How do people prove $\Gamma$-convergence in more complicated settings?

Question

This is a soft question, I guess. $\Gamma$-convergence is a notion of convergence of functionals so that if $F_n$ $\Gamma$-converges to $F$, then cluster points of $\arg\inf F_n$ are minimizers of $F$. This is especially helpful if you want to minimize $F$ but find it easier to minimize $F_n$.

However, if you look at the definition of $\Gamma$-convergence it becomes pretty untenable to prove in complicated settings. It requires you to perform 4 optimizations, and take two limits. In some simple cases one can prove $\Gamma$-convergence "by hand" but I was wondering - what is the state of the art for proving $\Gamma$-convergence in a complicated setting? I know that it is equivalent to convergence of the epi-graph, and also in first countable spaces there is an equivalent definition, that is still pretty hard to show.

What techniques exist to prove $\Gamma$-convergence?

I know you can add and subtract continuous functions.

Answer 1

I am mostly familiar with the simpler definition ("Definition in first-countable spaces" from the Wikipedia link):

Given the functionals $F_\varepsilon, F: X \to \overline{\Bbb{R}}$ (indexed for $\varepsilon>0$, we say that $F_\varepsilon$ $\Gamma$-converges to $F$ if the two properties hold:

(LI) For every $u \in X$ and every $u_\varepsilon \to u$ we have $\liminf_{\varepsilon \to 0} F_\varepsilon(u_\varepsilon) \geq F(u)$.

(LS) For every $u \in X$ there exists $u_\varepsilon \to u$ such that $\limsup_{\varepsilon \to 0} F_\varepsilon(u_\varepsilon) \leq F(u)$.

The techniques used to prove $\Gamma$-convergence in this case vary with the problem studied.

• Generally (LI) comes from some lower-semicontinuity result (see some examples below)

• (LS) is easy if the constant sequence can be chosen. However, in some cases, $F_\varepsilon$ and $F$ can have different domains in the sense that $F$ is $+\infty$ for any $u$ such that $F_\varepsilon(u)$ is finite. In such cases, recovery sequences realizing the equality $\lim_{\varepsilon \to 0}F_\varepsilon(u_\varepsilon)=F(u)$ need to be constructed by hand.

• the (LS) is usually proved on a dense subset of $X$ verifying additional regularity properties, facilitating the construction of the recovery sequences described above.

• the (LS) property is sometimes proved using a slicing procedure, i.e. the result is somehow reduced to the dimension one by considering the whole space as a product between a manifold of dimension $N-1$ and one dimensional spaces.

Examples and references

1. The first example I knew was the Modica-Mortola theorem. It gives an approximation of the perimeter using energies defined for smoother density functions. Consider $F, F_\varepsilon : L^1(D) \to \overline{\Bbb{R}}$: $$ F_\varepsilon(u) = \begin{cases} \int_D (\varepsilon |\nabla u|^2 +\frac{1}{\varepsilon}u^2(1-u)^2) & \text{ if } u \in H^1(D) \\ +\infty & \text{ otherwise} \end{cases} $$ and $$ F(u) = \begin{cases} \frac{1}{3} \text{Per}(E_1) & \text{ if } u \in BV(D,\{0,1\}), E_1 = u^{-1}(\{1\}) \\ +\infty & \text{ otherwise} \end{cases} $$ Then $F_\varepsilon$ $\Gamma$-converges to $F$. See my blog for more details and a proof. This is an example where the constant sequence cannot be chosen as a recovery sequence in (LS). Characteristic functions of finite perimeter sets are not in $H^1$. The (LI) part comes from the lower semicontinuity of the total variation for $BV$ functions (abstract, but standard result).

2. A similar setting can be used for the total perimeter of a partition. Notice however that this is not a consequence of the Modica-Mortola theorem, since $\Gamma$-convergence is not stable for the sum. This kind of argument was used for approximating numerically partitions minimizing the total perimeter in the following article. Similar approaches can be extended to Cheeger clusters, again with numerical implications. See this link. Reading these articles could give more insight about difficulties regarding proofs of properties (LI) or (LS) above. More references in this sense exist for finding Steiner trees, minimal surfaces and much more (take a look at publications related to $\Gamma$-convergence on E. Bretin's webpage; here is a link to the Wayback Machine).

3. More generally, the books of Andrea Brides on the subject are a must read. They go into more depth regarding the techniques involved. Consider for example: Gamma-convergence for beginners by A. Braides, Approximation of Free-Discontinuity Problems again by A. Braides. And this is just scratching the surface.

I hope this answer helps giving some insights regarding $\Gamma$-convergence results in some basic cases.