Bent function - Revision history

en>Wcherowi: /* Generalizations */ perhaps this is a better phrasing - ignore my last summary

2014-08-26T04:38:27Z

Generalizations: perhaps this is a better phrasing - ignore my last summary

← Older revision		Revision as of 06:38, 26 August 2014
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en>Trappist the monk: Fix CS1 deprecated coauthor parameter errors (test) using AWB

2014-02-18T14:11:13Z

Fix CS1 deprecated coauthor parameter errors (test) using AWB

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en>Monkbot: Fix CS1 deprecated date parameter errors

2014-01-15T02:17:39Z

Fix CS1 deprecated date parameter errors

← Older revision		Revision as of 04:17, 15 January 2014
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	~~Kareem Rascon~~ is title ~~my parents gave me~~ and ~~I totally dig~~ that ~~concept~~. ~~Ice skating~~ is ~~what I every week~~. ~~For years he~~'~~s been prevented Rhode Island and screwed up and try never take~~. ~~In her professional life she's~~ a ~~production and planning officer~~. ~~You can always find her website here~~: ~~http~~://~~www~~.~~topmathbooks~~.~~com~~/~~top~~-10-~~fifth~~-~~grade~~-math~~-books~~/<br><br>~~Feel free~~ to ~~surf~~ to ~~my site~~; [~~http~~://~~www~~.~~topmathbooks~~.~~com~~/~~top~~-10-~~fifth~~-~~grade~~-math~~-books~~/ ~~www~~.~~topmathbooks~~.~~com~~/]		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>

			== The method ==
			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>

			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 <math>J</math> must be bounded from below to have a minimizer. This means

			:<math>\inf\{J(u)\|u\in V\} > -\infty.\,</math>

			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>

			The direct method may broken into the following steps
			# Take a minimizing sequence <math>(u_n)</math> for <math>J</math>.
			# 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>.
			# Show that <math>J</math> is sequentially [[lower semi-continuous]] with respect to the topology <math>\tau</math>.

			To see that this shows the existence of a minimizer, consider the following characterization of sequentially lower-semicontinuous functions.
			:The function <math>J</math> is sequentially lower-semicontinuous if
			::<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>.

			The conclusions follows from
			:<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>,
			in other words
			:<math>J(u_0) = \inf\{J(u)\|u\in V\}</math>.

			== Details ==
			=== Banach spaces ===
			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
			# <math>J</math> is bounded from below,
			# any minimizing sequence for <math>J</math> is bounded, and
			# <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>.
			The second part is usually accomplished by showing that <math>J</math> admits some growth condition. An example is
			:<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>.
			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
			:<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.

			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.

			== Sequential lower semi-continuity of integrals ==
			As many functionals in the calculus of variations are of the form
			:<math>J(u) = \int_\Omega F(x, u(x), \nabla u(x))dx</math>,
			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.

			In general we have the following<ref>Dacorogna, pp. 74–79.</ref>
			:Assume that <math>F</math> is a function such that
			:# The function <math>(y, p) \mapsto F(x, y, p)</math> is continuous for [[almost every]] <math>x \in \Omega</math>,
			:# 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
			:# <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>).
			: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>,
			:then <math>J</math> is sequentially weakly lower semi-continuous.
			When <math>n = 1</math> or <math>m = 1</math> the following converse-like theorem holds<ref>Dacorogna, pp. 66–74.</ref>
			:Assume that <math>F</math> is continuous and satisfies
			::<math>\| F(x, y, p) \| \leq a(x, \| y \|, \| p \|)</math>
			: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.

			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>

			== Notes ==
			{{reflist}}

			== References and further reading ==
			* {{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 }}
			* {{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 }}

			{{DEFAULTSORT:Direct Method In The Calculus Of Variations}}
			[[Category:Calculus of variations]]

en>Helpful Pixie Bot: ISBNs (Build KE)

2012-05-11T02:07:13Z

ISBNs (Build KE)

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