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,^[1] 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.^[2]

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

Take a minimizing sequence $(u_n)$ for $J$ .
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 $V$ .
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

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

$J$ is bounded from below,
any minimizing sequence for $J$ is bounded, and
$J$ 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^[3]

Assume that

F

is a function such that

The function $(y, p) \mapsto F(x, y, p)$ is continuous for almost every $x \in \Omega$ ,
the function $x \mapsto F(x, y, p)$ is measurable for every $(y, p) \in \mathbb{R}^m \times \mathbb{R}^{mn}$ , and
$F(x, y, p) \geq a(x)\cdot p + b(x)$ 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^[4]

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.^[5]

Notes

↑ Dacorogna, pp. 1–43.
↑ I. M. Gelfand, S. V. Fomin (1991). Calculus of Variations. Dover Publications. ISBN 978-0-486-41448-5.
↑ Dacorogna, pp. 74–79.
↑ Dacorogna, pp. 66–74.
↑ Dacorogna, pp. 87–185.

References and further reading

Dacorogna, Bernard (1989). Direct Methods in the Calculus of Variations. Springer-Verlag. ISBN 0-387-50491-5.
Fonseca, Irene; Giovanni Leoni (2007). Modern Methods in the Calculus of Variations: $L^p$ Spaces. Springer. ISBN 978-0-387-35784-3.
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.
T. Roubíček (2000). "Direct method for parabolic problems". Adv. Math. Sci. Appl. 10. pp. 57–65. MR 1769181 (2001e:34126).

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