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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
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