Church encoding

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My name is Jestine (34 years old) and my hobbies are Origami and Microscopy.

Here is my web site; http://Www.hostgator1centcoupon.info/ (support.file1.com) 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 Fn:X[0,+) a sequence of functionals on X. Then Fn are said to Γ-converge to the Γ-limit F:X[0,+) if the following two conditions hold:

F(x)lim infnFn(xn).
  • Upper bound inequality: For every xX, there is a sequence xn converging to x such that
F(x)lim supnFn(xn)

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

Properties

  • Minimizers converge to minimizers: If Fn Γ-converge to F, and xn is a minimizer for Fn, then every cluster point of the sequence xn is a minimizer of F.
  • Γ-limits are always lower semicontinuous.
  • Γ-convergence is stable under continuous perturbations: If Fn Γ-converges to F and G:X[0,+) is continuous, then Fn+G will Γ-converge to F+G.
  • A constant sequence of functionals Fn=F does not necessarily Γ-converge to F, but to the relaxation of F, the largest lower semicontinuous functional below F.

Applications

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, 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.


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