Γ-convergence


In the calculus of variations, Γ-convergence is a notion of convergence for functionals. It was introduced by Ennio de Giorgi.

Definition

Let be a topological space and denote the set of all neighbourhoods of the point. Let further be a sequence of functionals on. The and the are defined as follows:
are said to -converge to, if there exist a functional such that.

Definition in first-countable spaces

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:
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

-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

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.