Relaxation

Abstract

Consider the functional

$$ \mathscr {F}[u] := \int _0^1 |u(x)|^2 + \bigl (|u'(x)|^2-1 \bigr )^2 \;\mathrm{d}x, \qquad u \in \mathrm {W}^{1,4}_0(0,1). $$

The gradient part of the integrand, $a \mapsto (a^2-1)^2$, see Figure 7.1, has two distinct minima, which makes it a double-well potential . Approximate minimizers of $\mathscr {F}$ try to satisfy $u' \in \{-1, 1\}$ as closely as possible, while at the same time staying close to zero because of the first term.

Appendices

Notes and Historical Remarks

Classically, one often defines the quasiconvex envelope through (7.3) and not as we have done through (7.1). In this case, (7.1) is usually called Dacorogna’s formula , see Section 6.3 in [76] for further references.

The construction of Lemma 7.3 and Example 7.7 are from [250], but our proof also uses some ellipticity arguments similar to those in Lemma 2.7 of [203]. A different proof of Lemma 7.3 can be found in Section 5.3.9 of [76]. We refer to [33] for some regularity properties of quasiconvex envelopes.

Further relaxation formulas can be found in Chapter 11 of the textbook [19]; historically, Dacorogna’s lecture notes [75] were also influential.

The conditions (i)–(iii) at the beginning of Section 7.4 are modeled on the concept of $\Gamma$-convergence (introduced by De Giorgi), see Chapter 13 for more on this topic.

The Kinderlehrer–Pedregal theorem is conceptually very important. In particular, it entails that if we could understand the class of quasiconvex functions, then we also could understand gradient Young measures and thus the asymptotic “shape” of gradients. Unfortunately, our knowledge of quasiconvex functions, and hence of gradient Young measures, is limited at present. There is some further discussion on this point throughout [222].

The truncation argument used in Zhang’s Lemma 7.18 seems to be due originally to Acerbi–Fusco [1, 3]. The book [177] makes use of this technique in regularity theory and also contains several refinements.

Problems

7.1.

Generalize Lemmas 7.1 and 7.2 to $h :\mathbb {R}^{m \times d} \rightarrow [0,\infty )$ with p-growth that are merely upper semicontinuous.

7.2.

Let $\varOmega \subset \mathbb {R}^d$ be a bounded Lipschitz domain and $F \in \mathbb {R}^{m \times d}$ with ${{\mathrm{rank}}}F = 1$. Consider the functional

$$ \mathscr {F}[u] := \int _\varOmega {{\mathrm{dist}}}( \nabla u(x), \{-F, F\} )^2 \;\mathrm{d}x, \qquad u \in \mathrm {W}^{1,\infty }(\varOmega ;\mathbb {R}^m). $$

Construct a sequence $(u_j) \subset \mathrm {W}^{1,\infty }(\varOmega ;\mathbb {R}^m)$ with $u_j \overset{*}{\rightharpoonup }0$ in $\mathrm {W}^{1,\infty }$ and $\mathscr {F}[u_j] = 0$ for all $j \in \mathbb {N}$. Conclude that $\mathscr {F}$ is not lower semicontinuous with respect to weak* convergence in $\mathrm {W}^{1,\infty }(\varOmega ;\mathbb {R}^m)$.

7.3.

Show that the integrand in the integral functional from the previous exercise is not rank-one convex.

7.4.

Let L be a linear subspace of $\mathbb {R}^{m \times d}$ with ${{\mathrm{rank}}}(A-B) \ge 2$ for all $A, B \in L$, let $K \subset L$ be compact, and let $p \in (1,\infty )$. Show that for

$$ h(A) := {{\mathrm{dist}}}(A, K)^p, \qquad A \in \mathbb {R}^{m \times d}, $$

it holds that Qh is not convex (at zero). Conclude that for $p < 2$ and K not convex, Qh cannot be polyconvex.

7.5.

Show that under the assumptions of Theorem 7.5 and assuming additionally that X is separable, it holds that

$$ \mathscr {F}_*[u] = \inf \, \biggl \{\, \liminf _{j\rightarrow \infty } \mathscr {F}[u_j] \ \ \mathbf{: }\ \ u_j \rightharpoonup u \text { in } X \,\biggr \}, \qquad u \in X. $$

7.6.

Let $h :\mathbb {R}^{m \times d} \rightarrow \!\mathbb {R}$ be twice continuously differentiable. Show that then h is rank-one convex if and only if h satisfies the Legendre–Hadamard condition, that is,

$$\begin{aligned} \mathrm {D}^2 h(A)[a \otimes b, a \otimes b]&= \frac{\mathrm{d}^2}{\mathrm{d}t^2} \, h(A + ta \otimes b) \bigg |_{t=0} \nonumber \\&= \sum _{i, k=1}^m \sum _{j, l=1}^d \frac{\partial ^2 h(A)}{\partial A^i_j \partial A^k_l} a^i b^j a^k b^l \nonumber \\&\ge 0 \end{aligned}$$

(7.26)

for all $A \in \mathbb {R}^{m \times d}$ and all $a \in \mathbb {R}^m$, $b \in \mathbb {R}^d$.

7.7.

Show that $\mathbf {I}^p(\mathbb {R}^{m \times d})$ (defined in (7.20)) is isomorphic to the (separable) space $\mathrm {C}(\alpha \mathbb {R}^{m \times d})$, where $\alpha \mathbb {R}^{m \times d}$ is the Alexandrov (one-point) compactification of $\mathbb {R}^{m \times d}$, or, equivalently, $\mathbf {I}^p(\mathbb {R}^{m \times d})$ is isometrically isomorphic to the set of all $\phi \in \mathrm {C}(\overline{\mathbb {B}^{m \times d}})$ with $\phi |_{\partial \mathbb {B}^{m \times d}}$ constant, where $\mathbb {B}^{m \times d}$ denotes the open unit ball in $\mathbb {R}^{m \times d}$ (with respect to the Frobenius norm). Conclude that $\mathbf {I}^p(\mathbb {R}^{m \times d})$ is separable.

7.8.

Show that in the Kinderlehrer–Pedregal Theorem 7.15 in the case $p \in (1,\infty )$ it suffices to verify (7.18) for all quasiconvex $h :\mathbb {R}^{m \times d} \rightarrow \mathbb {R}$ that are bounded from below and for which

$$ \displaystyle \lim _{|A|\rightarrow \infty } \frac{h(A)}{1+|A|}\; \text { exists (in}\,\,\mathbb {R}\text {).} $$

7.9.

Prove the claim of Remark 5.15.

7.10.

Let $K \subset \mathbb {R}^{m \times d}$ be compact and non-empty. Also assume there is a norm-bounded sequence $(u_j) \subset \mathrm {W}^{1,p}(\varOmega ;\mathbb {R}^m)$, $p \in (1,\infty )$ with ${{\mathrm{dist}}}(\nabla u_j, K) \rightarrow 0$ in measure. Show that then there exists a sequence $(v_j) \subset \mathrm {W}^{1,\infty }(\varOmega ;\mathbb {R}^m)$ such that also ${{\mathrm{dist}}}(\nabla v_j, K) \rightarrow 0$ in measure. Hint: Use Zhang’s Lemma 7.18.