Past exam of the calculus of variations course of the University of Oxford / 2009 / 3 / b / ii / Solution

Zero is neither a weak nor a strong local minimizer. For any nonzero smooth zero-trace $\phi$, Taylor expansion givesThe mixed integral $\int \phi {\phi }^{\prime }=0$ by the endpoint values. Writing $D=\int {\phi }^{\prime 2}$ and $U=\int {\phi }^2$, the quadratic coefficient is $-3D/8+U/2$. The elementary estimate $U\le D/2$ on $[0,1]$ makes this at most $-D/8<0$. Thus arbitrarily small variations decrease the energy. The separate coefficient one on $u^2$ is retained in this calculation.