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Γ-convergence

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In the field of mathematical analysis for the calculus of variations, Γ-convergence (Gamma-convergence) is a notion of convergence for functionals. It was introduced by Ennio De Giorgi.

Definition

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Let be a topological space and denote the set of all neighbourhoods of the point . Let further be a sequence of functionals on . The Γ-lower limit and the Γ-upper limit are defined as follows:

.

are said to -converge to , if there exist a functional such that .

Definition in first-countable spaces

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In first-countable spaces, the above definition can be characterized in terms of sequential -convergence in the following way. Let be a first-countable space and a sequence of functionals on . Then are said to -converge to the -limit if the following two conditions hold:

  • Lower bound inequality: For every sequence such that as ,
  • Upper bound inequality: For every , there is a sequence converging to such that

The first condition means that provides an asymptotic common lower bound for the . The second condition means that this lower bound is optimal.

Relation to Kuratowski convergence

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-convergence is connected to the notion of Kuratowski-convergence of sets. Let denote the epigraph of a function and let be a sequence of functionals on . Then

where denotes the Kuratowski limes inferior and the Kuratowski limes superior in the product topology of . In particular, -converges to in if and only if -converges to in . This is the reason why -convergence is sometimes called epi-convergence.

Properties

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  • Minimizers converge to minimizers: If -converge to , and is a minimizer for , then every cluster point of the sequence is a minimizer of .
  • -limits are always lower semicontinuous.
  • -convergence is stable under continuous perturbations: If -converges to and is continuous, then will -converge to .
  • A constant sequence of functionals does not necessarily -converge to , but to the relaxation of , the largest lower semicontinuous functional below .

Applications

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An important use for -convergence is in homogenization theory. It can also be used to rigorously justify the passage from discrete to continuum theories for materials, for example, in elasticity theory.

See also

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References

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  • A. Braides: Γ-convergence for beginners. Oxford University Press, 2002.
  • G. Dal Maso: An introduction to Γ-convergence. Birkhäuser, Basel 1993.