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, for example, in elasticity theory.

See also

• Mosco convergence

References

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