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}

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 be 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 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} ==References==

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