Direct method in the calculus of variations

In mathematics, the direct method in the calculus of variations is a general method for constructing a proof of the existence of a minimizer for a given functional,^[1] introduced by Stanisław 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 [math]\displaystyle{ J:V \to \bar{\mathbb{R}} }[/math], where [math]\displaystyle{ V }[/math] is some function space and [math]\displaystyle{ \bar{\mathbb{R}} = \mathbb{R} \cup \{\infty\} }[/math]. The main interest of the subject is to find minimizers for such functionals, that is, functions [math]\displaystyle{ v \in V }[/math] such that:[math]\displaystyle{ J(v) \leq J(u)\forall u \in V. }[/math]

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 [math]\displaystyle{ J }[/math] must be bounded from below to have a minimizer. This means

[math]\displaystyle{ \inf\{J(u)|u\in V\} \gt -\infty.\, }[/math]

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

The direct method may be broken into the following steps

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

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

The function [math]\displaystyle{ J }[/math] is sequentially lower-semicontinuous if

[math]\displaystyle{ \liminf_{n\to\infty} J(u_n) \geq J(u_0) }[/math] for any convergent sequence [math]\displaystyle{ u_n \to u_0 }[/math] in [math]\displaystyle{ V }[/math].

The conclusions follows from

[math]\displaystyle{ \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\} }[/math],

in other words

[math]\displaystyle{ J(u_0) = \inf\{J(u)|u\in V\} }[/math].

Details

Banach spaces

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

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

The second part is usually accomplished by showing that [math]\displaystyle{ J }[/math] admits some growth condition. An example is

[math]\displaystyle{ J(x) \geq \alpha \lVert x \rVert^q - \beta }[/math] for some [math]\displaystyle{ \alpha \gt 0 }[/math], [math]\displaystyle{ q \geq 1 }[/math] and [math]\displaystyle{ \beta \geq 0 }[/math].

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

[math]\displaystyle{ J(u) = \int_\Omega F(x, u(x), \nabla u(x))dx }[/math]

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

When deriving the Euler–Lagrange equation, the common approach is to assume [math]\displaystyle{ \Omega }[/math] has a [math]\displaystyle{ C^2 }[/math] boundary and let the domain of definition for [math]\displaystyle{ J }[/math] be [math]\displaystyle{ C^2(\Omega, \mathbb{R}^m) }[/math]. 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 [math]\displaystyle{ W^{1,p}(\Omega, \mathbb{R}^m) }[/math] with [math]\displaystyle{ p \gt 1 }[/math], which is a reflexive Banach space. The derivatives of [math]\displaystyle{ u }[/math] in the formula for [math]\displaystyle{ J }[/math] must then be taken as weak derivatives.

Another common function space is [math]\displaystyle{ W^{1,p}_g(\Omega, \mathbb{R}^m) }[/math] which is the affine sub space of [math]\displaystyle{ W^{1,p}(\Omega, \mathbb{R}^m) }[/math] of functions whose trace is some fixed function [math]\displaystyle{ g }[/math] in the image of the trace operator. This restriction allows finding minimizers of the functional [math]\displaystyle{ J }[/math] that satisfy some desired boundary conditions. This is similar to solving the Euler–Lagrange equation with Dirichlet boundary conditions. Additionally there are settings in which there are minimizers in [math]\displaystyle{ W^{1,p}_g(\Omega, \mathbb{R}^m) }[/math] but not in [math]\displaystyle{ W^{1,p}(\Omega, \mathbb{R}^m) }[/math]. The idea of solving minimization problems while restricting the values on the boundary can be further generalized by looking on function spaces where the trace is fixed only on a part of the boundary, and can be arbitrary on the rest.

The next section presents 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

[math]\displaystyle{ J(u) = \int_\Omega F(x, u(x), \nabla u(x))dx }[/math],

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

In general one has the following:^[3]

Assume that [math]\displaystyle{ F }[/math] is a function that has the following properties:

1. The function [math]\displaystyle{ F }[/math] is a Carathéodory function.
2. There exist [math]\displaystyle{ a\in L^q(\Omega, \mathbb{R}^{mn}) }[/math] with Hölder conjugate [math]\displaystyle{ q = \tfrac{p}{p-1} }[/math] and [math]\displaystyle{ b \in L^1(\Omega) }[/math] such that the following inequality holds true for almost every [math]\displaystyle{ x \in \Omega }[/math] and every [math]\displaystyle{ (y, A) \in \mathbb{R}^m \times \mathbb{R}^{mn} }[/math]: [math]\displaystyle{ F(x, y, A) \geq \langle a(x) , A \rangle + b(x) }[/math]. Here, [math]\displaystyle{ \langle a(x) , A \rangle }[/math] denotes the Frobenius inner product of [math]\displaystyle{ a(x) }[/math] and [math]\displaystyle{ A }[/math] in [math]\displaystyle{ \mathbb{R}^{mn} }[/math]).

If the function [math]\displaystyle{ A \mapsto F(x, y, A) }[/math] is convex for almost every [math]\displaystyle{ x \in \Omega }[/math] and every [math]\displaystyle{ y\in \mathbb{R}^m }[/math],

then [math]\displaystyle{ J }[/math] is sequentially weakly lower semi-continuous.

When [math]\displaystyle{ n = 1 }[/math] or [math]\displaystyle{ m = 1 }[/math] the following converse-like theorem holds^[4]

Assume that [math]\displaystyle{ F }[/math] is continuous and satisfies

[math]\displaystyle{ | F(x, y, A) | \leq a(x, | y |, | A |) }[/math]

for every [math]\displaystyle{ (x, y, A) }[/math], and a fixed function [math]\displaystyle{ a(x, |y|, |A|) }[/math] increasing in [math]\displaystyle{ |y| }[/math] and [math]\displaystyle{ |A| }[/math], and locally integrable in [math]\displaystyle{ x }[/math]. If [math]\displaystyle{ J }[/math] is sequentially weakly lower semi-continuous, then for any given [math]\displaystyle{ (x, y) \in \Omega \times \mathbb{R}^m }[/math] the function [math]\displaystyle{ A \mapsto F(x, y, A) }[/math] is convex.

In conclusion, when [math]\displaystyle{ m = 1 }[/math] or [math]\displaystyle{ n = 1 }[/math], the functional [math]\displaystyle{ J }[/math], assuming reasonable growth and boundedness on [math]\displaystyle{ F }[/math], is weakly sequentially lower semi-continuous if, and only if the function [math]\displaystyle{ A \mapsto F(x, y, A) }[/math] is convex.

However, there are many interesting cases where one cannot assume that [math]\displaystyle{ F }[/math] is convex. The following theorem^[5] proves sequential lower semi-continuity using a weaker notion of convexity:

Assume that [math]\displaystyle{ F: \Omega \times \mathbb{R}^m \times \mathbb{R}^{mn} \to [0, \infty) }[/math] is a function that has the following properties:

1. The function [math]\displaystyle{ F }[/math] is a Carathéodory function.
2. The function [math]\displaystyle{ F }[/math] has [math]\displaystyle{ p }[/math]-growth for some [math]\displaystyle{ p\gt 1 }[/math]: There exists a constant [math]\displaystyle{ C }[/math] such that for every [math]\displaystyle{ y \in \mathbb{R}^m }[/math] and for almost every [math]\displaystyle{ x \in \Omega }[/math] [math]\displaystyle{ | F(x, y, A) | \leq C(1+|y|^p + |A|^p) }[/math].
3. For every [math]\displaystyle{ y \in \mathbb{R}^m }[/math] and for almost every [math]\displaystyle{ x \in \Omega }[/math], the function [math]\displaystyle{ A \mapsto F(x, y, A) }[/math] is quasiconvex: there exists a cube [math]\displaystyle{ D \subseteq \mathbb{R}^n }[/math] such that for every [math]\displaystyle{ A \in \mathbb{R}^{mn}, \varphi \in W^{1,\infty}_0(\Omega, \mathbb{R}^m) }[/math] it holds:

[math]\displaystyle{ F(x, y, A) \leq |D|^{-1} \int_D F(x, y, A+ \nabla \varphi (z))dz }[/math]

where [math]\displaystyle{ |D| }[/math] is the volume of [math]\displaystyle{ D }[/math].

Then [math]\displaystyle{ J }[/math] is sequentially weakly lower semi-continuous in [math]\displaystyle{ W^{1,p}(\Omega,\mathbb{R}^m) }[/math].

A converse like theorem in this case is the following: ^[6]

Assume that [math]\displaystyle{ F }[/math] is continuous and satisfies

[math]\displaystyle{ | F(x, y, A) | \leq a(x, | y |, | A |) }[/math]

for every [math]\displaystyle{ (x, y, A) }[/math], and a fixed function [math]\displaystyle{ a(x, |y|, |A|) }[/math] increasing in [math]\displaystyle{ |y| }[/math] and [math]\displaystyle{ |A| }[/math], and locally integrable in [math]\displaystyle{ x }[/math]. If [math]\displaystyle{ J }[/math] is sequentially weakly lower semi-continuous, then for any given [math]\displaystyle{ (x, y) \in \Omega \times \mathbb{R}^m }[/math] the function [math]\displaystyle{ A \mapsto F(x, y, A) }[/math] is quasiconvex. The claim is true even when both [math]\displaystyle{ m, n }[/math] are bigger than [math]\displaystyle{ 1 }[/math] and coincides with the previous claim when [math]\displaystyle{ m = 1 }[/math] or [math]\displaystyle{ n = 1 }[/math], since then quasiconvexity is equivalent to convexity.

Notes

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: [math]\displaystyle{ L^p }[/math] Spaces. Springer. ISBN 978-0-387-35784-3. 
• Morrey, C. B., Jr.: Multiple Integrals in the Calculus of Variations. Springer, 1966 (reprinted 2008), Berlin ISBN:978-3-540-69915-6.
• 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. 
• Acerbi Emilio, Fusco Nicola. "Semicontinuity problems in the calculus of variations." Archive for Rational Mechanics and Analysis 86.2 (1984): 125-145

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