In the calculus of variations, Γ-convergence (Gamma-convergence) is a notion of convergence for functionals. It was introduced by Ennio de Giorgi.
Let
X
be a topological space and
F
n
:
X
→
[
0
,
+
∞
)
a sequence of functionals on
X
. Then
F
n
are said to
Γ
-converge to the
Γ
-limit
F
:
X
→
[
0
,
+
∞
)
if the following two conditions hold:
Lower bound inequality: For every sequence
x
n
∈
X
such that
x
n
→
x
as
n
→
+
∞
,
F
(
x
)
≤
lim inf
n
→
∞
F
n
(
x
n
)
.
Upper bound inequality: For every
x
∈
X
, there is a sequence
x
n
converging to
x
such that
F
(
x
)
≥
lim sup
n
→
∞
F
n
(
x
n
)
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.
Minimizers converge to minimizers: If
F
n
Γ
-converge to
F
, and
x
n
is a minimizer for
F
n
, then every cluster point of the sequence
x
n
is a minimizer of
F
.
Γ
-limits are always lower semicontinuous.
Γ
-convergence is stable under continuous perturbations: If
F
n
Γ
-converges to
F
and
G
:
X
→
[
0
,
+
∞
)
is continuous, then
F
n
+
G
will
Γ
-converge to
F
+
G
.
A constant sequence of functionals
F
n
=
F
does not necessarily
Γ
-converge to
F
, but to the relaxation of
F
, the largest lower semicontinuous functional below
F
.
An important use for
Γ
-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.
