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