Variational problem
= Variational problem
A variational problem seeks a stationary point or minimizer of a <functional> on a specified <function space>. For a symmetric <coercive bilinear form> $a$ and bounded <linear functional> $\ell$ on a real <Hilbert space>, minimizing $J(v)=a(v,v)/2-\ell(v)$ is equivalent to the <weak formulation> $a(u,v)=\ell(v)$ for every test vector. The <Lax-Milgram theorem> gives a unique solution, and $J(u+w)-J(u)=a(w,w)/2$ proves unique minimization. Specifying the admissible space and its boundary constraints is part of the problem.