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 $X$ be a topological space and ${𝒩}(x)$ denote the set of all neighbourhoods of the point $x\in X$. Let further $F_n:X\to \overline{\mathbb{ℝ}}$ be a sequence of functionals on $X$. The Γ-lower limit and the Γ-upper limit are defined as follows:

$\mathrm{\Gamma }\text{-}\underset{n\to \mathrm{\infty }}{\mathrm{lim\; inf}}{F}_n(x)=\underset{N_x\in {𝒩}(x)}{\sup }\underset{n\to \mathrm{\infty }}{\mathrm{lim\; inf}}\underset{y\in N_x}{\inf }{F}_n(y),$

$\mathrm{\Gamma }\text{-}\underset{n\to \mathrm{\infty }}{\mathrm{lim\; sup}}{F}_n(x)=\underset{N_x\in {𝒩}(x)}{\sup }\underset{n\to \mathrm{\infty }}{\mathrm{lim\; sup}}\underset{y\in N_x}{\inf }{F}_n(y)$.

$F_n$ are said to $\mathrm{\Gamma }$-converge to $F$, if there exist a functional $F$ such that $\mathrm{\Gamma }\text{-}\underset{n\to \mathrm{\infty }}{\mathrm{lim\; inf}}{F}_n=\mathrm{\Gamma }\text{-}\underset{n\to \mathrm{\infty }}{\mathrm{lim\; sup}}{F}_n=F$.

Definition in first-countable spaces

In first-countable spaces, the above definition can be characterized in terms of sequential $\mathrm{\Gamma }$-convergence in the following way. Let $X$ be a first-countable space and $F_n:X\to \overline{\mathbb{ℝ}}$ a sequence of functionals on $X$. Then $F_n$ are said to $\mathrm{\Gamma }$-converge to the $\mathrm{\Gamma }$-limit $F:X\to \overline{\mathbb{ℝ}}$ if the following two conditions hold:

• Lower bound inequality: For every sequence $x_n\in X$ such that $x_n\to x$ as $n\to +\mathrm{\infty }$,

$F(x)\le \underset{n\to \mathrm{\infty }}{\mathrm{lim\; inf}}{F}_n(x_n).$

• Upper bound inequality: For every $x\in X$, there is a sequence $x_n$ converging to $x$ such that

$F(x)\ge \underset{n\to \mathrm{\infty }}{\mathrm{lim\; sup}}{F}_n(x_n)$

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

$\mathrm{\Gamma }$-convergence is connected to the notion of Kuratowski-convergence of sets. Let $\text{epi}(F)$ denote the epigraph of a function $F$ and let $F_n:X\to \overline{\mathbb{ℝ}}$ be a sequence of functionals on $X$. Then

$\text{epi}(\mathrm{\Gamma }\text{-}\underset{n\to \mathrm{\infty }}{\mathrm{lim\; inf}}{F}_n)=\text{K}\text{-}\underset{n\to \mathrm{\infty }}{\mathrm{lim\; sup}}\text{epi}(F_n),$

$\text{epi}(\mathrm{\Gamma }\text{-}\underset{n\to \mathrm{\infty }}{\mathrm{lim\; sup}}{F}_n)=\text{K}\text{-}\underset{n\to \mathrm{\infty }}{\mathrm{lim\; inf}}\text{epi}(F_n),$

where $\text{K-}\mathrm{lim\; inf}$ denotes the Kuratowski limes inferior and $\text{K-}\mathrm{lim\; sup}$ the Kuratowski limes superior in the product topology of $X\times \mathbb{ℝ}$. In particular, $(F_n{)}_n$ $\mathrm{\Gamma }$-converges to $F$ in $X$ if and only if $(\text{epi}(F_n){)}_n$ $\text{K}$-converges to $\text{epi}(F)$ in $X\times \mathbb{ℝ}$. This is the reason why $\mathrm{\Gamma }$-convergence is sometimes called epi-convergence.

Properties

• Minimizers converge to minimizers: If $F_n$ $\mathrm{\Gamma }$-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 $F$.
• $\mathrm{\Gamma }$-limits are always lower semicontinuous.
• $\mathrm{\Gamma }$-convergence is stable under continuous perturbations: If $F_n$ $\mathrm{\Gamma }$-converges to $F$ and $G:X\to [0,+\mathrm{\infty })$ is continuous, then $F_n+G$ will $\mathrm{\Gamma }$-converge to $F+G$.
• A constant sequence of functionals $F_n=F$ does not necessarily $\mathrm{\Gamma }$-converge to $F$, but to the relaxation of $F$, the largest lower semicontinuous functional below $F$.

Applications

An important use for $\mathrm{\Gamma }$-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.