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:

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

Motivation

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

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

where the energy density function, $f:\mathrm{\Omega }\times {\mathbb{ℝ}}^{m\times n}\to [0,\mathrm{\infty })$ satisfies $p$-growth, i.e., $|f(x,A)|\le M(1+|A{|}^p)$ for some $M>0$ and $p\in (1,\mathrm{\infty })$. 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 $W^{1,p}(\mathrm{\Omega },{\mathbb{ℝ}}^m)$ is that $f(x,\cdot )$ is quasiconvex for almost every $x\in \mathrm{\Omega }$. With coercivity assumptions on $f$ and boundary conditions on $u$, this leads to the existence of minimizers for $E$ on $W^{1,p}(\mathrm{\Omega },{\mathbb{ℝ}}^m)$. 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 $+\mathrm{\infty }$ 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 $f:{\mathbb{ℝ}}^{m\times n}\to \mathbb{ℝ}$ is said to be polyconvex if there exists a convex function $\mathrm{\Phi }:{\mathbb{ℝ}}^{\tau (m,n)}\to \mathbb{ℝ}$ such that

$f(F)=\mathrm{\Phi }(T(F))$

where $T:{\mathbb{ℝ}}^{m\times n}\to {\mathbb{ℝ}}^{\tau (m,n)}$ 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 $F\in {\mathbb{ℝ}}^{m\times n}$, $2\le s\le m\wedge n:=\min (m,n)$ and

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

where $\sigma (s):=(\genfrac{}{}{0ex}{}{m}{s})(\genfrac{}{}{0ex}{}{n}{s})$.

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 $\mathbb{ℝ}\cup \{+\mathrm{\infty }\}$.

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 $\alpha \ge 0$ and $0\le p<2$ such that

$f(F)\le \alpha (1+|F{|}^p)$ 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:

$f(A)=\{\begin{array}{ll}\frac{1}{\det (A)},& \det (A)>0;\\ +\mathrm{\infty },& \det (A)\le 0;\end{array}$