Γ-convergence
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In the calculus of variations, Γ-convergence (Gamma-convergence) is a notion of convergence for functionals. It was introduced by Ennio de Giorgi.
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[edit] Definition
Let X be a topological space and Fn:X→[0,∞] a sequence of functionals on X. Then Fn are said to Γ-converge to the Γ-limit F:X→[0,∞] if the following two conditions hold:
- Lower bound inequality: For every sequence xn in X such that xn→x as n→∞,
- Upper bound inequality: For every x∈X, there is a sequence xn converging to x such that
The first condition means that F provides an asymptotic common lower bound for the Fn. The second condition means that this lower bound is optimal.
[edit] Properties
- Minimizers converge to minimizers: If Fn Γ-converge to F, and xn is a minimizer for Fn, then every cluster point of the sequence xn is a minimizer of F.
- Γ-limits are always lower semicontinuous.
- Γ-convergence is stable under continuous perturbations: If Fn Γ-converges to F and G:X→[0,∞] is continuous, then Fn+G will Γ-converge to F+G.
- A constant sequence of functionals Fn=F does not necessarily Γ-converge to F, but to the relaxation of F, the largest lower semicontinuous functional below F.
[edit] Applications
An important use for Γ-convergence is in homogenization theory. It can also be used to rigorously justify the passage from discrete to continuum theories for materials, e.g. in elasticity theory.
[edit] See also
[edit] References
- A. Braides: Γ-convergence for beginners. Oxford University Press, 2002.
- G. dal Maso: An introduction to Γ-convergence. Birkhäuser, Basel 1993.
