Γ-convergence

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

${\displaystyle \Gamma {\text{-}}\liminf _{n\to \infty }F_{n}(x)=\sup _{N_{x}\in {\mathcal {N}}(x)}\liminf _{n\to \infty }\inf _{y\in N_{x}}F_{n}(y),}$ 

${\displaystyle \Gamma {\text{-}}\limsup _{n\to \infty }F_{n}(x)=\sup _{N_{x}\in {\mathcal {N}}(x)}\limsup _{n\to \infty }\inf _{y\in N_{x}}F_{n}(y)}$ .

${\displaystyle F_{n}}$  are said to ${\displaystyle \Gamma }$ -converge to ${\displaystyle F}$ , if there exist a functional ${\displaystyle F}$  such that ${\displaystyle \Gamma {\text{-}}\liminf _{n\to \infty }F_{n}=\Gamma {\text{-}}\limsup _{n\to \infty }F_{n}=F}$ .

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

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

${\displaystyle {\text{epi}}(\Gamma {\text{-}}\liminf _{n\to \infty }F_{n})={\text{K}}{\text{-}}\limsup _{n\to \infty }{\text{epi}}(F_{n}),}$ 

${\displaystyle {\text{epi}}(\Gamma {\text{-}}\limsup _{n\to \infty }F_{n})={\text{K}}{\text{-}}\liminf _{n\to \infty }{\text{epi}}(F_{n}),}$ 

where ${\displaystyle {\text{K-}}\liminf }$  denotes the Kuratowski limes inferior and ${\displaystyle {\text{K-}}\limsup }$  the Kuratowski limes superior in the product topology of ${\displaystyle X\times \mathbb {R} }$ . In particular, ${\displaystyle (F_{n})_{n}}$  ${\displaystyle \Gamma }$ -converges to ${\displaystyle F}$  in ${\displaystyle X}$  if and only if ${\displaystyle ({\text{epi}}(F_{n}))_{n}}$  ${\displaystyle {\text{K}}}$ -converges to ${\displaystyle {\text{epi}}(F)}$  in ${\displaystyle X\times \mathbb {R} }$ . This is the reason why ${\displaystyle \Gamma }$ -convergence is sometimes called epi-convergence.

• 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 ${\displaystyle F}$ .

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.

• Mosco convergence
• Kuratowski convergence
• Epi-convergence

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