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:


 * $$f(A) = \begin{cases} \frac1{\det (A)}, & \det (A) > 0; \\ + \infty, & \det (A) \leq 0; \end{cases}$$