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: ${\displaystyle X}$ ${\displaystyle (F_{n})}$ ${\displaystyle F_{n}:X\rightarrow [0,\infty ]}$ ${\displaystyle X}$ ${\displaystyle (F_{n})}$ ${\displaystyle F:X\rightarrow [0,\infty ]}$

• For every convergent sequence in with limit, the following applies ${\displaystyle (x_{n})}$ ${\displaystyle X}$ ${\displaystyle x\in X}$

${\displaystyle F(x)\leq \liminf _{n\to \infty }F_{n}(x_{n}).}$

• For each there is a consequence in that converges against and ${\displaystyle x\in X}$ ${\displaystyle (x_{n})}$ ${\displaystyle X}$ ${\displaystyle x}$

${\displaystyle F(x)\geq \limsup _{n\to \infty }F_{n}(x_{n})}$

Fulfills.

The first condition means that there is a "common asymptotic lower bound" for ; the latter condition, on the other hand, guarantees optimality. ${\displaystyle F}$ ${\displaystyle F_{n}}$

properties

• Minimizers converge to minimizers: A sequence is called a minimum sequence for , if ${\displaystyle (x_{n})}$ ${\displaystyle (F_{n})}$

${\displaystyle \lim _{n\rightarrow \infty }\left(F_{n}(x_{n})-\inf\{F_{n}(y)\ |\ y\in X\}\right)=0}$.

If now converges to Γ and is a minimal sequence for , then every accumulation point of is a minimizer of , ie ${\displaystyle (F_{n})}$ ${\displaystyle F}$ ${\displaystyle (x_{n})}$ ${\displaystyle (F_{n})}$ ${\displaystyle x}$ ${\displaystyle (x_{n})}$ ${\displaystyle F}$

${\displaystyle F(x)=\inf\{F(y)\ |\ y\in X\}}$.

• 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 . ${\displaystyle (F_{n})}$ ${\displaystyle F}$ ${\displaystyle G:X\rightarrow [0,\infty )}$ ${\displaystyle (F_{n}+G)}$ ${\displaystyle F+G}$
• 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 . ${\displaystyle F_{n}\equiv F}$ ${\displaystyle F}$ ${\displaystyle F}$ ${\displaystyle F}$

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 .