Γ-convergence
In the field of mathematical analysis for the calculus of variations, Γ-convergence is a notion of convergence for functionals. It was introduced by Ennio De Giorgi.
Definition
Let be a topological space and denote the set of all neighbourhoods of the point. Let further be a sequence of functionals on. The Γ-lower limit and the Γ-upper limit are defined as follows:are said to -converge to a functional, if.
Definition in first-countable spaces
In first-countable spaces, the above definition can be characterized in terms of sequential -convergence in the following way.Let be a first-countable space and a sequence of functionals on. Then are said to -converge to the -limit if the following two conditions hold:
- Lower bound inequality: For every sequence such that as,
- Upper bound inequality: For every, there is a sequence converging to such that
Relation to Kuratowski convergence
-convergence is connected to the notion of Kuratowski-convergence of sets. Let denote the epigraph of a function and let be a sequence of functionals on. Thenwhere denotes the Kuratowski limes inferior and the Kuratowski limes superior in the product topology of. In particular, -converges to in if and only if -converges to in. This is the reason why -convergence is sometimes called epi-convergence.
Properties
- Minimizers converge to minimizers: If -converge to, and is a minimizer for, then every cluster point of the sequence is a minimizer of.
- -limits are always lower semicontinuous.
- -convergence is stable under continuous perturbations: If -converges to and is continuous, then will -converge to.
- A constant sequence of functionals does not necessarily -converge to, but to the relaxation of, the largest lower semicontinuous functional below.