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 ${\displaystyle X}$ 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:

• Lower bound inequality: For every sequence ${\displaystyle x_{n}\in X}$ such that ${\displaystyle x_{n}\to x}$ as ${\displaystyle n\to +\infty }$,
${\displaystyle F(x)\leq \liminf _{n\to \infty }F_{n}(x_{n}).}$
• Upper bound inequality: For every ${\displaystyle x\in X}$, there is a sequence ${\displaystyle x_{n}}$ converging to ${\displaystyle x}$ such that
${\displaystyle F(x)\geq \limsup _{n\to \infty }F_{n}(x_{n})}$

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.

Properties

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

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, for example, in elasticity theory.