Γ-convergence

In the calculus of variations called Γ - convergence (gamma - convergence ) a special Konvergenzart for functionals. It was introduced by Ennio De Giorgi. Originally it was called G- convergence, since it was developed for Green's functionals. The term Γ - convergence was created by the generalization of this convergence concept.

Definition

Be a topological space and a sequence of functionals on. The sequence converges in the sense of Γ - convergence to the Γ - limit, if the following two conditions:

• For any convergent sequence with limit in

• There is for each a sequence in which converges to and

The first condition implies that a "common asymptotic lower bound " for which; the latter condition, however, guarantees the optimality.

Properties

• Minimizers converge to minimizers: A sequence is called minimizing sequence for if

• Γ - limits are always lower semicontinuous.
• Γ - convergence is stable under continuous fault: If Γ - converges to and is continuous, then it is against Γ - convergent.
• A constant sequence of functionals does not necessarily have to Γ - converge, but against the relaxation of, namely, the largest of semi-continuous functional below.

Applications

An important application is the Γ - convergence in the Homogenisierungstheorie and dimension reduction. They can also be used to provide a rigorous explanation for the transition from discrete to the continuous models, for example in the Elasitizitätstheorie. Other applications are found in the field of phase transitions and program slicing.

Related concepts of convergence

A related notion of convergence in Banach spaces is the Mosco - convergence, which is equivalent with respect to simultaneous Γ - convergence of the norm topology and the weak topology.