Quasiconvexity

Abstract

We saw in the Tonelli–Serrin Theorem 2.6 that convexity of the integrand (in the gradient variable) implies the weak lower semicontinuity of the corresponding integral functional. Moreover, we proved in Proposition 2.9 that if $d = 1$ or $m = 1$, then convexity of the integrand is also necessary for weak lower semicontinuity. In the vectorial case ($d, m > 1$), however, it turns out that one can find weakly lower semicontinuous integral functionals whose integrands are non-convex.

Appendices

Notes and Historical Remarks

The notion of quasiconvexity was first introduced in Morrey’s seminal paper [195]. Lemma 5.6 is originally due to Morrey [196]; we follow the presentation in [33]. The results about null-Lagrangians, in particular Lemmas 5.8 and 5.10, go back to Morrey [196] and Ball [25]. The pivotal proof idea that certain combinations of derivatives might have good convergence properties even if the individual derivatives do not, is also the starting point for the theory of compensated compactness (see Section 8.8). A more general result on why convexity is inadmissible for realistic problems in nonlinear elasticity can be found in Section 4.8 of [64].

The convexity properties of quadratic forms have received considerable attention because they correspond to linear Euler–Lagrange equations. In this case, quasiconvexity and rank-one convexity are the same, see Problem 5.7. Moreover, for quadratic forms, even polyconvexity (see the next chapter) is equivalent to rank-one convexity if $d = 2$ or $m = 2$, but this does not hold for $d, m \ge 3$. These results together with pointers to the literature can be found in Section 5.3.2 of [76].

The result that rank-one convex functions are locally Lipschitz continuous, Lemma 5.6, is well-known for convex functions, see, for example, Corollary 2.4 in [106] and an adapted version for rank-one convex (even separately convex) functions is in Theorem  2.31 of [76]. Our proof with a quantitative bound is from Lemma 2.2 in [33]. A more general version of this statement can be found in Lemma 2.3 of [162].

The Ball–James Rigidity Theorem  5.13 is from [30]. We will see much more general rigidity results in Chapter 8.

It is possible to prove Morrey’s Theorem  5.16 without the use of Young measures, see, for instance, Chapter 8 in [76] for such an approach. However, many of the ideas are essentially the same, they are just carried out directly without the Young measure intermediary (which obscures them somewhat). More on lower semicontinuity and Young measures can be found in the book [222].

All results in this chapter are formulated for Carathéodory integrands, but many continue to hold for $f :\varOmega \times \mathbb {R}^N \rightarrow \mathbb {R}$ that are Borel-measurable and lower semicontinuous in the second argument, so called normal integrands , see [39] and [122].

An argument by Kružík [170], which was refined by Müller, shows the curious fact that for a quasiconvex $h :\mathbb {R}^{m \times d} \rightarrow \mathbb {R}$ with $m \ge 3$, $d \ge 2$ the function $A \mapsto h(A^T)$ may not be quasiconvex. The proof can be found in Section 4.7 of [203]; it is based on Šverák’s example of a rank-one convex function that is not quasiconvex (for the same dimensions as above), which we will present in Example 7.10 in Chapter 7.

For minimization problems where the integrand can take negative values one needs to look carefully at the negative part of the integrand, see Problem 5.6. If the integrand has critical negative growth, then lower semicontinuity only holds if the boundary values are fixed along a sequence or if one imposes quasiconvexity at the boundary, see [37] for a recent survey article discussing this topic.

Problems

5.1.

For non-convex domains, statement (i) (b) of the Ball–James Rigidity Theorem  5.13 is false. Construct a counterexample.

5.2.

Define, with $D := (0,1)^d \subset \mathbb {R}^d$,

$$ \mathrm {W}^{1,\infty }_{\mathrm {per}}(D;\mathbb {R}^m) := \bigl \{\, u \in \mathrm {W}^{1,\infty }(\mathbb {R}^d;\mathbb {R}^m) \ \ \mathbf{: }\ \ { u }(x+\text {e}_i) = u(x), x \in \mathbb {R}^d, i = 1,\ldots , d \,\bigr \}. $$

A locally bounded Borel-measurable function $h :\mathbb {R}^{m \times d} \rightarrow \mathbb {R}$ is called periodic quasiconvex if

$$ h(A) \le \int _D h(A + \nabla \psi (z)) \;\text {d}z \qquad \text { for all }A \in \mathbb {R}^{m \times d} \text { and all }\psi \in \mathrm {W}^{1,\infty }_{\mathrm {per}}(D;\mathbb {R}^m). $$

Show that periodic quasiconvexity and the usual quasiconvexity are equivalent. Hint: Let $\psi \in \mathrm {W}^{1,\infty }_{\mathrm {per}}(D;\mathbb {R}^m)$ and define for $k\in \mathbb {N}$ the function $\psi _k :\mathbb {R}^d \rightarrow \mathbb {R}^m$ as

$$ \psi _k(x) := \frac{1}{k} \psi (kx), \qquad x \in \mathbb {R}^d. $$

Prove that

$$ \int _D h(A + \nabla \psi (z)) \;\text {d}z = \int _D h(A + \nabla \psi _k(z)) \;\text {d}z \qquad \text { for all } A \in \mathbb {R}^{m \times d}, k \in \mathbb {N}, $$

and that $\psi _k \in \mathrm {W}^{1,\infty }_{\mathrm {per}}(D;\mathbb {R}^m)$. You will also need a cut-off argument close to the boundary $\partial D$.

5.3.

Denote by $B := \overline{B(0,1)}$ the closed unit ball in $\mathbb {R}^d$.

• (i) Let $w :B \rightarrow \partial B$ be smooth (a “retraction”). Use $|w(x)|^2 = 1$ for every $x \in B$ to show that $\det \nabla w = 0$ in B.

• (ii) Use the fact that the determinant is a null-Lagrangian to conclude that there exists at least one $x \in \partial \varOmega$ with $w(x) \ne x$. Hint: Use an argument by contradiction.

• (iii) Derive a smooth version of the Brouwer fixed point theorem: Let $u :B \rightarrow B$ be smooth. Then u has a fixed point $x_* \in B$, that is, $u(x_*) = x_*$. Hint: Argue by contradiction and consider the ray emanating from u(x) and passing through x for all $x \in \varOmega$ and reduce to (ii).

• (iv) Extend the proof to also apply to merely continuous u.

5.4.

Let $f :\mathbb {R}^{m \times d} \rightarrow \mathbb {R}$ be Borel-measurable and strongly quasiconvex, that is, there exists a $\gamma > 0$ such that $A \mapsto f(A) - \gamma |A|^2$ is quasiconvex. Assume furthermore that $|f(A)| \le M(1+|A|^2)$ for some $M > 0$ and all $A \in \mathbb {R}^{m \times d}$. Show that the functional

$$ \mathscr {F}[u] := \int _\varOmega f(\nabla u(x)) \;\text {d}x $$

attains its minimum on $\mathrm {W}^{1,2}_{Fx}(\varOmega ;\mathbb {R}^m)$ for any $F \in \mathbb {R}^{m \times d}$.

5.5.

Show that for $\varOmega := (-1,1)^2 \subset \mathbb {R}^2$ the functional

$$ \mathscr {F}[u] := \int _\varOmega \det (\nabla u_j(x)) \;\text {d}x $$

is not weakly lower semicontinuous on $\mathrm {W}^{1,2}(\varOmega ;\mathbb {R}^2)$ by considering the sequence

$$ u_j(x, y) := \frac{(1-|x_2|)^j}{\sqrt{j}} \bigl ( \sin (jx),\cos (jx) \bigr ). $$

5.6.

Show that we may extend Morrey’s Theorem  5.16 to Carathéodory integrands $f :\varOmega \times \mathbb {R}^{m \times d} \rightarrow \mathbb {R}$ that take negative values as long as

$$ -M^{-1}|A|^q - M \le f(A) \le M(1+|A|^p), \qquad A \in \mathbb {R}^{m \times d}, $$

where $q \in (0,p)$ and $M > 0$. Hint: Observe that the family of negative parts $\{f(\nabla u_j)\}_j$ is equiintegrable.

5.7.

Prove that every quadratic form $q :\mathbb {R}^{m \times d} \rightarrow \mathbb {R}$, that is, $q(A) = b(A, A)$ for a bilinear $b :\mathbb {R}^{m \times d} \times \mathbb {R}^{m \times d} \rightarrow \mathbb {R}$, is quasiconvex if and only if it is rank-one convex. Hint: Use Plancherel’s identity (A.4).

5.8.

Complete the proof of Lemma 5.10 for higher dimensions. Hint: Use the multilinear algebra formulation with differential forms and an induction over the dimension.

5.9.

Show that a weaker version of Lemma 5.10 is true if $r = p$, where we only have the convergence of the minors in the sense of distributions.

5.10.

A locally bounded Borel-measurable function $h :\mathbb {R}^{m \times d} \rightarrow \mathbb {R}$ is called $\mathrm {W}^{1,p}$ -closed-quasiconvex if

for all $F \in \mathbb {R}^{m \times d}$ and for all homogeneous $\mathrm {W}^{1,p}$-gradient Young measures $\nu \in \mathbf {GY}^p(B(0,1);\mathbb {R}^{m \times d})$ with $[\nu ] = F$. Show that for all continuous h that satisfy the p-growth condition $|h(A)| \le M(1+|A|^p)$, $\mathrm {W}^{1,p}$-closed-quasiconvexity is equivalent to $\mathrm {W}^{1,p}$-quasiconvexity.