---
tag: 23RK8EZ20P
kind: statement
role: theorem
name: Fundamental theorem of Γ-convergence
msc: [49J45]
uses: [NW9MM370A5, 0VKD1TP7F3, 0R6EWFKRCF, 0GSZ7B0CGR]
about: [NW9MM370A5]
sources:
  - url: https://en.wikipedia.org/wiki/Γ-convergence
---

## Statement

Let $(X, d)$ be a [[metric space|0VKD1TP7F3]].
Let $F_n \colon X \to [-\infty, +\infty]$ be functions that [[Γ-converge|NW9MM370A5]] to $F \colon X \to [-\infty, +\infty]$.
Assume that the sequence is equi-coercive in the following sense: there is a nonempty [[compact set|0R6EWFKRCF]] $K \subseteq X$ with $\inf_X F_n = \inf_K F_n$ for every $n$.
Then:

1. $F$ attains its minimum on $X$, and
$$\min_X F = \lim_{n \to \infty} \inf_X F_n.$$
2. If $(x_n)$ is a sequence in $X$ with $\lim_n F_n(x_n) = \lim_n \inf_X F_n$, then every limit of a subsequence of $(x_n)$ is a minimum point of $F$.

## Remarks

For the proof, pick almost minimizers $x_n \in K$ of $F_n$ and a subsequence along which $\inf_X F_n$ tends to its limit inferior and $x_n$ converges to some $x \in K$.
The lower bound inequality gives $F(x) \le \liminf_n \inf_X F_n$.
For any $y \in X$, a recovery sequence $y_n \to y$ gives $\limsup_n \inf_X F_n \le \limsup_n F_n(y_n) \le F(y)$.
Together these show that $x$ minimizes $F$ and that the infima converge; part 2 follows from the same lower bound.

Example: for $F_n(x) = \sin(nx) + x^2$ on $\R$, the Γ-limit is $F(x) = x^2 - 1$, since adding a continuous real function commutes with Γ-convergence.
Since $F_n(x) \ge x^2 - 1 > 0 = F_n(0)$ for $\abs{x} > 1$, the infimum of $F_n$ over $\R$ equals its infimum over $K = [-1, 1]$.
The theorem gives $\min F_n \to -1$, and the minimizers of $F_n$ tend to $0$, the minimizer of $F$.

The coercivity hypothesis cannot be dropped.
Let $F_n(x) = -\max(0, 1 - \abs{x - n})$ on $\R$, a downward tent centred at $n$.
Its Γ-limit is $F = 0$, but $\inf F_n = -1$ for every $n$, so the infima do not converge to $\min F = 0$; the minimizers $x_n = n$ escape to infinity.

## Explanations

### L1 human

Suppose a hard minimization problem is close, in the sense of Γ-convergence, to a simpler limit problem.
The theorem says that the smallest values of the hard problems approach the smallest value of the limit problem.
It also says that their best points can only accumulate at best points of the limit problem.
The one extra condition is that the best points do not run away to infinity.

### L2 human

This theorem is the reason Γ-convergence is defined the way it is.
The lower bound inequality controls minimum values from below, and recovery sequences control them from above.
In applications one proves the two inequalities separately: the lower bound often by a lower semicontinuity argument, the upper bound by an explicit construction.
Equi-coercivity usually comes from a compactness theorem, such as the Rellich compactness theorem for Sobolev spaces.
Results such as the Modica–Mortola theorem on phase transitions are then read as statements about the behaviour of minimizers.