Γ-convergence

From Wikipedia, the free encyclopedia

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 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)\leq \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)\geq \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

  • 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

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

References

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


This article is issued from Wikipedia. The text is available under the Creative Commons Attribution/Share Alike; additional terms may apply for the media files.