Fundamental theorem of Γ-convergence

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.

Going deeper

For a student

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.