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

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The calculus of variations deals with functionals  , where   is some function space and  . The main interest of the subject is to find minimizers for such functionals, that is, functions   such that   for all  .

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   must be bounded from below to have a minimizer. This means

 

This condition is not enough to know that a minimizer exists, but it shows the existence of a minimizing sequence, that is, a sequence   in   such that  

The direct method may be broken into the following steps

  1. Take a minimizing sequence   for  .
  2. Show that   admits some subsequence  , that converges to a   with respect to a topology   on  .
  3. Show that   is sequentially lower semi-continuous with respect to the topology  .

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

The function   is sequentially lower-semicontinuous if
  for any convergent sequence   in  .

The conclusions follows from

 ,

in other words

 .

Details

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Banach spaces

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The direct method may often be applied with success when the space   is a subset of a separable reflexive Banach space  . In this case the sequential Banach–Alaoglu theorem implies that any bounded sequence   in   has a subsequence that converges to some   in   with respect to the weak topology. If   is sequentially closed in  , so that   is in  , the direct method may be applied to a functional   by showing

  1.   is bounded from below,
  2. any minimizing sequence for   is bounded, and
  3.   is weakly sequentially lower semi-continuous, i.e., for any weakly convergent sequence   it holds that  .

The second part is usually accomplished by showing that   admits some growth condition. An example is

  for some  ,   and  .

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

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The typical functional in the calculus of variations is an integral of the form

 

where   is a subset of   and   is a real-valued function on  . The argument of   is a differentiable function  , and its Jacobian   is identified with a  -vector.

When deriving the Euler–Lagrange equation, the common approach is to assume   has a   boundary and let the domain of definition for   be  . 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   with  , which is a reflexive Banach space. The derivatives of   in the formula for   must then be taken as weak derivatives.

Another common function space is   which is the affine sub space of   of functions whose trace is some fixed function   in the image of the trace operator. This restriction allows finding minimizers of the functional   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   but not in  . 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

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As many functionals in the calculus of variations are of the form

 ,

where   is open, theorems characterizing functions   for which   is weakly sequentially lower-semicontinuous in   with   is of great importance.

In general one has the following:[3]

Assume that   is a function that has the following properties:
  1. The function   is a Carathéodory function.
  2. There exist   with Hölder conjugate   and   such that the following inequality holds true for almost every   and every  :  . Here,   denotes the Frobenius inner product of   and   in  ).
If the function   is convex for almost every   and every  ,
then   is sequentially weakly lower semi-continuous.

When   or   the following converse-like theorem holds[4]

Assume that   is continuous and satisfies
 
for every  , and a fixed function   increasing in   and  , and locally integrable in  . If   is sequentially weakly lower semi-continuous, then for any given   the function   is convex.

In conclusion, when   or  , the functional  , assuming reasonable growth and boundedness on  , is weakly sequentially lower semi-continuous if, and only if the function   is convex.

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

Assume that   is a function that has the following properties:
  1. The function   is a Carathéodory function.
  2. The function   has  -growth for some  : There exists a constant   such that for every   and for almost every    .
  3. For every   and for almost every  , the function   is quasiconvex: there exists a cube   such that for every   it holds:

 

where   is the volume of  .
Then   is sequentially weakly lower semi-continuous in  .

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

Assume that   is continuous and satisfies
 
for every  , and a fixed function   increasing in   and  , and locally integrable in  . If   is sequentially weakly lower semi-continuous, then for any given   the function   is quasiconvex. The claim is true even when both   are bigger than   and coincides with the previous claim when   or  , since then quasiconvexity is equivalent to convexity.

Notes

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

References and further reading

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  • 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:   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. Vol. 10. pp. 57–65. MR 1769181.
  • Acerbi Emilio, Fusco Nicola. "Semicontinuity problems in the calculus of variations." Archive for Rational Mechanics and Analysis 86.2 (1984): 125-145