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 $\Gamma {\text{-lower limit}}$ and the $\Gamma {\text{-upper limit}}$ 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 $n\to +\infty$,

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

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

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

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.

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.