Quasiconvex functions can be approximated by quasiconvex polynomials
ESAIM: Control, Optimisation and Calculus of Variations (2008)
- Volume: 14, Issue: 4, page 795-801
- ISSN: 1292-8119
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topHeinz, 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
top- J.-J. Alibert and B. Dacorogna, An example of a quasiconvex function that is not polyconvex in two dimensions. Arch. Rational Mech. Anal.117 (1992) 155–166.
- 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 . Max Planck Institute for Mathematics in the Sciences, Preprint N° 60 (2006).
- S. Gutiérrez, A necessary condition for the quasiconvexity of polynomials of degree four. J. Convex Anal.13 (2006) 51–60.
- T. Iwaniec, Nonlinear Cauchy-Riemann operators in . Trans. Amer. Math. Soc.354 (2002) 1961–1995.
- T. Iwaniec and J. Kristensen, A construction of quasiconvex functions. Rivista di Matematica Università di Parma4 (2005) 75–89.
- J. Kristensen, On the non-locality of quasiconvexity. Ann. Inst. H. Poincaré Anal. Non Linéaire16 (1999) 1–13.
- C.B. Morrey, Quasi-convexity and the lower semicontinuity of multiple integrals. Pacific J. Math.2 (1952) 25–53.
- S. Müller, A sharp version of Zhang's theorem on truncating sequences of gradients. Trans. Amer. Math. Soc.351 (1999) 4585–4597.
- S. Müller, Rank-one convexity implies quasiconvexity on diagonal matrices. Internat. Math. Res. Not.20 (1999) 1087–1095.
- F. Sauvigny, Partial differential equations, Foundations and Integral Representations1. Springer-Verlag (2006).
- V. Šverák, Rank-one convexity does not imply quasiconvexity. Proc. Roy. Soc. Edinburgh120A (1992) 185–189.
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