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

  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 EulerLagrange 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
  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[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. 143.
  2. I. M. Gelfand, S. V. Fomin (1991). Calculus of Variations. Dover Publications. ISBN 978-0-486-41448-5.
  3. Dacorogna, pp. 7479.
  4. Dacorogna, pp. 6674.
  5. Dacorogna, pp. 87185.

References and further reading

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