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

Let X be a topological space and denote the set of all neighbourhoods of the point x\in X. Let further be a sequence of functionals on X. The Γ-lower limit and the Γ-upper limit are defined as follows: F_n are said to \Gamma-converge to F, if there exist a functional F such that.

Definition in first-countable spaces

In first-countable spaces, the above definition can be characterized in terms of sequential \Gamma-convergence in the following way. Let X be a first-countable space and a sequence of functionals on X. Then F_n are said to \Gamma-converge to the \Gamma-limit if the following two conditions hold: The first condition means that F provides an asymptotic common lower bound for the F_n. The second condition means that this lower bound is optimal.

Relation to Kuratowski convergence

\Gamma-convergence is connected to the notion of Kuratowski-convergence of sets. Let denote the epigraph of a function F and let be a sequence of functionals on X. Then where denotes the Kuratowski limes inferior and the Kuratowski limes superior in the product topology of. In particular, (F_n)_n \Gamma-converges to F in X if and only if \text{K}-converges to in. This is the reason why \Gamma-convergence is sometimes called epi-convergence.

Properties

Applications

An important use for \Gamma-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.