Past exam of the partial differential equations for pure and applied mathematicians and calculus of variations course of the University of Oxford / 2008 / B / 6 / b / ii / Solution

The inequality $\cos t\ge 1-t^2/2$ givesThe elementary endpoint estimate $|u(x){|}^2\le x{\int }_0^1{u}^{\prime 2}$ implies $\int u^2\le \frac{1}{2}\int u^{\prime 2}$. ConsequentlyThus zero is the unique global minimizer, and in particular a strict strong and strict weak local minimizer.