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In the [[calculus of variations]], the notion of '''polyconvexity''' is a generalization of the notion of [[convex function|convexity]] for [[function (mathematics)|functions]] defined on spaces of [[matrix (mathematics)|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]].<ref>{{cite journal
|last1=Ball
|first1=John M.
|author-link=John M. Ball
|year=1976
|title=Convexity conditions and existence theorems in nonlinear elasticity
|journal=Archive for Rational Mechanics and Analysis
|volume=63
|issue=4
|pages=337–403
|publisher=Springer
|doi=10.1007/BF00279992
|bibcode=1976ArRMA..63..337B
|url=https://people.maths.ox.ac.uk/ball/Papers/Ball77.pdf
}}</ref> It is satisfied by a large class of [[hyperelasticity|hyperelastic]] stored energy densities, such as [[Mooney-Rivlin solid|Mooney-Rivlin]] and [[Ogden hyperelastic model|Ogden]] materials. The notion of polyconvexity is related to the notions of convexity, [[quasiconvexity (calculus of variations)|quasiconvexity]] and rank-one convexity through the following diagram:<ref>{{cite book
|last=Dacorogna
|first=Bernard
|author-link=Bernard Dacorogna
|title=Direct Methods in the Calculus of Variations
|publisher=Springer Science+Business Media, LLC
|series= Applied mathematical sciences
|year=2008
|volume=78
|doi=10.1007/978-0-387-55249-1
|isbn=978-0-387-35779-9
|edition= 2nd
|page=156
|url=http://infoscience.epfl.ch/record/129683
}}</ref>

:<math>f\text{ convex}\implies f\text{ polyconvex}\implies f\text{ quasiconvex}\implies f\text{ rank-one convex}</math>

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

:<math>E[u]=\int_\Omega f(x,\nabla u(x))dx</math>,

where the energy density function, <math>f:\Omega\times\mathbb{R}^{m\times n}\rightarrow[0,\infty)</math> satisfies <math>p</math>-growth, i.e., <math>|f(x,A)|\leq M(1+|A|^p)</math> for some <math>M>0</math> and <math>p\in(1,\infty)</math>. It is well-known from a theorem of [[Charles Morrey|Morrey]] and Acerbi-Fusco that a necessary and sufficient condition for <math>E</math> to [[weak topology|weakly]] [[semi-continuity|lower-semicontinuous]] on <math>W^{1,p}(\Omega,\mathbb{R}^m)</math> is that <math>f(x,\cdot)</math> is quasiconvex for [[Almost everywhere|almost every]] <math>x\in\Omega</math>. With [[coercive function|coercivity]] assumptions on <math>f</math> and boundary conditions on <math>u</math>, this leads to the existence of minimizers for <math>E</math> on <math>W^{1,p}(\Omega,\mathbb{R}^m)</math>.<ref>{{cite book
|last=Rindler
|first=Filip
|title=Calculus of Variations
|publisher=Springer International Publishing AG
|series=Universitext
|year=2018
|doi=10.1007/978-3-319-77637-8
|isbn=978-3-319-77636-1
|page=124-125
}}</ref> However, in many applications, the assumption of <math>p</math>-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 <math>+\infty</math> 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 <math>f:\mathbb{R}^{m\times n}\rightarrow\mathbb{R}</math> is said to be polyconvex<ref>{{cite book
|last=Dacorogna
|first=Bernard
|author-link=Bernard Dacorogna
|title=Direct Methods in the Calculus of Variations
|publisher=Springer Science+Business Media, LLC
|series= Applied mathematical sciences
|year=2008
|volume=78
|doi=10.1007/978-0-387-55249-1
|isbn=978-0-387-35779-9
|edition= 2nd
|page=157
|url=http://infoscience.epfl.ch/record/129683
}}</ref> if there exists a ''convex'' function <math>\Phi:\mathbb{R}^{\tau(m,n)}\rightarrow\mathbb{R}</math> such that

:<math> f(F)=\Phi(T(F))</math>

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

:<math>T(F):=(F,\text{adj}_2(F),...,\text{adj}_{m\wedge n}(F)).</math>

Here, <math>\text{adj}_s</math> stands for the matrix of all <math>s\times s</math> [[Minor (linear algebra)|minors]] of the matrix <math>F\in\mathbb{R}^{m\times n}</math>, <math>2\leq s\leq m\wedge n:=\min(m,n)</math> and

:<math>\tau(m,n):=\sum_{s=1}^{m\wedge n}\sigma(s),</math>

where <math>\sigma(s):=\binom{m}{s}\binom{n}{s}</math>.

When <math>m=n=2</math>, <math>T(F)=(F,\det F)</math> and when <math>m=n=3</math>, <math>T(F)=(F,\text{cof}\,F,\det F)</math>, where <math>\text{cof}\,F</math> denotes the [[adjugate matrix|cofactor matrix]] of <math>F</math>.

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

==Properties==
* If <math>f</math> 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 <math>m=1</math> or <math>n=1</math>, then polyconvexity reduces to convexity.

* If <math>f</math> is polyconvex, then it is [[Lipschitz function|locally Lipschitz]].

* Polyconvex functions with subquadratic growth must be convex, i.e., if there exists <math>\alpha\geq 0</math> and <math>0\leq p<2</math> such that
:<math> f(F)\leq\alpha (1+|F|^p)</math> for every <math> F\in \mathbb{R}^{m\times n}</math>, then <math>f</math> is convex.

==Examples==
* Every convex function is polyconvex.
* For the case <math>m=n</math>, 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:

:<math>f(A) = \begin{cases} \frac1{\det (A)}, & \det (A) > 0; \\ + \infty, & \det (A) \leq 0; \end{cases}</math>

==References==
{{Reflist}}

[[Category:Convex analysis]]
[[Category:Calculus of variations]]
[[Category:Matrices (mathematics)]]
[[Category:Types of functions]]

Polyconvex function - Revision history

imported>Citation bot: Added url. | Use this bot. Report bugs. | Suggested by Dominic3203 | Category:Matrices (mathematics) | #UCB_Category 162/234