Γ-convergence

{{ safesubst:#invoke:Unsubst||$N=Context |date=__DATE__ |$B= {{#invoke:Message box|ambox}} }} 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 ${\displaystyle X}$ be a topological space and ${\displaystyle F_{n}:X\to [0,+\infty )}$ a sequence of functionals on ${\displaystyle X}$. Then ${\displaystyle F_{n}}$ are said to ${\displaystyle \Gamma }$-converge to the ${\displaystyle \Gamma }$-limit ${\displaystyle F:X\to [0,+\infty )}$ if the following two conditions hold:

${\displaystyle F(x)\leq \liminf _{n\to \infty }F_{n}(x_{n}).}$
${\displaystyle F(x)\geq \limsup _{n\to \infty }F_{n}(x_{n})}$

The first condition means that ${\displaystyle F}$ provides an asymptotic common lower bound for the ${\displaystyle F_{n}}$. The second condition means that this lower bound is optimal.

Applications

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