2
\geq\left(\frac12-\frac1{\pi^2}\right)\int_0^1(u')^2\geq0.
$$
The coefficient is positive, and equality forces $u'=0$, hence $u=0$ by its endpoint values. Thus
$$
\boxed{u=0\text{ is the unique strict global minimizer}.}
$$
It is therefore \b[both a strict strong and a strict weak local minimizer].