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.
be a topological space and
denote the set of all neighbourhoods of the point
be a sequence of functionals on
The Γ-lower limit and the Γ-upper limit are defined as follows:
, if there exist a functional
lim inf
lim sup
In first-countable spaces, the above definition can be characterized in terms of sequential
be a first-countable space and
a sequence of functionals on
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.
epi
{\displaystyle {\text{epi}}(F)}
denote the epigraph of a function
be a sequence of functionals on
lim inf
denotes the Kuratowski limes inferior and
lim sup
the Kuratowski limes superior in the product topology of
-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.