- Γ-convergence
In the
calculus of variations , Γ-convergence (Gamma-convergence) is a notion of convergence for functionals. It was introduced byEnnio de Giorgi .Definition
Let "X" be a
topological space and "F""n":"X"→[ 0,∞] a sequence of functionals on "X". Then "F""n" are said to "Γ-converge" to the "Γ-limit" "F":"X"→[ 0,∞] if the following two conditions hold:*Lower bound inequality: For every sequence "x""n" in "X" such that "x""n"→"x" as "n"→∞, :
*Upper bound inequality: For every "x"∈"X", there is a sequence "x""n" converging to "x" such that: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" Γ-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".
*Γ-limits are alwayslower semicontinuous .
*Γ-convergence is stable under continuous perturbations: If "F""n" Γ-converges to "F" and "G":"X"→[ 0,∞] is continuous, then "F""n"+"G" will Γ-converge to "F"+"G".
*A constant sequence of functionals "F""n"="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.
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