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 $$\mathcal{N}(x)$$ 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:


 * $$\Gamma\text{-}\liminf_{n\to\infty} F_n(x)=\sup_{N_x\in\mathcal{N}(x)}\liminf_{n\to\infty}\inf_{y\in N_x}F_n(y),$$


 * $$\Gamma\text{-}\limsup_{n\to\infty} F_n(x)=\sup_{N_x\in\mathcal{N}(x)}\limsup_{n\to\infty}\inf_{y\in N_x}F_n(y)$$.

$$F_n$$ are said to $$\Gamma$$-converge to $$F$$, if there exist a functional $$F$$ such that $$\Gamma\text{-}\liminf_{n\to\infty} F_n=\Gamma\text{-}\limsup_{n\to\infty} F_n=F$$.

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 $$F_n:X\to\overline{\mathbb{R}}$$ a sequence of functionals on $$X$$. Then $$F_n$$ are said to $$\Gamma$$-converge to the $$\Gamma$$-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 $$n\to+\infty$$,
 * $$F(x)\le\liminf_{n\to\infty} F_n(x_n).$$


 * 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.

Relation to Kuratowski convergence
$$\Gamma$$-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


 * $$\text{epi} ( \Gamma\text{-}\liminf_{n\to\infty} F_n ) = \text{K}\text{-}\limsup_{n\to\infty} \text{epi}(F_n),$$
 * $$ \text{epi} ( \Gamma\text{-}\limsup_{n\to\infty} F_n ) = \text{K}\text{-}\liminf_{n\to\infty} \text{epi}(F_n),$$

where $$\text{K-}\liminf$$ denotes the Kuratowski limes inferior and $$\text{K-}\limsup$$ the Kuratowski limes superior in the product topology of $$ X\times \mathbb{R}$$. In particular, $$(F_n)_n$$ $$\Gamma$$-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 $$\Gamma$$-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 $$G:X\to[0,+\infty)$$ 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.