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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 [ edit ]
Let
X
{\displaystyle X}
be a topological space and
N
(
x
)
{\displaystyle {\mathcal {N}}(x)}
denote the set of all neighbourhoods of the point
x
∈
X
{\displaystyle x\in X}
. Let further
F
n
:
X
→
R
¯
{\displaystyle F_{n}:X\to {\overline {\mathbb {R} }}}
be a sequence of functionals on
X
{\displaystyle X}
. The
Γ
-lower limit
{\displaystyle \Gamma {\text{-lower limit}}}
and the
Γ
-upper limit
{\displaystyle \Gamma {\text{-upper limit}}}
are defined as follows:
Γ
-
lim inf
n
→
∞
F
n
(
x
)
=
sup
N
x
∈
N
(
x
)
lim inf
n
→
∞
inf
y
∈
N
x
F
n
(
y
)
,
{\displaystyle \Gamma {\text{-}}\liminf _{n\to \infty }F_{n}(x)=\sup _{N_{x}\in {\mathcal {N}}(x)}\liminf _{n\to \infty }\inf _{y\in N_{x}}F_{n}(y),}
Γ
-
lim sup
n
→
∞
F
n
(
x
)
=
sup
N
x
∈
N
(
x
)
lim sup
n
→
∞
inf
y
∈
N
x
F
n
(
y
)
{\displaystyle \Gamma {\text{-}}\limsup _{n\to \infty }F_{n}(x)=\sup _{N_{x}\in {\mathcal {N}}(x)}\limsup _{n\to \infty }\inf _{y\in N_{x}}F_{n}(y)}
.
F
n
{\displaystyle F_{n}}
are said to
Γ
{\displaystyle \Gamma }
-converge to
F
{\displaystyle F}
, if there exist a functional
F
{\displaystyle F}
such that
Γ
-
lim inf
n
→
∞
F
n
=
Γ
-
lim sup
n
→
∞
F
n
=
F
{\displaystyle \Gamma {\text{-}}\liminf _{n\to \infty }F_{n}=\Gamma {\text{-}}\limsup _{n\to \infty }F_{n}=F}
.
Definition in first-countable spaces [ edit ]
In first-countable spaces , the above definition can be characterized in terms of sequential
Γ
{\displaystyle \Gamma }
-convergence in the following way.
Let
X
{\displaystyle X}
be a first-countable space and
F
n
:
X
→
R
¯
{\displaystyle F_{n}:X\to {\overline {\mathbb {R} }}}
a sequence of functionals on
X
{\displaystyle X}
. Then
F
n
{\displaystyle F_{n}}
are said to
Γ
{\displaystyle \Gamma }
-converge to the
Γ
{\displaystyle \Gamma }
-limit
F
:
X
→
R
¯
{\displaystyle F:X\to {\overline {\mathbb {R} }}}
if the following two conditions hold:
Lower bound inequality: For every sequence
x
n
∈
X
{\displaystyle x_{n}\in X}
such that
x
n
→
x
{\displaystyle x_{n}\to x}
as
n
→
+
∞
{\displaystyle n\to +\infty }
,
F
(
x
)
≤
lim inf
n
→
∞
F
n
(
x
n
)
.
{\displaystyle F(x)\leq \liminf _{n\to \infty }F_{n}(x_{n}).}
Upper bound inequality: For every
x
∈
X
{\displaystyle x\in X}
, there is a sequence
x
n
{\displaystyle x_{n}}
converging to
x
{\displaystyle x}
such that
F
(
x
)
≥
lim sup
n
→
∞
F
n
(
x
n
)
{\displaystyle F(x)\geq \limsup _{n\to \infty }F_{n}(x_{n})}
The first condition means that
F
{\displaystyle F}
provides an asymptotic common lower bound for the
F
n
{\displaystyle F_{n}}
. The second condition means that this lower bound is optimal.
Relation to Kuratowski convergence [ edit ]
Γ
{\displaystyle \Gamma }
-convergence is connected to the notion of Kuratowski-convergence of sets. Let
epi
(
F
)
{\displaystyle {\text{epi}}(F)}
denote the epigraph of a function
F
{\displaystyle F}
and let
F
n
:
X
→
R
¯
{\displaystyle F_{n}:X\to {\overline {\mathbb {R} }}}
be a sequence of functionals on
X
{\displaystyle X}
. Then
epi
(
Γ
-
lim inf
n
→
∞
F
n
)
=
K
-
lim sup
n
→
∞
epi
(
F
n
)
,
{\displaystyle {\text{epi}}(\Gamma {\text{-}}\liminf _{n\to \infty }F_{n})={\text{K}}{\text{-}}\limsup _{n\to \infty }{\text{epi}}(F_{n}),}
epi
(
Γ
-
lim sup
n
→
∞
F
n
)
=
K
-
lim inf
n
→
∞
epi
(
F
n
)
,
{\displaystyle {\text{epi}}(\Gamma {\text{-}}\limsup _{n\to \infty }F_{n})={\text{K}}{\text{-}}\liminf _{n\to \infty }{\text{epi}}(F_{n}),}
where
K-
lim inf
{\displaystyle {\text{K-}}\liminf }
denotes the Kuratowski limes inferior and
K-
lim sup
{\displaystyle {\text{K-}}\limsup }
the Kuratowski limes superior in the product topology of
X
×
R
{\displaystyle X\times \mathbb {R} }
. In particular,
(
F
n
)
n
{\displaystyle (F_{n})_{n}}
Γ
{\displaystyle \Gamma }
-converges to
F
{\displaystyle F}
in
X
{\displaystyle X}
if and only if
(
epi
(
F
n
)
)
n
{\displaystyle ({\text{epi}}(F_{n}))_{n}}
K
{\displaystyle {\text{K}}}
-converges to
epi
(
F
)
{\displaystyle {\text{epi}}(F)}
in
X
×
R
{\displaystyle X\times \mathbb {R} }
. This is the reason why
Γ
{\displaystyle \Gamma }
-convergence is sometimes called epi-convergence .
Properties [ edit ]
Minimizers converge to minimizers: If
F
n
{\displaystyle F_{n}}
Γ
{\displaystyle \Gamma }
-converge to
F
{\displaystyle F}
, and
x
n
{\displaystyle x_{n}}
is a minimizer for
F
n
{\displaystyle F_{n}}
, then every cluster point of the sequence
x
n
{\displaystyle x_{n}}
is a minimizer of
F
{\displaystyle F}
.
Γ
{\displaystyle \Gamma }
-limits are always lower semicontinuous .
Γ
{\displaystyle \Gamma }
-convergence is stable under continuous perturbations: If
F
n
{\displaystyle F_{n}}
Γ
{\displaystyle \Gamma }
-converges to
F
{\displaystyle F}
and
G
:
X
→
[
0
,
+
∞
)
{\displaystyle G:X\to [0,+\infty )}
is continuous, then
F
n
+
G
{\displaystyle F_{n}+G}
will
Γ
{\displaystyle \Gamma }
-converge to
F
+
G
{\displaystyle F+G}
.
A constant sequence of functionals
F
n
=
F
{\displaystyle F_{n}=F}
does not necessarily
Γ
{\displaystyle \Gamma }
-converge to
F
{\displaystyle F}
, but to the relaxation of
F
{\displaystyle F}
, the largest lower semicontinuous functional below
F
{\displaystyle F}
.
Applications [ edit ]
An important use for
Γ
{\displaystyle \Gamma }
-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 [ edit ]
References [ edit ]
A. Braides: Γ-convergence for beginners . Oxford University Press, 2002.
G. Dal Maso: An introduction to Γ-convergence . Birkhäuser, Basel 1993.