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 $X$ be a topological space and $\mathcal(x)$ denote the set of all neighbourhoods of the point $x\in X$. Let further $F_n:X\to\overline$ be a sequence of functionals on $X$. The $\Gamma\text$ and the $\Gamma\text$ are defined as follows:

:$\Gamma\text\liminf_ F_n(x)=\sup_\liminf_\inf_F_n(y),$

:$\Gamma\text\limsup_ F_n(x)=\sup_\limsup_\inf_F_n(y)$.

$F_n$ are said to $\Gamma$-converge to $F$, if there exist a functional $F$ such that $\Gamma\text\liminf_ F_n=\Gamma\text\limsup_ F_n=F$.

Definition in first-countable spaces

In first-countable spaces, the above definition can be characterized in terms of sequential $\Gamma$-convergence in the following way.
Let $X$ be a first-countable space and $F_n:X\to\overline$ a sequence of functionals on $X$. Then $F_n$ are said to $\Gamma$-converge to the $\Gamma$-limit $F:X\to\overline$ if the following two conditions hold:
* Lower bound inequality: For every sequence $x_n\in X$ such that $x_n\to x$ as $n\to+\infty$,
: $F(x)\le\liminf_ F_n(x_n).$
* Upper bound inequality: For every $x\in X$, there is a sequence $x_n$ converging to $x$ such that
: $F(x)\ge\limsup_ F_n(x_n)$

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.

Relation to Kuratowski convergence

$\Gamma$-convergence is connected to the notion of Kuratowski-convergence of sets. Let $\text (F)$ denote the epigraph of a function $F$ and let $F_n:X\to\overline$ be a sequence of functionals on $X$. Then

: $\text ( \Gamma\text\liminf_ F_n ) = \text\text\limsup_ \text(F_n),$
: $\text ( \Gamma\text\limsup_ F_n ) = \text\text\liminf_ \text(F_n),$

where $\text\liminf$ denotes the Kuratowski limes inferior and $\text\limsup$ the Kuratowski limes superior in the product topology of $X\times \mathbb$. In particular, $(F_n)_n$ $\Gamma$-converges to $F$ in $X$ if and only if $(\text(F_n))_n$ $\text$-converges to $\text(F)$ in $X\times\mathbb R$. This is the reason why $\Gamma$-convergence is sometimes called epi-convergence.

Properties

* Minimizers converge to minimizers: If $F_n$ $\Gamma$-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 $F$.
* $\Gamma$-limits are always lower semicontinuous.
* $\Gamma$-convergence is stable under continuous perturbations: If $F_n$ $\Gamma$-converges to $F$ and G:X\to 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

* Elasticity (physics)">elasticity theory.

See also

* Mosco convergence
* Kuratowski convergence">Mosco convergence">Elasticity (physics)">elasticity theory.

See also

* Mosco convergence
* Kuratowski convergence
* Epi-convergence

References

* A. Braides: ''Γ-convergence for beginners''. Oxford University Press, 2002.
* G. Dal Maso: ''An introduction to Γ-convergence''. Birkhäuser, Basel 1993.

Calculus of variations
Variational analysis
Convergence (mathematics)

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