Γ-convergence

In 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 $F_n:X\to[0,+\infty)$ a sequence of functionals on $X$ . Then $F_n$ are said to $\Gamma$ -converge to the $\Gamma$ -limit $F:X\to[0,+\infty)$ 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_{n\to\infty} 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_{n\to\infty} 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.

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 $G:X\to[0,+\infty)$ 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, e.g. in elasticity theory.

References

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

Γ-convergence

Definition

Properties

Applications

See also

References