Polyconvex function

In the calculus of variations, the notion of polyconvexity is a generalization of the notion of convexity for functions defined on spaces of matrices. The notion of polyconvexity was introduced by John M. Ball as a sufficient conditions for proving the existence of energy minimizers in nonlinear elasticity theory. It is satisfied by a large class of hyperelastic stored energy densities, such as Mooney-Rivlin and Ogden materials. The notion of polyconvexity is related to the notions of convexity, quasiconvexity and rank-one convexity through the following diagram:

$f{\text{ convex}}\implies f{\text{ polyconvex}}\implies f{\text{ quasiconvex}}\implies f{\text{ rank-one convex}}$

Motivation

Let $\Omega \subset \mathbb {R} ^{n}$ be an open bounded domain, $u:\Omega \rightarrow \mathbb {R} ^{m}$ and $W^{1,p}(\Omega ,\mathbb {R} ^{m})$ denote the Sobolev space of mappings from $\Omega$ to $\mathbb {R} ^{m}$. A typical problem in the calculus of variations is to minimize a functional, $E:W^{1,p}(\Omega ,\mathbb {R} ^{m})\rightarrow \mathbb {R}$ of the form

$E[u]=\int _{\Omega }f(x,\nabla u(x))dx$,

where the energy density function, $f:\Omega \times \mathbb {R} ^{m\times n}\rightarrow [0,\infty )$ satisfies $p$-growth, i.e., $|f(x,A)|\leq M(1+|A|^{p})$ for some $M>0$ and $p\in (1,\infty )$. It is well-known from a theorem of Morrey and Acerbi-Fusco that a necessary and sufficient condition for $E$ to weakly lower-semicontinuous on $W^{1,p}(\Omega ,\mathbb {R} ^{m})$ is that $f(x,\cdot )$ is quasiconvex for almost every $x\in \Omega$. With coercivity assumptions on $f$ and boundary conditions on $u$, this leads to the existence of minimizers for $E$ on $W^{1,p}(\Omega ,\mathbb {R} ^{m})$. However, in many applications, the assumption of $p$-growth on the energy density is often too restrictive. In the context of elasticity, this is because the energy is required to grow unboundedly to $+\infty$ as local measures of volume approach zero. This led Ball to define the more restrictive notion of polyconvexity to prove the existence of energy minimizers in nonlinear elasticity.

Definition

A function $f:\mathbb {R} ^{m\times n}\rightarrow \mathbb {R}$ is said to be polyconvex if there exists a convex function $\Phi :\mathbb {R} ^{\tau (m,n)}\rightarrow \mathbb {R}$ such that

$f(F)=\Phi (T(F))$

where $T:\mathbb {R} ^{m\times n}\rightarrow \mathbb {R} ^{\tau (m,n)}$ is such that

$T(F):=(F,{\text{adj}}_{2}(F),...,{\text{adj}}_{m\wedge n}(F)).$

Here, ${\text{adj}}_{s}$ stands for the matrix of all $s\times s$ minors of the matrix $F\in \mathbb {R} ^{m\times n}$, $2\leq s\leq m\wedge n:=\min(m,n)$ and

$\tau (m,n):=\sum _{s=1}^{m\wedge n}\sigma (s),$

where $\sigma (s):={\binom {m}{s}}{\binom {n}{s}}$.

When $m=n=2$, $T(F)=(F,\det F)$ and when $m=n=3$, $T(F)=(F,{\text{cof}}\,F,\det F)$, where ${\text{cof}}\,F$ denotes the cofactor matrix of $F$.

In the above definitions, the range of $f$ can also be extended to $\mathbb {R} \cup \{+\infty \}$.

Properties

• If $f$ takes only finite values, then polyconvexity implies quasiconvexity and thus leads to the weak lower semicontinuity of the corresponding integral functional on a Sobolev space.

• If $m=1$ or $n=1$, then polyconvexity reduces to convexity.

• If $f$ is polyconvex, then it is locally Lipschitz.

• Polyconvex functions with subquadratic growth must be convex, i.e., if there exists $\alpha \geq 0$ and $0\leq p<2$ such that

$f(F)\leq \alpha (1+|F|^{p})$ for every $F\in \mathbb {R} ^{m\times n}$, then $f$ is convex.

Examples

• Every convex function is polyconvex.
• For the case $m=n$, the determinant function is polyconvex, but not convex. In particular, the following type of function that commonly appears in nonlinear elasticity is polyconvex but not convex:

$f(A)={\begin{cases}{\frac {1}{\det(A)}},&\det(A)>0;\\+\infty ,&\det(A)\leq 0;\end{cases}}$