# 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.

## Definition

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:

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

## Properties

- 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

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, e.g. in elasticity theory.

## See also

## References

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