Γ-convergence explained

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 l{N}(x) denote the set of all neighbourhoods of the point x\in X . Let further F _{n:X\to\overline{R} } be a sequence of functionals on X . The Γ-lower limit and the Γ-upper limit are defined as follows:\Gamma - \liminf _{n\toinfty} F _{n(x)=\sup} ^{N _{x\inl{N}} (x)}\liminf _{n\toinfty} inf _{y\in N _{x}} F _{n(y),} \Gamma - \limsup _{n\toinfty} F _{n(x)=\sup} ^{N _{x\inl{N}} (x)}\limsup _{n\toinfty} inf _{y\in N _{x}} F _{n(y)} .F _{n} are said to \Gamma -converge to F , if there exist a functional F such that \Gamma - \liminf _{n\toinfty} F _{n=\Gamma-\limsup} _{n\toinfty} 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{R} } a sequence of functionals on X . Then F _{n} are said to \Gamma -converge to the \Gamma -limit F:X\to\overline{ R } 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 +inf ty ,F(x)\le\liminf _{n\toinfty} F _{n(x} _{n).}

• Upper bound inequality: For every

x\in X , there is a sequence x _{n} converging to x such thatF(x)\ge\limsup _{n\toinfty} 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 epi (F) denote the epigraph of a function F and let F _{n:X\to\overline{R} } be a sequence of functionals on X . Thenepi ( \Gamma - \liminf _{n\toinfty} F _{n} ) = K - \limsup _{n\toinfty} epi (F _{n),} epi ( \Gamma - \limsup _{n\toinfty} F _{n} ) = K - \liminf _{n\toinfty} epi (F _{n),}

where

K- \liminf denotes the Kuratowski limes inferior and K- \limsup the Kuratowski limes superior in the product topology of X x R . In particular, (F _{n)} _{n} \Gamma -converges to F in X if and only if ( epi (F _{n))} _{n} K -converges to epi (F) in X x 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[ 0 , +inf ty) is continuous, then F _{n+G} will \Gamma -converge to F + G .

• A constant sequence of functionals

F _{n=F} does not necessarily \Gamma -converge to F , but to the relaxation of F , the largest lower semicontinuous functional below F .

Applications

An important use for

\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

• 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.