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 ${\displaystyle {\mathcal {N}}(x)}$ 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 ${\displaystyle \Gamma {\text{-lower limit}}}$ and the ${\displaystyle \Gamma {\text{-upper limit}}}$ 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 ${\displaystyle \Gamma {\text{-}}\liminf _{n\to \infty }F_{n}=\Gamma {\text{-}}\limsup _{n\to \infty }F_{n}=F}$.

Definition in first-countable spaces

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.

Relation to Kuratowski convergence

${\displaystyle \Gamma }$-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.

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.

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.