---
tag: 410CS90H3R
kind: definition
name: Calculus of variations
msc: [49-XX, 49K10, 35A15]
defines: PQ25NHV0KS
uses: [M4CEZ5K9HP, 05AXG3QRKR]
sources:
  - url: https://en.wikipedia.org/wiki/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|M4CEZ5K9HP]] $W^{1,q}(\Omega)$.
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.

## Remarks

The same words are used for functionals with higher derivatives, for vector-valued $u$, and for constraints such as fixed volume or fixed $L^2$ norm.

Example: for $n = 2$, the Dirichlet energy $L(x, z, p) = \frac{1}{2}\abs{p}^2$ gives $J(u) = \frac{1}{2} \int_\Omega \abs{\nabla u}^2 \, dx$.
Its minimisers with boundary values $g$ are the harmonic functions with these boundary values, which is Dirichlet's principle.

Example: the area of the graph of $u$ has Lagrangian $L(x, z, p) = \sqrt{1 + \abs{p}^2}$, and its critical points solve the [[minimal surface equation|3GV2E2YTZX]].

Non-example: minimising $u \mapsto \max_{\overline\Omega} \abs{u}$ is an optimisation over functions.
But this functional is not the integral of a Lagrangian, so it is not a variational problem in this sense.

## Explanations

### L1 human

A variational problem asks for the function that makes an integral quantity, such as energy or area, as small as possible.
The calculus of variations is the theory of these problems.