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:$J(v) \leq J(u)\text{ for every }u \in V.$

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

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

It 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 τ on V.
3. Show that J is sequentially lower semi-continuous with respect to the topology τ.

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 reflexive Banach space W. In this case the Banach–Alaoglu theorem implies, that any bounded sequence (u_n) in V has a subsequence that converges to some u₀ in W with respect to the weak topology. If V is sequentially closed in W, so that u₀ 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 $\liminf_{n\to\infty} J(x_n) \geq J(y)$.

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 α > 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 Ω 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 Ω has a C² 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 spaces. 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

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 $a\in L ^q(\Omega, \mathbb{R}^m)$ 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

1. ^ Dacorogna, pp. 1–43.
2. ^ Calculus of Variations. Dover Publications. 1991. ISBN 978-0-486-41448-5. 
3. ^ Dacorogna, pp. 74–79.
4. ^ Dacorogna, pp. 66–74.
5. ^ 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.