Quasiconvex functions can be approximated by quasiconvex polynomials

Sebastian Heinz

ESAIM: Control, Optimisation and Calculus of Variations (2008)

  • Volume: 14, Issue: 4, page 795-801
  • ISSN: 1292-8119

Abstract

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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.

How to cite

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Heinz, Sebastian. "Quasiconvex functions can be approximated by quasiconvex polynomials." ESAIM: Control, Optimisation and Calculus of Variations 14.4 (2008): 795-801. <http://eudml.org/doc/250277>.

@article{Heinz2008,
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. },
author = {Heinz, Sebastian},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {Stone-Weierstrass theorem; locally uniform convergence},
language = {eng},
month = {1},
number = {4},
pages = {795-801},
publisher = {EDP Sciences},
title = {Quasiconvex functions can be approximated by quasiconvex polynomials},
url = {http://eudml.org/doc/250277},
volume = {14},
year = {2008},
}

TY - JOUR
AU - Heinz, Sebastian
TI - Quasiconvex functions can be approximated by quasiconvex polynomials
JO - ESAIM: Control, Optimisation and Calculus of Variations
DA - 2008/1//
PB - EDP Sciences
VL - 14
IS - 4
SP - 795
EP - 801
AB - 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.
LA - eng
KW - Stone-Weierstrass theorem; locally uniform convergence
UR - http://eudml.org/doc/250277
ER -

References

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  8. J. Kristensen, On the non-locality of quasiconvexity. Ann. Inst. H. Poincaré Anal. Non Linéaire16 (1999) 1–13.  Zbl0932.49015
  9. C.B. Morrey, Quasi-convexity and the lower semicontinuity of multiple integrals. Pacific J. Math.2 (1952) 25–53.  Zbl0046.10803
  10. S. Müller, A sharp version of Zhang's theorem on truncating sequences of gradients. Trans. Amer. Math. Soc.351 (1999) 4585–4597.  Zbl0942.49013
  11. S. Müller, Rank-one convexity implies quasiconvexity on diagonal matrices. Internat. Math. Res. Not.20 (1999) 1087–1095.  Zbl1055.49506
  12. F. Sauvigny, Partial differential equations, Foundations and Integral Representations1. Springer-Verlag (2006).  Zbl1198.35002
  13. V. Šverák, Rank-one convexity does not imply quasiconvexity. Proc. Roy. Soc. Edinburgh120A (1992) 185–189.  Zbl0777.49015

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