Polyconvexity

Abstract

At the beginning of the previous chapter we saw that convexity cannot hold concurrently with frame-indifference (and a mild non-degeneracy condition). Thus, we were led to consider quasiconvex integrands. However, while quasiconvexity is of tremendous importance in the theory of the calculus of variations, our Lower Semicontinuity Theorem 5.16 has one major drawback: we needed to require the p-growth bound

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

Appendices

Notes and Historical Remarks

Theorem 6.7 is a refined version of Ball’s original result [25] due to Ball, Currie & Olver [29]; also see [30, 31] for further reading. This theorem and its applications to elasticity theory are described in great detail in [64].

Many questions about polyconvex integral functionals remain open to this day. In particular, the regularity of solutions and the validity of the Euler–Lagrange equations are largely unknown in the general case. Note that the regularity theory from the previous chapter is not in general applicable, at least if we do not assume the upper p-growth. These questions are even open for the more restricted situation of nonlinear elasticity theory. See [28] for a survey on the current state of the art and a collection of challenging open problems. We note that since the publication of [28] counterexamples to uniqueness have been found, see [245].

As for the almost injectivity (for $p > 3$), this is in fact sometimes automatic, as shown by Ball [26], but the arguments do not apply to all situations. Tang [266] extended this to $p>2$, but since then one has to deal with non-continuous functions, it is not even obvious how to define $u(\varOmega )$.

Theorem 6.10 is from [26]. More general results can be found in [248]. The questions of injectivity, invertibility, and regularity are intimately connected with cavitation and fracture phenomena, see, for instance, [204] and the recent [148], which also contains a large bibliography.

Finally, we mention in passing the alternative so-called intrinsic approach to elasticity, as pioneered by Ciarlet, see [66].

Problems

6.1

Define $h :\mathbb {R}^{3 \times 3} \rightarrow \mathbb {R}$ via

$$ h(A) := \bigl (|A|^6 + |{{\mathrm{{cof}\,}}}A|^6\bigr )^{1/2} + g(\det A), \qquad A \in \mathbb {R}^{3 \times 3}, $$

where $g :\mathbb {R}\rightarrow [0,\infty ]$ is convex and continuous. Prove that h is polyconvex.

6.2

Show that the function

$$ h(A) := {\left\{ \begin{array}{ll} \displaystyle |A| \ln (1+|A|) + \frac{1}{\det A} &{}\text {if}\ \det A > 0, \\ +\infty &{}\text {if}\ \det A \le 0, \end{array}\right. } \qquad A \in \mathbb {R}^{3 \times 3}, $$

is polyconvex.

6.3

Show that for $u \in (\mathrm {W}^{1,3} \cap \mathrm {C}^2)(\varOmega ;\mathbb {R}^3)$ it holds that

$$ \det \nabla u(x) = \sum _{l=1}^3 \partial _l(u^1 ({{\mathrm{{cof}\,}}}\nabla u(x))^1_l), \qquad x \in \varOmega . $$

Hint: Use the Piola identity.

6.4

Let $u_j, u \in (\mathrm {W}^{1,3} \cap \mathrm {C}^2)(\varOmega ;\mathbb {R}^3)$, $j \in \mathbb {N}$, with

$$ {{\mathrm{{cof}\,}}}\nabla u_j, \; {{\mathrm{{cof}\,}}}\nabla u \in \mathrm {L}^3(\varOmega ;\mathbb {R}^{3 \times 3}). $$

Prove that if

$$ u_j \rightharpoonup u \quad \text { in}\ \mathrm {W}^{1,3} \qquad \text {and}\qquad {{\mathrm{{cof}\,}}}\nabla u_j \rightharpoonup {{\mathrm{{cof}\,}}}\nabla u \quad \text { in}\ \mathrm {L}^3, $$

then $\det \nabla u_j, \det \nabla u \in \mathrm {L}^{3/2}(\varOmega )$ and $\det \nabla u_j \rightharpoonup \det \nabla u$ in $\mathrm {L}^{3/2}$.

6.5

In this problem we will construct a rank-one convex function that is not polyconvex.

• (i) Find $A_1,A_2,A_3 \in \mathbb {R}^{2 \times 2}$ and $\theta _1, \theta _2, \theta _3 \in (0,1)$ such that simultaneously

  • (a) $\theta _1 + \theta _2 + \theta _3 = 1$;

  • (b) $\displaystyle \sum _{k=1}^3 \theta _i \det A_i = \det \left[ \sum _{k=1}^3 \theta _i A_i \right]$;

  • (c) $\det (A_1 - A_2) \ne 0$, $\det (A_1 - A_3) \ne 0$, $\det (A_2 - A_3) \ne 0$;

  • (d) $\displaystyle \sum _{k=1}^3 \theta _i A_i \notin \{A_1,A_2,A_3\}$.

• (ii) Define with the $A_1, A_2, A_3$ from (i) the function $f :\mathbb {R}^{2 \times 2} \rightarrow \mathbb {R}\cup \{+\infty \}$ as

  $$ f(A) := {\left\{ \begin{array}{ll} 0 &{} \text {if}\ A \in \{A_1,A_2,A_3\},\\ +\infty &{} \text {otherwise.} \end{array}\right. } $$

  Show that f is rank-one convex (extending the definition of rank-one convexity in a suitable way to $(\mathbb {R}\cup \{+\infty \})$-valued functions).

• (iii) Show that f is not polyconvex, i.e., there exists no convex function $F :\mathbb {R}^{2 \times 2} \times \mathbb {R}\rightarrow \mathbb {R}\cup \{+\infty \}$ such that

  $$ f(A) = F(A,\det A), \qquad A \in \mathbb {R}^{2 \times 2}. $$

6.6

Show that the function f from Example 6.4 (Ogden materials) is polyconvex. Hint: Use Cramer’s rule to see that ${{\mathrm{{cof}\,}}}(AB) = {{\mathrm{{cof}\,}}}(A) {{\mathrm{{cof}\,}}}(B)$.

6.7

Set

$$\begin{aligned} \mathrm {GL}(d)&:= \bigl \{\, A \in \mathbb {R}^{d \times d} \ \ \mathbf : \ \ \det A \ne 0 \,\bigr \} \qquad \text {and} \\ \mathrm {GL}^+(d)&:= \bigl \{\, A \in \mathbb {R}^{d \times d} \ \ \mathbf : \ \ \det A > 0 \,\bigr \}. \end{aligned}$$

Prove that the convex hull $\mathrm {GL}^+(3)^{**}$ of $\mathrm {GL}^+(3)$ is equal to $\mathrm {GL}(3)$.

6.8

With the notation from the previous problem, define

$$ U := \bigl \{\, (A, {{\mathrm{{cof}\,}}}A, \det A) \in \mathrm {GL}(3) \times \mathrm {GL}(3) \times \mathbb {R} \ \ \mathbf : \ \ A \in \mathrm {GL}^+(3) \,\bigr \} $$

and show that

$$ U^{**} = \mathrm {GL}(3) \times \mathrm {GL}(3) \times (0,\infty ). $$

Hint: Show first:

• (i) $(A, H,\delta ) \in U$ and $G \in \mathrm {GL}^+(3)$ implies that $(GA,({{\mathrm{{cof}\,}}}G)H,(\det G)\delta ) \in U^{**}$;

• (ii) $(\pm \mathrm {Id}, 0,\delta ) \in U$ and $(0,\pm \mathrm {Id},\delta ) \in U^{**}$ for all $\delta > 0$;

• (iii) $(A, 0,\delta ), (0,H,\delta ) \in U^{**}$ for all $A, H \in \mathrm {GL}(3)$ and all $\delta > 0$ (see the previous problem).

6.9

Let $\varPhi :[0,\infty )^d \rightarrow \mathbb {R}$ be symmetric, jointly convex, and increasing (in every variable). Denote by $\sigma _1(A), \ldots , \sigma _d(A) \ge 0$ the singular values of a matrix $A \in \mathbb {R}^{d \times d}$. Then, show that

$$ g(A) := \varPhi (\sigma _1(A), \ldots , \sigma _d(A)), \qquad A \in \mathbb {R}^{d \times d}, $$

is convex.

6.10

Let $h :\mathbb {R}^{3 \times 3} \rightarrow \mathbb {R}\cup \{+\infty \}$ be of the form

$$ h(A) := {\left\{ \begin{array}{ll} \varTheta \bigl ( \sigma _1, \sigma _2, \sigma _3, \sigma _1\sigma _2, \sigma _2\sigma _3, \sigma _3\sigma _1, \sigma _1\sigma _2\sigma _3 \bigr ) &{}\text {if}\ \det A > 0,\\ +\infty &{}\text {if}\ \det A \le 0, \end{array}\right. } $$

where $\sigma _1,\sigma _2,\sigma _3$ are the three singular values of A and the function $\varTheta :[0,\infty )^7 \times (0,\infty ) \rightarrow \mathbb {R}$ is jointly convex, increasing in the first six variables, and

$$ \varTheta (x_1,x_2,x_3,y_1,y_2,y_3,z) = \varTheta (x_{\gamma (1)}, x_{\gamma (2)}, x_{\gamma (3)}, y_{\eta (1)}, y_{\eta (2)}, y_{\eta (3)}, z) $$

for all permutations $\gamma ,\eta :\{1,2,3\} \rightarrow \{1,2,3\}$ and all $x_1,x_2,x_3,y_1,y_2,y_3 \in [0,\infty )$, $z \in (0,\infty )$. Show that h is polyconvex. Hint: Use the previous problem.