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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:

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

Motivation

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

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

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

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

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

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

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

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

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

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

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

Properties

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

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

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

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

Examples

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

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

References

1. Ball, John M. (1976). "Convexity conditions and existence theorems in nonlinear elasticity" (PDF). Archive for Rational Mechanics and Analysis. 63 (4). Springer: 337–403. doi:10.1007/BF00279992.
2. Dacorogna, Bernard (2008). Direct Methods in the Calculus of Variations. Applied mathematical sciences. Vol. 78 (2nd ed.). Springer Science+Business Media, LLC. p. 156. doi:10.1007/978-0-387-55249-1. ISBN 978-0-387-35779-9.
3. Rindler, Filip (2018). Calculus of Variations. Universitext. Springer International Publishing AG. p. 124-125. doi:10.1007/978-3-319-77637-8. ISBN 978-3-319-77636-1.
4. Dacorogna, Bernard (2008). Direct Methods in the Calculus of Variations. Applied mathematical sciences. Vol. 78 (2nd ed.). Springer Science+Business Media, LLC. p. 157. doi:10.1007/978-0-387-55249-1. ISBN 978-0-387-35779-9.

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