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 X be a topological space and F_n:X\to[0,+\infty) a sequence of functionals on X. Then F_n are said to \Gamma-converge to the \Gamma-limit F:X\to[0,+\infty) 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)\le\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)\ge\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.

Properties[edit]

  • 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[edit]

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, e.g. 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.