Calculus of variations

Definition

Let $\Omega \subseteq \R^n$ be a bounded open set and let $L \colon \Omega \times \R \times \R^n \to \R$ be a continuous function, the Lagrangian, written $L(x, z, p)$. Let $\mathcal{A}$ be a class of functions $u \colon \Omega \to \R$ with prescribed values $u = g$ on $\partial\Omega$, for which the integral below makes sense. For example, $\mathcal{A}$ may lie in $C^1(\overline{\Omega})$ or in a Sobolev space $W^{1,q}(\Omega)$. The integral functional with Lagrangian $L$ is $$J(u) = \int_\Omega L\big(x, u(x), \nabla u(x)\big) \, dx, \qquad u \in \mathcal{A}.$$ A variational problem asks for a minimiser of $J$ on $\mathcal{A}$, or more generally for its critical points. A minimiser is a $u_0 \in \mathcal{A}$ with $J(u_0) \le J(u)$ for all $u \in \mathcal{A}$. The calculus of variations is the study of such problems: existence of minimisers, necessary and sufficient conditions for a minimum, and properties of minimisers.