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A polyconvexity condition in dimension two. (English) Zbl 0839.49013
An important criterion for $$f\in C^1(R^{2\times 2})$$ to be polyconvex has been formulated: If $$h: R^{2\times 2}\times R^{2\times 2}\times R\to R$$ is defined by $h(X, A, \sigma)= f(X)- f(A)- \langle\nabla f(A), X- A\rangle- \sigma\text{ det}(X- A)$ then the necessary and sufficient condition for the polyconvexity of $$f$$ is that for every $$A\in R^{2\times 2}$$ there exists $$\sigma(A)\in R$$ such that $\inf_X h(X, A, \sigma(A))= h(A, A, \sigma(A))= 0.$ This criterion has been applied to the known polyconvex function $$f(X)= |X|^2(|X|^2- 2\text{ det }X)$$ for obtaining its convex representation $$g: R^{2\times 2}\times R\to R$$ ($$g$$ is convex with $$g(X, \text{det } X)= f(X)$$ for all $$X\in R^{2\times 2}$$) which turned out to be of the form $g(X, z) = \begin{cases} (|X|^2+ 2\text{ det } X- 2z)(|X|^2\text{ det } X- 4z)\quad & \text{if}\quad |X|^2+ 2\text{ det } X\geq 4z\\ 0\quad & \text{elsewhere}.\end{cases}.$

##### MSC:
 49J45 Methods involving semicontinuity and convergence; relaxation
##### Keywords:
polyconvex function; convex representation
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##### References:
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