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)\leq \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)\geq \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, for example, 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.

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Γ-convergence

Definition

Properties

Applications

See also

References