$\Gamma$-Convergence

Abstract

Often, a functional of interest depends on the value of a parameter, say a small $\varepsilon > 0$, as we have seen with the functionals $\mathscr {F}_\varepsilon$ from the examples on phase transitions and composite elastic materials in Sections 1.9 and 1.10, respectively. In these cases the goal often lies not in minimizing $\mathscr {F}_\varepsilon$ for one particular value of $\varepsilon$, but in determining the asymptotic limit of the minimization problems as $\varepsilon \downarrow 0$. Concretely, we need to identify, if possible, a limit functional $\mathscr {F}_0$ such that the minimizers and minimum values of the $\mathscr {F}_\varepsilon$ (if they exist) converge to the minimizers and minimum values of $\mathscr {F}_0$ as $\varepsilon \downarrow 0$.

Appendices

Notes and Historical Remarks

The theory of $\Gamma$-convergence was founded by Ennio De Giorgi in the 1970s and nowadays is widely used in the calculus of variations. All results from Section 13.1 are essentially due to De Giorgi, see [89]. The book [51] provides a first introduction to $\Gamma$-convergence with many applications to homogenization theory, phase transition, and free discontinuity problems. The encyclopedic work [82] treats $\Gamma$-convergence in general topological spaces and also considers many applications to integral functionals.

For the Modica–Mortola Theorem 13.6 we closely follow the original works [189, 190], with some minor modifications. Our proof of Theorem 13.13 is the one in [199], which was obtained independently of Braides’ proof in [50].

Another powerful technique in the theory of $\Gamma$-convergence is the compactness method: It can often be shown by a compactness argument that the $\Gamma$-limit of a (sub)sequence of integral functionals exists as some abstract functional. The task is then to identify this $\Gamma$-limit. For this, one can use a technique, which seems to be due to De Giorgi–Letta, Fusco, and Braides, where one first shows that the $\Gamma$-limit parametrized on the domain is a measure, which then in a further step is seen to be given by an integral. The book [52] uses this technique to present many homogenization results in a unified framework. In fact, Chapter 14 of [52] proves Theorem 13.13 and several extensions using this technique.

In a non-variational context, that is, on the level of PDEs, several techniques have been developed for homogenization, which can also cope with completely non-periodic situations. We only mention G-convergence (see [82]), H-convergence (see [8]) and two-scale convergence (see [216] and also [8]). Here also the Div-curl Lemma 8.32 finds its natural home.

Problems

13.1.

Let X be a complete metric space. Show that the $\Gamma$-limit of the functionals $\mathscr {F}_k :X \rightarrow \mathbb {R}$, $k \in \mathbb {N}$, if it exists, is uniquely determined.

13.2.

Show that for a sequence $(u_j) \subset \mathrm {BV}(\varOmega )$ and $u \in \mathrm {L}^1(\varOmega )$ with $u_j \rightarrow u$ in $\mathrm {L}^1$ it holds that

$$ |Du|(\varOmega ) \le \liminf _{j\rightarrow \infty } |Du|(\varOmega ). $$

Also show that the perimeter ${{\mathrm{Per}}}_\varOmega (E)$ is lower semicontinuous with respect to the convergence of sets $E_j \rightarrow E$ defined as $\mathbbm {1}_{E_j} \rightarrow \mathbbm {1}_E$ in $\mathrm {L}^1(\varOmega )$. Hint: Write the total variation norm and the perimeter as a supremum over $\mathrm {L}^1$-continuous functionals.

13.3.

Let $\mathscr {F}_k :X \rightarrow \mathbb {R}\cup \{+\infty \}$, $k \in \mathbb {N}$, be equicoercive functionals on a complete metric space. Prove that ${{\mathrm{\Gamma {\mathrm{-lim\, inf}}}}}_k \mathscr {F}_k$ admits a minimizer and that

$$ \min _X \, {{\mathrm{\Gamma {\mathrm{-lim\, inf}}}}}_k \mathscr {F}_k = \liminf _{k\rightarrow \infty } \, \inf _X \mathscr {F}_k. $$

Also show that

$$ \limsup _{k\rightarrow \infty } \, \inf _X \mathscr {F}_k \le \inf _X \, {{\mathrm{\Gamma {\mathrm{-lim\, sup}}}}}_k \mathscr {F}_k. $$

Find a sequence of equicoercive functionals such that the $\Gamma$-upper limit does not fulfill the reverse inequality (this is contrary to the situation for the $\Gamma$-lower limit).

13.4.

Find a non-separable and complete metric space (X, d) and a sequence of functionals $\mathscr {F}_k :X \rightarrow \mathbb {R}\cup \{+\infty \}$ such that no subsequence of the $\mathscr {F}_k$’s $\Gamma$-converges. Hint: Consider $X := \{-1,1\}^\mathbb {N}$ and observe that $\Gamma$-convergence in X is equivalent to pointwise convergence.

13.5.

Prove that if $g \in \mathrm {L}^p_\mathrm {loc}(\mathbb {R}^d)$ is k-periodic for some $k \in \mathbb {N}$, then the maps $h_\varepsilon (x) := g(x/\varepsilon )$ converge weakly in $\mathrm {L}^1_\mathrm {loc}$ to the constant map

Hint: Inspect the proof of Lemma 4.15.

13.6.

Let $f :(0,\infty ) \times (0,\infty ) \rightarrow \mathbb {R}$. Show that there exists a function $\delta :(0,\infty ) \rightarrow (0,\infty )$ such that $\delta (\varepsilon ) \rightarrow 0$ as $\varepsilon \downarrow 0$ and

$$ \limsup _{\varepsilon \downarrow 0} f(\varepsilon ,\delta (\varepsilon )) \le \limsup _{\delta \downarrow 0} \limsup _{\varepsilon \downarrow 0} f(\varepsilon ,\delta ). $$

13.7.

Let f be as in Marcellini’s Homogenization Theorem 13.18. Show that the problem

$$ \left\{ \begin{aligned}&\text {Minimize} \quad \int _{(0,1)^d} f(x, F + \nabla \varphi (x)) \;\mathrm{d}x\\&\text {over all} \quad \varphi \in \mathrm {W}^{1,p}_\mathrm {per}((0,1)^d;\mathbb {R}^m) \end{aligned} \right. $$

has a solution $\varphi _* \in \mathrm {W}^{1,p}_\mathrm {per}((0,1)^d;\mathbb {R}^m)$. Moreover, prove that $\varphi _*$ is a weak solution of the Euler–Lagrange equation

$$ - {{\mathrm{div\,}}}\bigl [ \mathrm {D}_A f(x, F + \nabla \varphi _*(x)) \bigr ] = 0, \qquad x \in (0,1)^d, $$

that is,

$$ \int _\varOmega \mathrm {D}_A f(x, F + \nabla \varphi _*(x)) : \nabla \psi (x) \;\mathrm{d}x = 0 \qquad \text { for all }\psi \in \mathrm {W}^{1,p}_\mathrm {per}((0,1)^d;\mathbb {R}^m)\text {.} $$

13.8.

Let $\varOmega \subset \mathbb {R}^d$ be a bounded Lipschitz domain and let $f :\mathbb {R}^d \times \mathbb {R}^{m \times d} \rightarrow [0,\infty )$ be a Carathéodory integrand. Define the partial regularization  $f_\delta :\mathbb {R}^d \times \mathbb {R}^{m \times d} \rightarrow [0,\infty )$ of f as

$$ f_\delta (x,A) := \int \eta _\delta (A-B) f(x,B) \;\mathrm{d}B, \qquad (x, A) \in \mathbb {R}^d \times \mathbb {R}^{m \times d}, $$

where $(\eta _\delta )_{\delta > 0}$ is a radially symmetric and positive family of mollifiers on $\mathbb {R}^{m \times d}$. Show that $f_\delta \rightarrow f$ pointwise. Show also that if f satisfies any of the following conditions, then so does $f_\delta$:

• (i) $\mu |A|^p \le f(x, A) \le M(1+|A|^p)$ for all $(x, A) \in \mathbb {R}^d \times \mathbb {R}^{m \times d}$ and some $p \in (1,\infty )$, $\mu , M > 0$;

• (ii) $x \mapsto f(x, A)$ is 1-periodic for all $A \in \mathbb {R}^{m \times d}$;

• (iii) $|f(x,A)-f(x, B)| \le C(1+|A|^{p-1}+|B|^{p-1})|A-B|$ for all $x \in \varOmega$, $A, B \in \mathbb {R}^{m \times d}$ and some $C > 0$.

13.9.

In the situation of Marcellini’s Homogenization Theorem 13.18, show that if $m = 1$ then

$$ f_{\mathrm {hom}}(A) = \inf _{\varphi \in \mathrm {W}^{1,p}_\mathrm {per}((0,k)^d)} \int _{(0,k)^d} f^{**}(x, A + \nabla \varphi (x)) \;\mathrm{d}x, $$

where $f^{**}$ denotes the convex envelope of f with respect to the second argument. Conclude that for $m = 1$ Marcellini’s Homogenization Theorem 13.18 also holds without assuming the convexity of f in the second argument. Hint: Extend a relaxation theorem to periodic integrands.

13.10.

Prove Lemma 13.22.