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