Past exam of the approximation theory and finite element methods course of the University of Oxford / 2008 / B / 6 / a

For any $v\in H^1(\Omega )$, expand $J(u+ϵv)$ in $ϵ$. Its linear term must vanish at a minimizer, givingThus the required functionals areThe form is continuous, symmetric, and coercive, with $a(v,v)=\Vert v{\Vert }_{H^1}^2$. Conversely, this weak equation implies $J(u+w)=J(u)+\frac{1}{2}a(w,w)$, so its unique solution is indeed the global minimizer.