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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:
  Dacorogna, Proc. Roy. Soc. Edinburgh Sect. A 114 pp 135– (1990) · Zbl 0722.49018  Dacorogna, Direct Methods in the Calculus of Variations (1989) · Zbl 0703.49001  Ball, Arch. Rational Mech. Anal. 64 pp 337– (1977)  DOI: 10.1007/BF00387763 · Zbl 0761.26009  Šverák, Proc. Roy. Soc. Edinburgh Sect. A 120 pp 185– (1992) · Zbl 0777.49015  DOI: 10.1098/rspa.1991.0073 · Zbl 0741.49016  Dacorogna, Ann. Fac. Sci. Toulouse II pp 163– (1993) · Zbl 0828.49016  DOI: 10.1002/cpa.3160390305 · Zbl 0694.49004  Morrey, Pacific J. Math. 2 pp 25– (1952) · Zbl 0046.10803  DOI: 10.1007/BF00411477 · Zbl 0793.58002  Hartwig, Proceedings of the IVth International Workshop on Generalized Convexity, Pécs, 1992 (1994)  Dacorogna, Material Instabilities in Continuum Mechanics. Proceedings pp 77– (1988)  Morrey, Multiple Integrals in the Calculus of Variations (1966) · Zbl 0142.38701
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