Γ-convergence
In the calculus of variations , Γ-convergence ( gamma-convergence ) denotes a special type of convergence for functionals . It was introduced by Ennio de Giorgi . It was originally referred to as G-convergence, as it was developed for Green functionals . The term Γ-convergence emerged from the generalization of this concept of convergence.
definition
Be a topological space and a series of functional on . The sequence converges in the sense of the Γ-convergence towards the Grenz-limit value if the following two conditions hold:
- For every convergent sequence in with limit, the following applies
- For each there is a consequence in that converges against and
- Fulfills.
The first condition means that there is a "common asymptotic lower bound" for ; the latter condition, on the other hand, guarantees optimality.
properties
- Minimizers converge to minimizers: A sequence is called a minimum sequence for , if
- .
- If now converges to Γ and is a minimal sequence for , then every accumulation point of is a minimizer of , ie
- .
- Γ-limits are always lower continuous .
- Γ-convergence is stable under continuous perturbation: If it converges to Γ and is continuous, then Γ-convergent to .
- A constant sequence of functionals does not necessarily have to converge to Γ, but to the relaxation of , namely the largest sub-semicontinuous functional below .
Applications
The Γ-convergence finds an important application in the homogenization theory and the dimension reduction. It can also be used to provide a rigorous rationale for the transition from discrete to continuous models, for example in elasticity theory . Further areas of application can be found in the area of phase transitions and program slicing .
Related convergence terms
A convergence term related to Banach spaces is the Mosco convergence , which is equivalent to simultaneous Γ-convergence with regard to the norm topology and the weak topology .
literature
- Andrea Braides : Γ-convergence for beginners . In: Oxford Lecture Series in Mathematics and Its Applications. Volume 22, Oxford University Press, 2002, ISBN 0-19-850784-4 .
- Gianni Dal Maso : An Introduction to Γ-Convergence . In: Progress in Nonlinear Differential Equations and Their Applications. Volume 8, Birkhäuser, Basel 1993, ISBN 978-0-8176-3679-1 .