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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 (mathematics)|functional]],<ref>Dacorogna, pp. 1–43.</ref> 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.<ref>{{cite book |title=Calculus of Variations |authors=I. M. Gelfand, S. V. Fomin |year=1991 |publisher=Dover Publications |isbn=978-0-486-41448-5}}</ref>
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| == The method ==
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| The calculus of variations deals with functionals <math>J:V \to \bar{\mathbb{R}}</math>, where <math>V</math> is some [[function space]] and <math>\bar{\mathbb{R}} = \mathbb{R} \cup \{\infty\}</math>. The main interest of the subject is to find ''minimizers'' for such functionals, that is, functions <math>v \in V</math> such that:<math>J(v) \leq J(u)\text{ for every }u \in V. </math>
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| 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.
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| The functional <math>J</math> must be bounded from below to have a minimizer. This means
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| :<math>\inf\{J(u)|u\in V\} > -\infty.\,</math>
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| This condition is not enough to know that a minimizer exists, but it shows the existence of a ''minimizing sequence'', that is, a sequence <math>(u_n)</math> in <math>V</math> such that <math>J(u_n) \to \inf\{J(u)|u\in V\}.</math>
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| The direct method may broken into the following steps
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| # Take a minimizing sequence <math>(u_n)</math> for <math>J</math>.
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| # Show that <math>(u_n)</math> admits some [[subsequence]] <math>(u_{n_k})</math>, that converges to a <math>u_0\in V</math> with respect to a topology <math>\tau</math> on <math>V</math>.
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| # Show that <math>J</math> is sequentially [[lower semi-continuous]] with respect to the topology <math>\tau</math>.
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| To see that this shows the existence of a minimizer, consider the following characterization of sequentially lower-semicontinuous functions.
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| :The function <math>J</math> is sequentially lower-semicontinuous if
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| ::<math>\liminf_{n\to\infty} J(u_n) \geq J(u_0)</math> for any convergent sequence <math>u_n \to u_0</math> in <math>V</math>.
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| The conclusions follows from
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| :<math>\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\}</math>,
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| in other words
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| :<math>J(u_0) = \inf\{J(u)|u\in V\}</math>.
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| == Details ==
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| === Banach spaces ===
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| The direct method may often be applied with success when the space <math>V</math> is a subset of a [[reflexive space|reflexive]] [[Banach space]] <math>W</math>. In this case the [[Banach–Alaoglu theorem]] implies, that any bounded sequence <math>(u_n)</math> in <math>V</math> has a subsequence that converges to some <math>u_0</math> in <math>W</math> with respect to the [[weak topology]]. If <math>V</math> is sequentially closed in <math>W</math>, so that <math>u_0</math> is in <math>V</math>, the direct method may be applied to a functional <math>J:V\to\bar{\mathbb{R}}</math> by showing
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| # <math>J</math> is bounded from below,
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| # any minimizing sequence for <math>J</math> is bounded, and
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| # <math>J</math> is weakly sequentially lower semi-continuous, i.e., for any weakly convergent sequence <math>u_n \to u_0</math> it holds that <math>\liminf_{n\to\infty} J(x_n) \geq J(y)</math>.
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| The second part is usually accomplished by showing that <math>J</math> admits some growth condition. An example is
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| :<math>J(x) \geq \alpha \lVert x \rVert^q - \beta</math> for some <math>\alpha > 0</math>, <math>q \geq 1</math> and <math>\beta \geq 0</math>.
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| 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.
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| === Sobolev spaces ===
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| The typical functional in the calculus of variations is an integral of the form
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| :<math>J(u) = \int_\Omega F(x, u(x), \nabla u(x))dx</math> | |
| where <math>\Omega</math> is a subset of <math>\mathbb{R}^n</math> and <math>F</math> is a real-valued function on <math>\Omega \times \mathbb{R}^m \times \mathbb{R}^{mn}</math>. The argument of <math>J</math> is a differentiable function <math>u:\Omega \to \mathbb{R}^m</math>, and its [[Jacobian matrix and determinant|Jacobian]] <math>\nabla u(x)</math> is identified with a <math>mn</math>-vector.
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| When deriving the Euler–Lagrange equation, the common approach is to assume <math>\Omega</math> has a <math>C^2</math> boundary and let the domain of definition for <math>J</math> be <math>C^2(\Omega, \mathbb{R}^m)</math>. 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]] <math>W^{1,p}(\Omega, \mathbb{R}^m)</math> with <math>p > 1</math>, which is a reflexive Banach space. The derivatives of <math>u</math> in the formula for <math>J</math> must then be taken as [[weak derivative]]s. The next section presents two theorems regarding weak sequential lower semi-continuity of functionals of the above type.
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| == Sequential lower semi-continuity of integrals ==
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| As many functionals in the calculus of variations are of the form
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| :<math>J(u) = \int_\Omega F(x, u(x), \nabla u(x))dx</math>,
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| where <math>\Omega \subseteq \mathbb{R}^n</math> is open, theorems characterizing functions <math>F</math> for which <math>J</math> is weakly sequentially lower-semicontinuous in <math>W^{1,p}(\Omega, \mathbb{R}^m)</math> is of great importance.
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| In general we have the following<ref>Dacorogna, pp. 74–79.</ref>
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| :Assume that <math>F</math> is a function such that
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| :# The function <math>(y, p) \mapsto F(x, y, p)</math> is continuous for [[almost every]] <math>x \in \Omega</math>,
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| :# the function <math>x \mapsto F(x, y, p)</math> is [[measurable]] for every <math>(y, p) \in \mathbb{R}^m \times \mathbb{R}^{mn}</math>, and
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| :# <math>F(x, y, p) \geq a(x)\cdot p + b(x)</math> for a fixed <math>a\in L ^q(\Omega, \mathbb{R}^m)</math> where <math>1/q + 1/p = 1</math>, a fixed <math>b \in L^1(\Omega)</math>, for a.e. <math>x \in \Omega</math> and every <math>(y, p) \in \mathbb{R}^m \times \mathbb{R}^{mn}</math> (here <math>a(x) \cdot p</math> means the inner product of <math>a(x)</math> and <math>p</math> in <math>\mathbb{R}^{mn}</math>).
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| :The following holds. If the function <math>p \mapsto F(x, y, p)</math> is convex for a.e. <math>x \in \Omega</math> and every <math>y\in \mathbb{R}^m</math>,
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| :then <math>J</math> is sequentially weakly lower semi-continuous.
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| When <math>n = 1</math> or <math>m = 1</math> the following converse-like theorem holds<ref>Dacorogna, pp. 66–74.</ref>
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| :Assume that <math>F</math> is continuous and satisfies
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| ::<math>| F(x, y, p) | \leq a(x, | y |, | p |)</math>
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| :for every <math>(x, y, p)</math>, and a fixed function <math>a(x, y, p)</math> increasing in <math>y</math> and <math>p</math>, and locally integrable in <math>x</math>. It then holds, if <math>J</math> is sequentially weakly lower semi-continuous, then for any given <math>(x, y) \in \Omega \times \mathbb{R}^m</math> the function <math>p \mapsto F(x, y, p)</math> is convex.
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| In conclusion, when <math>m = 1</math> or <math>n = 1</math>, the functional <math>J</math>, assuming reasonable growth and boundedness on <math>F</math>, is weakly sequentially lower semi-continuous if, and only if, the function <math>p \mapsto F(x, y, p)</math> is convex. If both <math>n</math> and <math>m</math> are greater than 1, it is possible to weaken the necessity of convexity to generalizations of convexity, namely [[polyconvex function|polyconvexity]] and quasiconvexity.<ref>Dacorogna, pp. 87–185.</ref>
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| == Notes ==
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| {{reflist}}
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| == References and further reading ==
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| * {{cite book | first = Bernard | last = Dacorogna | year = 1989 | title = Direct Methods in the Calculus of Variations | publisher = Springer-Verlag | id = ISBN 0-387-50491-5 }}
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| * {{cite book | first = Irene | last = Fonseca | authorlink = Irene Fonseca | coauthors = Giovanni Leoni | year = 2007 | title = Modern Methods in the Calculus of Variations: <math>L^p</math> Spaces | publisher = Springer | id = ISBN 978-0-387-35784-3 }}
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| {{DEFAULTSORT:Direct Method In The Calculus Of Variations}}
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| [[Category:Calculus of variations]]
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Have you been told about the new sci-fi movie called "District 9"? It could be the one big breakout, unexpected box office hit of 2009. Here a couple of Web sites that demonstrate the trailer for that movie.
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The gadget can easily work on 2G and 3G networking systems. It is very light in weight, hence it is readily carryable. Truly weighs 97 grams, so you can portable and easily to find yourself the divots.
There are excellent leaders out exist for you be successful. You just have to destroy the spammers, noise makers, and fruitcakes. Creating good content, natural conversations, and simply being real are not going now. Social Media Optimization makes sense if done right. Thousands of Twitter followers that mean nothing. I would rather target my niche area of real people having similar interests. Because learn tips on how to target these followers, your leads improve and your own sales.
The first video comes to from back 2004. It stars an audio lesson called "Dragostea Din Tei" sang by www.topmathbooks.com/ Moldovian pop group called O-Zone and a fat man doing alittle dance to your song. Film is usually referred to as "Numa Numa" because that is an element of the non-English lyrics of the songs. Numa Numa was such an enormous hit how the guy returned to twice daily . sequel. "New Numa Numa." THe second one, though, is along with graphics various other things create it less random and much less funny than first. Someone also was launched with the Numa Numa guy on American Idol which was pretty intriguing.
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