Γ-convergence
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  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:
 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. 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
 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, for example, in elasticity theory. See also
 Mosco convergence
 Kuratowski convergence
 Epi-convergence

References
 A. Braides: Γ-convergence for beginners. Oxford University Press, 2002. G. Dal Maso: An introduction to Γ-convergence. Birkhäuser, Basel 1993. Calculus of variations
Variational analysis
Convergence (mathematics)