In the field of mathematical analysis for the calculus of variations, ÃÂ-convergence (Gamma-convergence) is a notion of convergence for functionals. It was introduced by Ennio De Giorgi.
Let be a topological space and denote the set of all neighbourhoods of the point . Let further be a sequence of functionals on . The ÃÂ-lower limit and the ÃÂ-upper limit are defined as follows:
are said to -converge to a functional , if .
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.
-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.
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.