Bent function

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, 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.

The method

The calculus of variations deals with functionals $J:V\to \overline{\mathbb{ℝ}}$, where $V$ is some function space and $\overline{\mathbb{ℝ}}=\mathbb{ℝ}\cup \{\mathrm{\infty }\}$. The main interest of the subject is to find minimizers for such functionals, that is, functions $v\in V$ such that:$J(v)\le J(u)\text{ for every }u\in V.$

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

$\inf \{J(u)|u\in V\}>-\mathrm{\infty }.$

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

${\mathrm{lim\; inf}}_{n\to \mathrm{\infty }}J(u_n)\ge J(u_0)$ for any convergent sequence $u_n\to u_0$ in $V$.

The conclusions follows from

$\inf \{J(u)|u\in V\}=\underset{n\to \mathrm{\infty }}{\lim }J(u_n)=\underset{k\to \mathrm{\infty }}{\lim }J(u_{n_k})\ge J(u_0)\ge \inf \{J(u)|u\in V\}$,

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 reflexive Banach space $W$. In this case the 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 $J:V\to \overline{\mathbb{ℝ}}$ 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 ${\mathrm{lim\; inf}}_{n\to \mathrm{\infty }}J(x_n)\ge J(y)$.

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

$J(x)\ge \alpha \Vert x{\Vert }^q-\beta$ for some $\alpha >0$, $q\ge 1$ and $\beta \ge 0$.

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

$J(u)=\underset{\mathrm{\Omega }}{\int }F(x,u(x),\mathrm{\nabla }u(x))dx$

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

When deriving the Euler–Lagrange equation, the common approach is to assume $\mathrm{\Omega }$ has a $C^2$ boundary and let the domain of definition for $J$ be $C^2(\mathrm{\Omega },{\mathbb{ℝ}}^m)$. 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 $W^{1,p}(\mathrm{\Omega },{\mathbb{ℝ}}^m)$ 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

$J(u)=\underset{\mathrm{\Omega }}{\int }F(x,u(x),\mathrm{\nabla }u(x))dx$,

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

In general we have the following

Assume that $F$ is a function such that

1. The function $(y,p)\mapsto F(x,y,p)$ is continuous for almost every $x\in \mathrm{\Omega }$,
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. $F(x,y,p)\ge a(x)\cdot p+b(x)$ for a fixed $a\in L^q(\mathrm{\Omega },{\mathbb{ℝ}}^m)$ where $1/q+1/p=1$, a fixed $b\in L^1(\mathrm{\Omega })$, for a.e. $x\in \mathrm{\Omega }$ 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. $x\in \mathrm{\Omega }$ 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

Assume that $F$ is continuous and satisfies

$|F(x,y,p)|\le a(x,|y|,|p|)$

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 $(x,y)\in \mathrm{\Omega }\times {\mathbb{ℝ}}^m$ 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.

Notes

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References and further reading

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• 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