|
|
| Line 1: |
Line 1: |
| {{Context|date=September 2011}}
| | Alyson is what my spouse loves to contact me but I don't like when people use my full title. To play lacross is the thing I adore most of all. Since he was 18 he's been working as an information officer but he ideas on altering it. For years he's been residing in Alaska and he doesn't plan on changing it.<br><br>Here is my site :: clairvoyants ([http://www.chk.woobi.co.kr/xe/?document_srl=346069 www.chk.woobi.co.kr]) |
| 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]].
| |
| | |
| ==Definition==
| |
| 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:
| |
| * 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>,
| |
| : <math>F(x)\le\liminf_{n\to\infty} F_n(x_n).</math> | |
| * 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
| |
| : <math>F(x)\ge\limsup_{n\to\infty} F_n(x_n)</math> | |
| | |
| 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.
| |
| | |
| ==Properties==
| |
| * 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>.
| |
| * <math>\Gamma</math>-limits are always [[Semi-continuity|lower semicontinuous]].
| |
| * <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>.
| |
| * 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>.
| |
| | |
| ==Applications==
| |
| 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.
| |
| | |
| ==See also==
| |
| * [[Mosco convergence]]
| |
| | |
| ==References==
| |
| * A. Braides: ''Γ-convergence for beginners''. Oxford University Press, 2002.
| |
| * G. Dal Maso: ''An introduction to Γ-convergence''. Birkhäuser, Basel 1993.
| |
| | |
| {{DEFAULTSORT:Gamma-Convergence}}
| |
| [[Category:Calculus of variations]]
| |
| [[Category:Variational analysis]]
| |
| | |
| | |
| {{Mathanalysis-stub}}
| |
Alyson is what my spouse loves to contact me but I don't like when people use my full title. To play lacross is the thing I adore most of all. Since he was 18 he's been working as an information officer but he ideas on altering it. For years he's been residing in Alaska and he doesn't plan on changing it.
Here is my site :: clairvoyants (www.chk.woobi.co.kr)