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 formula_1 be a topological space and formula_2 denote the set of all neighbourhoods of the point formula_3. Let further formula_4 be a sequence of functionals on formula_1. The formula_6 and the formula_7 are defined as follows: formula_10 are said to formula_11-converge to formula_12, if there exist a functional formula_12 such that formula_14. Definition in first-countable spaces. In first-countable spaces, the above definition can be  characterized in terms of sequential formula_11-convergence in the following way. Let formula_1 be a first-countable space and formula_4 a sequence of functionals on formula_1. Then formula_10 are said to formula_11-converge to the formula_11-limit formula_22 if the following two conditions hold: The first condition means that formula_12 provides an asymptotic common lower bound for the formula_10. The second condition means that this lower bound is optimal. Relation to Kuratowski convergence. formula_11-convergence is connected to the notion of Kuratowski-convergence of sets. Let formula_34 denote the epigraph of a function formula_12 and let formula_4 be a sequence  of functionals on formula_1. Then where formula_40 denotes the Kuratowski limes inferior and formula_41 the Kuratowski limes superior in the product topology of formula_42. In particular, formula_43 formula_11-converges to formula_12 in formula_1 if and only if formula_47 formula_48-converges to formula_49 in formula_50. This is the reason why formula_11-convergence is sometimes called epi-convergence. Applications. An important use for formula_11-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.