= Minimum Dirichlet energy at fixed L2 norm on an interval
{title2=$\min_{y\in H_0^1(0,L),\ \|y\|_2=1}\int_0^L|y'|^2=\pi^2/L^2$}
Expand a real function with zero endpoint values in its <Fourier sine series>, $y=\sum_{n\geq1}a_n\sin(n\pi x/L)$. The <Parseval identity> gives $\int y^2=(L/2)\sum a_n^2$. For its <weak derivative>, the <Fourier cosine series> has coefficients $(n\pi/L)a_n$ and zero constant term, so
$$
\int_0^L|y'|^2\,dx=\frac L2\sum_{n\geq1}\left(\frac{n\pi}L\right)^2a_n^2
\geq\frac{\pi^2}{L^2}\int_0^Ly^2\,dx.
$$
The derivative coefficient identity follows by <integration by parts>, rather than by assuming pointwise differentiation of an arbitrary series. This proves the <Wirtinger inequality> and its equality case. Under unit <L2 norm>, the only real minimizers are $y=\pm\sqrt{2/L}\sin(\pi x/L)$.
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