zbMATH — the first resource for mathematics

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 \[ g(X, z) = \begin{cases} (|X|^2+ 2\text{ det } X- 2z)(|X|^2\text{ det } X- 4z)\quad & \text{if}\quad |X|^2+ 2\text{ det } X\geq 4z\\ 0\quad & \text{elsewhere}.\end{cases}. \]

49J45 Methods involving semicontinuity and convergence; relaxation
Full Text: DOI
[1] Dacorogna, Proc. Roy. Soc. Edinburgh Sect. A 114 pp 135– (1990) · Zbl 0722.49018
[2] Dacorogna, Direct Methods in the Calculus of Variations (1989) · Zbl 0703.49001
[3] Ball, Arch. Rational Mech. Anal. 64 pp 337– (1977)
[4] DOI: 10.1007/BF00387763 · Zbl 0761.26009
[5] Šverák, Proc. Roy. Soc. Edinburgh Sect. A 120 pp 185– (1992) · Zbl 0777.49015
[6] DOI: 10.1098/rspa.1991.0073 · Zbl 0741.49016
[7] Dacorogna, Ann. Fac. Sci. Toulouse II pp 163– (1993) · Zbl 0828.49016
[8] DOI: 10.1002/cpa.3160390305 · Zbl 0694.49004
[9] Morrey, Pacific J. Math. 2 pp 25– (1952) · Zbl 0046.10803
[10] DOI: 10.1007/BF00411477 · Zbl 0793.58002
[11] Hartwig, Proceedings of the IVth International Workshop on Generalized Convexity, Pécs, 1992 (1994)
[12] Dacorogna, Material Instabilities in Continuum Mechanics. Proceedings pp 77– (1988)
[13] Morrey, Multiple Integrals in the Calculus of Variations (1966) · Zbl 0142.38701
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.