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 convex ⟹ f polyconvex ⟹ f quasiconvex ⟹ f rank-one convex {\displaystyle f{\text{ convex}}\implies f{\text{ polyconvex}}\implies f{\text{ quasiconvex}}\implies f{\text{ rank-one convex}}}
Motivation Let Ω ⊂ R n {\displaystyle \Omega \subset \mathbb {R} ^{n}} be an open bounded domain, u : Ω → R m {\displaystyle u:\Omega \rightarrow \mathbb {R} ^{m}} and W 1 , p ( Ω , R m ) {\displaystyle W^{1,p}(\Omega ,\mathbb {R} ^{m})} denote the Sobolev space of mappings from Ω {\displaystyle \Omega } to R m {\displaystyle \mathbb {R} ^{m}} . A typical problem in the calculus of variations is to minimize a functional, E : W 1 , p ( Ω , R m ) → R {\displaystyle E:W^{1,p}(\Omega ,\mathbb {R} ^{m})\rightarrow \mathbb {R} } of the form
E [ u ] = ∫ Ω f ( x , ∇ u ( x ) ) d x {\displaystyle E[u]=\int _{\Omega }f(x,\nabla u(x))dx} , where the energy density function, f : Ω × R m × n → [ 0 , ∞ ) {\displaystyle f:\Omega \times \mathbb {R} ^{m\times n}\rightarrow [0,\infty )} satisfies p {\displaystyle p} -growth, i.e., | f ( x , A ) | ≤ M ( 1 + | A | p ) {\displaystyle |f(x,A)|\leq M(1+|A|^{p})} for some M > 0 {\displaystyle M>0} and p ∈ ( 1 , ∞ ) {\displaystyle p\in (1,\infty )} . It is well-known from a theorem of Morrey and Acerbi-Fusco that a necessary and sufficient condition for E {\displaystyle E} to be weakly lower-semicontinuous on W 1 , p ( Ω , R m ) {\displaystyle W^{1,p}(\Omega ,\mathbb {R} ^{m})} is that f ( x , ⋅ ) {\displaystyle f(x,\cdot )} is quasiconvex for almost every x ∈ Ω {\displaystyle x\in \Omega } . With coercivity assumptions on f {\displaystyle f} and boundary conditions on u {\displaystyle u} , this leads to the existence of minimizers for E {\displaystyle E} on W 1 , p ( Ω , R m ) {\displaystyle W^{1,p}(\Omega ,\mathbb {R} ^{m})} . However, in many applications, the assumption of 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 } 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.
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