---
tag: NW9MM370A5
kind: concept
name: Γ-convergence
msc: [49J45]
sources:
  - url: https://en.wikipedia.org/wiki/Γ-convergence
same_as:
  wikidata: Q278549
  wikipedia: Γ-convergence
---

## Summary

Γ-convergence is a notion of convergence for functionals that is designed for minimization problems.
A sequence $F_n$ Γ-converges to $F$ when two things hold.
A lower bound inequality holds along every convergent sequence of points, and the bound is reached along some sequence.
Under a compactness condition, minimum values and minimizers of $F_n$ then converge to those of $F$.
It was introduced by Ennio De Giorgi.
It is the standard tool to pass from a family of variational problems to a limit problem, as in homogenization and phase transitions.
The [[definition|NNAR1HN9BQ]] gives the precise conditions.

## Explanations

### L1 human

Pointwise convergence of functions does not control where their minima go.
Γ-convergence asks instead for two things.
The limit $F$ is never larger than what the $F_n$ give near a point.
And every value of $F$ can be approached by values of $F_n$ at nearby points.
Think of a landscape with finer and finer ripples: the ripples have no pointwise limit, but the lowest points of the landscape settle down.
For example, $F_n(x) = \sin(nx)$ on $\R$ has no pointwise limit, yet it Γ-converges to the constant $-1$, because near every point there are points where $\sin(nx) = -1$.