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

Γ - lim inf n → ∞ F n ( x ) = sup N x ∈ N ( x ) lim inf n → ∞ inf y ∈ N x F n ( y ) , {\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),}

Γ - lim sup n → ∞ F n ( x ) = sup N x ∈ N ( x ) lim sup n → ∞ inf y ∈ 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)}.

F n {\displaystyle F_{n}} are said to Γ {\displaystyle \Gamma }-converge to a functional F {\displaystyle F}, if Γ - lim inf n → ∞ F n = Γ - lim sup n → ∞ F n = F {\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 Γ {\displaystyle \Gamma }-convergence in the following way. Let X {\displaystyle X} be a first-countable space and F n : X → R ¯ {\displaystyle F_{n}:X\to {\overline {\mathbb {R} }}} a sequence of functionals on X {\displaystyle X}. Then F n {\displaystyle F_{n}} are said to Γ {\displaystyle \Gamma }-converge to the Γ {\displaystyle \Gamma }-limit F : X → R ¯ {\displaystyle F:X\to {\overline {\mathbb {R} }}} if the following two conditions hold:

Lower bound inequality: For every sequence x n ∈ X {\displaystyle x_{n}\in X} such that x n → x {\displaystyle x_{n}\to x} as n → + ∞ {\displaystyle n\to +\infty },

F ( x ) ≤ lim inf n → ∞ F n ( x n ) . {\displaystyle F(x)\leq \liminf _{n\to \infty }F_{n}(x_{n}).}

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

F ( x ) ≥ lim sup n → ∞ F n ( x n ) {\displaystyle F(x)\geq \limsup _{n\to \infty }F_{n}(x_{n})}

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

Relation to Kuratowski convergence

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

epi ( Γ - lim inf n → ∞ F n ) = K - lim sup n → ∞ epi ( F n ) , {\displaystyle {\text{epi}}(\Gamma {\text{-}}\liminf _{n\to \infty }F_{n})={\text{K}}{\text{-}}\limsup _{n\to \infty }{\text{epi}}(F_{n}),}

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

Properties

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

Applications