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| {{Context|date=September 2011}}
| | The writer's title is Andera and she believes it seems fairly great. The favorite hobby for him and his children is to play lacross and he'll be beginning some thing else alongside with it. Invoicing is my occupation. For many years he's been living in Alaska and he doesn't strategy on altering it.<br><br>Feel free to surf to my web blog :: psychic readings; [http://breenq.com/index.php?do=/profile-1144/info/ breenq.com], |
| In the [[calculus of variations]], '''Γ-convergence''' ('''Gamma-convergence''') is a notion of convergence for [[Functional (mathematics)|functionals]]. It was introduced by [[Ennio de Giorgi]].
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| ==Definition==
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| Let <math>X</math> be a [[topological space]] and <math>F_n:X\to[0,+\infty)</math> a sequence of functionals on <math>X</math>. Then <math>F_n</math> are said to <math>\Gamma</math>-converge to the <math>\Gamma</math>-limit <math>F:X\to[0,+\infty)</math> if the following two conditions hold:
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| * Lower bound inequality: For every sequence <math>x_n\in X</math> such that <math>x_n\to x</math> as <math>n\to+\infty</math>,
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| : <math>F(x)\le\liminf_{n\to\infty} F_n(x_n).</math>
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| * Upper bound inequality: For every <math>x\in X</math>, there is a sequence <math>x_n</math> converging to <math>x</math> such that
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| : <math>F(x)\ge\limsup_{n\to\infty} F_n(x_n)</math>
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| The first condition means that <math>F</math> provides an asymptotic common lower bound for the <math>F_n</math>. The second condition means that this lower bound is optimal. | |
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| ==Properties==
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| * Minimizers converge to minimizers: If <math>F_n</math> <math>\Gamma</math>-converge to <math>F</math>, and <math>x_n</math> is a minimizer for <math>F_n</math>, then every cluster point of the sequence <math>x_n</math> is a minimizer of <math>F</math>.
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| * <math>\Gamma</math>-limits are always [[Semi-continuity|lower semicontinuous]].
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| * <math>\Gamma</math>-convergence is stable under continuous perturbations: If <math>F_n</math> <math>\Gamma</math>-converges to <math>F</math> and <math>G:X\to[0,+\infty)</math> is continuous, then <math>F_n+G</math> will <math>\Gamma</math>-converge to <math>F+G</math>.
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| * A constant sequence of functionals <math>F_n=F</math> does not necessarily <math>\Gamma</math>-converge to <math>F</math>, but to the ''relaxation'' of <math>F</math>, the largest lower semicontinuous functional below <math>F</math>.
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| ==Applications==
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| An important use for <math>\Gamma</math>-convergence is in [[homogenization (mathematics)|homogenization theory]]. It can also be used to rigorously justify the passage from discrete to continuum theories for materials, e.g. in [[Elasticity (physics)|elasticity]] theory.
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| ==See also==
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| * [[Mosco convergence]]
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| ==References==
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| * A. Braides: ''Γ-convergence for beginners''. Oxford University Press, 2002.
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| * G. Dal Maso: ''An introduction to Γ-convergence''. Birkhäuser, Basel 1993.
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| {{DEFAULTSORT:Gamma-Convergence}}
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| [[Category:Calculus of variations]]
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| [[Category:Variational analysis]]
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| {{Mathanalysis-stub}}
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The writer's title is Andera and she believes it seems fairly great. The favorite hobby for him and his children is to play lacross and he'll be beginning some thing else alongside with it. Invoicing is my occupation. For many years he's been living in Alaska and he doesn't strategy on altering it.
Feel free to surf to my web blog :: psychic readings; breenq.com,