Minimum Dirichlet energy at fixed L2 norm on an interval (${\min }_{y\in H_0^1(0,L),\Vert y{\Vert }_2=1}{\int }_0^L|y^{\prime }{|}^2={\pi }^2/L^2$)

Expand a real function with zero endpoint values in its Fourier sine series, $y={\sum }_{n\ge 1}{a}_n\sin (n\pi x/L)$. 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, soThe 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 $y=\pm \sqrt{2/L}\sin (\pi x/L)$.