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.
Let
be a topological space and
denote the set of all neighbourhoods of the point
. Let further
be a sequence of functionals on
. The Γ-lower limit and the Γ-upper limit are defined as follows:
![{\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),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d0787a99f6859963ae17e7355278a3ebb4228c51)
.
are said to
-converge to
, if there exist a functional
such that
.
Definition in first-countable spaces
[edit] 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
,
![{\displaystyle F(x)\leq \liminf _{n\to \infty }F_{n}(x_{n}).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e9ba3b929c3916cf064f802a73b246a266d321e1)
- Upper bound inequality: For every
, there is a sequence
converging to
such that
![{\displaystyle F(x)\geq \limsup _{n\to \infty }F_{n}(x_{n})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d5d60ad5e17c5986da4f29178b611ebcef2a749e)
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
[edit]
-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
![{\displaystyle {\text{epi}}(\Gamma {\text{-}}\liminf _{n\to \infty }F_{n})={\text{K}}{\text{-}}\limsup _{n\to \infty }{\text{epi}}(F_{n}),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1d638081c488244290a29e0c924a38f006f79e12)
![{\displaystyle {\text{epi}}(\Gamma {\text{-}}\limsup _{n\to \infty }F_{n})={\text{K}}{\text{-}}\liminf _{n\to \infty }{\text{epi}}(F_{n}),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/12f8625d058b90b66efad2a2bd5d12cbf41b6422)
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.
- 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
.
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.
- A. Braides: Γ-convergence for beginners. Oxford University Press, 2002.
- G. Dal Maso: An introduction to Γ-convergence. Birkhäuser, Basel 1993.