In the calculus of variations, Γ-convergence (Gamma-convergence) is a notion of convergence for functionals. It was introduced by Ennio de Giorgi.
Let X be a topological space and F n : X → [ 0 , + ∞ ) a sequence of functionals on X . Then F n are said to Γ -converge to the Γ -limit F : X → [ 0 , + ∞ ) if the following two conditions hold:
Lower bound inequality: For every sequence x n ∈ X such that x n → x as n → + ∞ , F ( x ) ≤ lim inf n → ∞ F n ( x n ) . Upper bound inequality: For every x ∈ X , there is a sequence x n converging to x such that F ( x ) ≥ lim sup n → ∞ 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.
Minimizers converge to minimizers: If F n Γ -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 . Γ -limits are always lower semicontinuous. Γ -convergence is stable under continuous perturbations: If F n Γ -converges to F and G : X → [ 0 , + ∞ ) is continuous, then F n + G will Γ -converge to F + G .A constant sequence of functionals F n = F does not necessarily Γ -converge to F , but to the relaxation of F , the largest lower semicontinuous functional below F .An important use for Γ -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.
