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Quasiconvex functions can be approximatedby quasiconvex polynomials

Published online by Cambridge University Press:  30 January 2008

Sebastian Heinz
Sebastian Heinz*
Affiliation:
Humboldt-Universität zu Berlin, Mathematisch-Naturwissenschaftliche Fakultät II, Institut für Mathematik, DFG-Graduiertenkolleg 1128, Germany; [email protected]
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Abstract

Let W be a function from the real m×n-matrices to the real numbers. If W is quasiconvex in the sense of the calculus of variations, then we show that W can be approximated locally uniformly by quasiconvex polynomials.

Keywords

Stone-Weierstrass theoremlocally uniform convergence
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 14 , Issue 4 , October 2008 , pp. 795 - 801
Copyright
© EDP Sciences, SMAI, 2008

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References

Alibert, J.-J. and Dacorogna, B., An example of a quasiconvex function that is not polyconvex in two dimensions. Arch. Rational Mech. Anal. 117 (1992) 155–166. CrossRef
J.M. Ball, Some open problems in elasticity, in Geometry, Mechanics, and Dynamics – Volume in Honor of the 60th Birthday of J.E. Marsden, P. Newton, P. Holmes and A. Weinstein Eds., Springer-Verlag (2002) 3–59.
B. Dacorogna, Direct Methods in the Calculus of Variations. Springer-Verlag (1989).
D. Faraco and L. Székelyhidi, Tartar's conjecture and localization of the quasiconvex hull in $\mathbb{R}^{2\times 2}$ . Max Planck Institute for Mathematics in the Sciences, Preprint N° 60 (2006).
Gutiérrez, S., A necessary condition for the quasiconvexity of polynomials of degree four. J. Convex Anal. 13 (2006) 51–60.
Iwaniec, T., Nonlinear Cauchy-Riemann operators in $\mathbb{R}^n$ . Trans. Amer. Math. Soc. 354 (2002) 1961–1995. CrossRef
Iwaniec, T. and Kristensen, J., A construction of quasiconvex functions. Rivista di Matematica Università di Parma 4 (2005) 75–89.
Kristensen, J., On the non-locality of quasiconvexity. Ann. Inst. H. Poincaré Anal. Non Linéaire 16 (1999) 1–13. CrossRef
Morrey, C.B., Quasi-convexity and the lower semicontinuity of multiple integrals. Pacific J. Math. 2 (1952) 25–53. CrossRef
Müller, S., A sharp version of Zhang's theorem on truncating sequences of gradients. Trans. Amer. Math. Soc. 351 (1999) 4585–4597. CrossRef
Müller, S., Rank-one convexity implies quasiconvexity on diagonal matrices. Internat. Math. Res. Not. 20 (1999) 1087–1095. CrossRef
F. Sauvigny, Partial differential equations, Foundations and Integral Representations 1. Springer-Verlag (2006).
Šverák, V., Rank-one convexity does not imply quasiconvexity. Proc. Roy. Soc. Edinburgh 120A (1992) 185–189. CrossRef