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.^[1] 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:^[2]

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

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

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

where the energy density function, ${\displaystyle f:\Omega \times \mathbb {R} ^{m\times n}\rightarrow [0,\infty )}${\displaystyle f:\Omega \times \mathbb {R} ^{m\times n}\rightarrow [0,\infty )} satisfies ${\displaystyle p}${\displaystyle p}-growth, i.e., ${\displaystyle |f(x,A)|\leq M(1+|A|^{p})}${\displaystyle |f(x,A)|\leq M(1+|A|^{p})} for some ${\displaystyle M>0}${\displaystyle M>0} and ${\displaystyle p\in (1,\infty )}${\displaystyle p\in (1,\infty )}. It is well-known from a theorem of Morrey and Acerbi-Fusco that a necessary and sufficient condition for ${\displaystyle E}${\displaystyle E} to weakly lower-semicontinuous on ${\displaystyle W^{1,p}(\Omega ,\mathbb {R} ^{m})}${\displaystyle W^{1,p}(\Omega ,\mathbb {R} ^{m})} is that ${\displaystyle f(x,\cdot )}${\displaystyle f(x,\cdot )} is quasiconvex for almost every ${\displaystyle x\in \Omega }${\displaystyle x\in \Omega }. With coercivity assumptions on ${\displaystyle f}${\displaystyle f} and boundary conditions on ${\displaystyle u}${\displaystyle u}, this leads to the existence of minimizers for ${\displaystyle E}${\displaystyle E} on ${\displaystyle W^{1,p}(\Omega ,\mathbb {R} ^{m})}${\displaystyle W^{1,p}(\Omega ,\mathbb {R} ^{m})}.^[3] However, in many applications, the assumption of ${\displaystyle p}${\displaystyle 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 ${\displaystyle +\infty }${\displaystyle +\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.

A function ${\displaystyle f:\mathbb {R} ^{m\times n}\rightarrow \mathbb {R} }${\displaystyle f:\mathbb {R} ^{m\times n}\rightarrow \mathbb {R} } is said to be polyconvex^[4] if there exists a convex function ${\displaystyle \Phi :\mathbb {R} ^{\tau (m,n)}\rightarrow \mathbb {R} }${\displaystyle \Phi :\mathbb {R} ^{\tau (m,n)}\rightarrow \mathbb {R} } such that

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

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

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

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

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

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

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

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

• If ${\displaystyle f}${\displaystyle 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 ${\displaystyle m=1}${\displaystyle m=1} or ${\displaystyle n=1}${\displaystyle n=1}, then polyconvexity reduces to convexity.

• If ${\displaystyle f}${\displaystyle f} is polyconvex, then it is locally Lipschitz.

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

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

• Every convex function is polyconvex.
• For the case ${\displaystyle m=n}${\displaystyle 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:

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

• Convex analysis
• Calculus of variations
• Matrices
• Types of functions