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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. Contents 1 Definition 2 Properties 3 Applications 4 See also 5 References 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: Lower bound inequality: For every sequence xn in X such that xn→x as n→∞, Upper bound inequality: For every x∈X, there is a sequence xn converging to x such that 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 Mosco convergence References A. Braides: Γ-convergence for beginners. Oxford University Press, 2002. G. Dal Maso: An introduction to Γ-convergence. Birkhäuser, Basel 1993. This mathematical analysis–related article is a stub. You can help Wikipedia by expanding it.v · d · e