Γ-convergence

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In the calculus of variations, Γ-convergence (Gamma-convergence) is a notion of convergence for functionals. It was introduced by Ennio de Giorgi.

Definition[edit]

Let be a topological space and a sequence of functionals on . Then are said to -converge to the -limit if the following two conditions hold:

  • Lower bound inequality: For every sequence such that as ,
  • Upper bound inequality: For every , there is a sequence converging to such that

The first condition means that provides an asymptotic common lower bound for the . The second condition means that this lower bound is optimal.

Properties[edit]

  • Minimizers converge to minimizers: If -converge to , and is a minimizer for , then every cluster point of the sequence is a minimizer of .
  • -limits are always lower semicontinuous.
  • -convergence is stable under continuous perturbations: If -converges to and is continuous, then will -converge to .
  • A constant sequence of functionals does not necessarily -converge to , but to the relaxation of , the largest lower semicontinuous functional below .

Applications[edit]

An important use for -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[edit]

References[edit]

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