Direct method in the calculus of variations explained

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

J:V \to \bar{ R }, where V is some function space and \bar{ R } = \mathbb \cup \ . 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) for all 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 meansinf \{J(u)|u\in V\} > -inf ty.

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 be broken into the following steps

1. Take a minimizing sequence

(u _{n)} for J .

1. 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 .

1. 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\toinfty} 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\toinfty} J(u _{n)} = \lim _{k\to} J(u _{n _{k}} ) \geq J(u _{0)} \geq inf \{J(u)|u\in V\} ,in other wordsJ(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{ R } by showingJ is bounded from below,

1. any minimizing sequence for

J is bounded, andJ is weakly sequentially lower semi-continuous, i.e., for any weakly convergent sequence u _{n} \to u _{0} it holds that \liminf _{n\toinfty} J(u _{n)} \geq J(u _{0)} .The second part is usually accomplished by showing that J admits some growth condition. An example isJ(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 R ^{n} and F is a real-valued function on \Omega x R ^{m} x R ^{mn} . The argument of J is a differentiable function u:\Omega \to 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,} 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, 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.

Another common function space is

W ^{1 ,p} _{g(\Omega,} R ^{m)} which is the affine sub space of W ^{1,p} (\Omega, R ^{m)} of functions whose trace is some fixed function g in the image of the trace operator. This restriction allows finding minimizers of the functional J 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 W ^{1 ,p} _{g(\Omega,} R ^{m)} but not in W ^{1,p} (\Omega, R ^{m)} .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

J(u) = \int _{\Omega} F(x, u(x), \nabla u(x))dx ,where \Omega \subseteq R ^{n} is open, theorems characterizing functions F for which J is weakly sequentially lower-semicontinuous in W ^{1,p} (\Omega, R ^{m)} with p \geq 1 is of great importance.

In general one has the following:^[3]

Assume that

F is a function that has the following properties:

1. The function

F is a Carathéodory function.

1. There exist

a\in L ^{q(\Omega,} R ^{mn} ) with Hölder conjugate q = \tfrac{p}{p -1 } and b \in L ^{1(\Omega)} such that the following inequality holds true for almost every x \in \Omega and every (y, A) \in R ^{m} x R ^{mn} : F(x, y, A) \geq \langle a(x) , A \rangle + b(x) . Here, \langle a(x) , A \rangle denotes the Frobenius inner product of a(x) and A in R ^{mn} ).

If the function

A \mapsto F(x, y, A) is convex for almost every x \in \Omega and every y\in 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, A) | \leq a(x, | y |, | A |)

for every

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

However, there are many interesting cases where one cannot assume that

F is convex. The following theorem^[5] proves sequential lower semi-continuity using a weaker notion of convexity:

Assume that

F: \Omega x R ^{m} x R ^{mn} \to [ 0 , inf ty) is a function that has the following properties:

1. The function

F is a Carathéodory function.

1. The function

F has p -growth for some p> 1 : There exists a constant C such that for every y \in R ^{m} and for almost every x \in \Omega | F(x, y, A) | \leq C( 1+ |y| ^{p} + |A| ^{p)} .

1. For every

y \in R ^{m} and for almost every x \in \Omega , the function A \mapsto F(x, y, A) is quasiconvex: there exists a cube D \subseteq R ^{n} such that for every A \in R ^{mn} , \varphi \in W ^{1 , inf ty} _{0 (\Omega,} R ^{m)} it holds:$$F(x,\; y,\; A)\; \backslash leq\; |D|^\; \backslash int\_D\; F(x,\; y,\; A+\; \backslash nabla\; \backslash varphi\; (z))dz$$

where

|D| is the volume of D .

Then

J is sequentially weakly lower semi-continuous in W ^{1,p} (\Omega, R ^{m)} .

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

Assume that

F is continuous and satisfies| F(x, y, A) | \leq a(x, | y |, | A |)

for every

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

References and further reading

• Book: Dacorogna, Bernard . 1989 . Direct Methods in the Calculus of Variations . Springer-Verlag . 0-387-50491-5 .
• Book: Fonseca, Irene . Irene Fonseca

. Irene Fonseca . Giovanni Leoni . 2007 . Modern Methods in the Calculus of Variations:

L ^{p} Spaces . Springer . 978-0-387-35784-3 .

• Morrey, C. B., Jr.: Multiple Integrals in the Calculus of Variations. Springer, 1966 (reprinted 2008), Berlin .
• Jindřich Nečas: Direct Methods in the Theory of Elliptic Equations. (Transl. from French original 1967 by A.Kufner and G.Tronel), Springer, 2012, .
• News: T. Roubíček. Direct method for parabolic problems. Adv. Math. Sci. Appl.. 10. 2000. 57–65. 1769181.
• Acerbi Emilio, Fusco Nicola. "Semicontinuity problems in the calculus of variations." Archive for Rational Mechanics and Analysis 86.2 (1984): 125-145

Notes and References

1. Dacorogna, pp. 1 - 43.
2. Book: Calculus of Variations . I. M. Gelfand . S. V. Fomin . 1991 . Dover Publications . 978-0-486-41448-5.
3. Dacorogna, pp. 74 - 79.
4. Dacorogna, pp. 66 - 74.
5. Acerbi-Fusco
6. Dacorogna, pp. 156.