Hostname: page-component-cd9895bd7-p9bg8 Total loading time: 0 Render date: 2024-12-25T05:20:33.327Z Has data issue: false hasContentIssue false
  • English
  • Français

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]
Get access

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

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