Direct method in the calculus of variations

In the calculus of variations, a topic in mathematics, the direct method is a general method for constructing a proof of the existence of a minimizer for a given functional, introduced by Zaremba and David Hilbert around 1900. The method relies on methods of functional analysis and topology. As well as being used to prove the existence of a solution, direct methods may be used to compute the solution to desired accuracy.

The method

The calculus of variations deals with functionals $J:V \to \bar{\mathbb{R}}$, where $V$ is some function space and $\bar{\mathbb{R}} = \mathbb{R} \cup \{\infty\}$. The main interest of the subject is to find minimizers for such functionals, that is, functions $v \in V$ such that:$J(v) \leq J(u)\forall u \in V.$

The standard tool for obtaining necessary conditions for a function to be a minimizer is the Euler–Lagrange equation. But seeking a minimizer amongst functions satisfying these may lead to false conclusions if the existence of a minimizer is not established beforehand.

The functional $J$ must be bounded from below to have a minimizer. This means

$\inf\{J(u)|u\in V\} > -\infty.\,$

This condition is not enough to know that a minimizer exists, but it shows the existence of a minimizing sequence, that is, a sequence $(u_n)$ in $V$ such that $J(u_n) \to \inf\{J(u)|u\in V\}.$

The direct method may broken into the following steps

1. Take a minimizing sequence $(u_n)$ for $J$.
2. Show that $(u_n)$ admits some subsequence $(u_{n_k})$, that converges to a $u_0\in V$ with respect to a topology $\tau$ on $V$.
3. Show that $J$ is sequentially lower semi-continuous with respect to the topology $\tau$.

To see that this shows the existence of a minimizer, consider the following characterization of sequentially lower-semicontinuous functions.

The function $J$ is sequentially lower-semicontinuous if

$\liminf_{n\to\infty} J(u_n) \geq J(u_0)$ for any convergent sequence $u_n \to u_0$ in $V$.

The conclusions follows from

$\inf\{J(u)|u\in V\} = \lim_{n\to\infty} J(u_n) = \lim_{k\to \infty} J(u_{n_k}) \geq J(u_0) \geq \inf\{J(u)|u\in V\}$,

in other words

$J(u_0) = \inf\{J(u)|u\in V\}$.

Details

Banach spaces

The direct method may often be applied with success when the space $V$ is a subset of a separable reflexive Banach space $W$. In this case the sequential Banach–Alaoglu theorem implies that any bounded sequence $(u_n)$ in $V$ has a subsequence that converges to some $u_0$ in $W$ with respect to the weak topology. If $V$ is sequentially closed in $W$, so that $u_0$ is in $V$, the direct method may be applied to a functional $J:V\to\bar{\mathbb{R}}$ by showing

1. $J$ is bounded from below,
2. any minimizing sequence for $J$ is bounded, and
3. $J$ is weakly sequentially lower semi-continuous, i.e., for any weakly convergent sequence $u_n \to u_0$ it holds that $\liminf_{n\to\infty} J(u_n) \geq J(u_0)$.

The second part is usually accomplished by showing that $J$ admits some growth condition. An example is

$J(x) \geq \alpha \lVert x \rVert^q - \beta$ for some $\alpha > 0$, $q \geq 1$ and $\beta \geq 0$.

A functional with this property is sometimes called coercive. Showing sequential lower semi-continuity is usually the most difficult part when applying the direct method. See below for some theorems for a general class of functionals.

Sobolev spaces

The typical functional in the calculus of variations is an integral of the form

$J(u) = \int_\Omega F(x, u(x), \nabla u(x))dx$

where $\Omega$ is a subset of $\mathbb{R}^n$ and $F$ is a real-valued function on $\Omega \times \mathbb{R}^m \times \mathbb{R}^{mn}$. The argument of $J$ is a differentiable function $u:\Omega \to \mathbb{R}^m$, and its Jacobian $\nabla u(x)$ is identified with a $mn$-vector.

When deriving the Euler–Lagrange equation, the common approach is to assume $\Omega$ has a $C^2$ boundary and let the domain of definition for $J$ be $C^2(\Omega, \mathbb{R}^m)$. This space is a Banach space when endowed with the supremum norm, but it is not reflexive. When applying the direct method, the functional is usually defined on a Sobolev space $W^{1,p}(\Omega, \mathbb{R}^m)$ with $p > 1$, which is a reflexive Banach space. The derivatives of $u$ in the formula for $J$ must then be taken as weak derivatives. The next section presents two theorems regarding weak sequential lower semi-continuity of functionals of the above type.

Sequential lower semi-continuity of integrals

As many functionals in the calculus of variations are of the form

$J(u) = \int_\Omega F(x, u(x), \nabla u(x))dx$,

where $\Omega \subseteq \mathbb{R}^n$ is open, theorems characterizing functions $F$ for which $J$ is weakly sequentially lower-semicontinuous in $W^{1,p}(\Omega, \mathbb{R}^m)$ is of great importance.

In general we have the following

Assume that $F$ is a function such that

1. The function $(y, p) \mapsto F(x, y, p)$ is continuous for almost every $x \in \Omega$,
2. the function $x \mapsto F(x, y, p)$ is measurable for every $(y, p) \in \mathbb{R}^m \times \mathbb{R}^{mn}$, and
3. $F(x, y, p) \geq a(x)\cdot p + b(x)$ for a fixed $a\in L ^q(\Omega, \mathbb{R}^{mn})$ where $1/q + 1/p = 1$, a fixed $b \in L^1(\Omega)$, for a.e. $x \in \Omega$ and every $(y, p) \in \mathbb{R}^m \times \mathbb{R}^{mn}$ (here $a(x) \cdot p$ means the inner product of $a(x)$ and $p$ in $\mathbb{R}^{mn}$).

The following holds. If the function $p \mapsto F(x, y, p)$ is convex for a.e. $x \in \Omega$ and every $y\in \mathbb{R}^m$,

then $J$ is sequentially weakly lower semi-continuous.

When $n = 1$ or $m = 1$ the following converse-like theorem holds

Assume that $F$ is continuous and satisfies

$| F(x, y, p) | \leq a(x, | y |, | p |)$

for every $(x, y, p)$, and a fixed function $a(x, y, p)$ increasing in $y$ and $p$, and locally integrable in $x$. It then holds, if $J$ is sequentially weakly lower semi-continuous, then for any given $(x, y) \in \Omega \times \mathbb{R}^m$ the function $p \mapsto F(x, y, p)$ is convex.

In conclusion, when $m = 1$ or $n = 1$, the functional $J$, assuming reasonable growth and boundedness on $F$, is weakly sequentially lower semi-continuous if, and only if, the function $p \mapsto F(x, y, p)$ is convex. If both $n$ and $m$ are greater than 1, it is possible to weaken the necessity of convexity to generalizations of convexity, namely polyconvexity and quasiconvexity.

References and further reading

• Jindřich Nečas: Direct Methods in the Theory of Elliptic Equations. (Transl. from French original 1967 by A.Kufner and G.Tronel), Springer, 2012, ISBN 978-3-642-10455-8.