Bent function

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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¯, where V is some function space and ¯={}. The main interest of the subject is to find minimizers for such functionals, that is, functions vV such that:J(v)J(u) for every uV.

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)|uV}>.

This condition is not enough to know that a minimizer exists, but it shows the existence of a minimizing sequence, that is, a sequence (un) in V such that J(un)inf{J(u)|uV}.

The direct method may broken into the following steps

  1. Take a minimizing sequence (un) for J.
  2. Show that (un) admits some subsequence (unk), that converges to a u0V 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
lim infnJ(un)J(u0) for any convergent sequence unu0 in V.

The conclusions follows from

inf{J(u)|uV}=limnJ(un)=limkJ(unk)J(u0)inf{J(u)|uV},

in other words

J(u0)=inf{J(u)|uV}.

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 (un) 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¯ 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 unu0 it holds that lim infnJ(xn)J(y).

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

J(x)αxqβ for some α>0, q1 and β0.

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)=ΩF(x,u(x),u(x))dx

where Ω is a subset of n and F is a real-valued function on Ω×m×mn. The argument of J is a differentiable function u:Ωm, and its Jacobian 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 C2(Ω,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 W1,p(Ω,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)=ΩF(x,u(x),u(x))dx,

where Ωn is open, theorems characterizing functions F for which J is weakly sequentially lower-semicontinuous in W1,p(Ω,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)F(x,y,p) is continuous for almost every xΩ,
  2. the function xF(x,y,p) is measurable for every (y,p)m×mn, and
  3. F(x,y,p)a(x)p+b(x) for a fixed aLq(Ω,m) where 1/q+1/p=1, a fixed bL1(Ω), for a.e. xΩ and every (y,p)m×mn (here a(x)p means the inner product of a(x) and p in mn).
The following holds. If the function pF(x,y,p) is convex for a.e. xΩ and every ym,
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)|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)Ω×m the function pF(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 pF(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

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References and further reading

  • 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

    My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
  • 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

    My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
  1. Dacorogna, pp. 1–43.
  2. 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

    My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
  3. Dacorogna, pp. 74–79.
  4. Dacorogna, pp. 66–74.
  5. Dacorogna, pp. 87–185.