Quasiconvex functions can be approximated by quasiconvex polynomials
ESAIM: Control, Optimisation and Calculus of Variations, Tome 14 (2008) no. 4, pp. 795-801.

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.

DOI : 10.1051/cocv:2008010
Classification : 49J45, 41A10
Mots-clés : Stone-Weierstrass theorem, locally uniform convergence 
@article{COCV_2008__14_4_795_0,
     author = {Heinz, Sebastian},
     title = {Quasiconvex functions can be approximated by quasiconvex polynomials},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     pages = {795--801},
     publisher = {EDP-Sciences},
     volume = {14},
     number = {4},
     year = {2008},
     doi = {10.1051/cocv:2008010},
     mrnumber = {2451797},
     zbl = {1148.49012},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/cocv:2008010/}
}
TY  - JOUR
AU  - Heinz, Sebastian
TI  - Quasiconvex functions can be approximated by quasiconvex polynomials
JO  - ESAIM: Control, Optimisation and Calculus of Variations
PY  - 2008
SP  - 795
EP  - 801
VL  - 14
IS  - 4
PB  - EDP-Sciences
UR  - http://www.numdam.org/articles/10.1051/cocv:2008010/
DO  - 10.1051/cocv:2008010
LA  - en
ID  - COCV_2008__14_4_795_0
ER  - 
%0 Journal Article
%A Heinz, Sebastian
%T Quasiconvex functions can be approximated by quasiconvex polynomials
%J ESAIM: Control, Optimisation and Calculus of Variations
%D 2008
%P 795-801
%V 14
%N 4
%I EDP-Sciences
%U http://www.numdam.org/articles/10.1051/cocv:2008010/
%R 10.1051/cocv:2008010
%G en
%F COCV_2008__14_4_795_0
Heinz, Sebastian. Quasiconvex functions can be approximated by quasiconvex polynomials. ESAIM: Control, Optimisation and Calculus of Variations, Tome 14 (2008) no. 4, pp. 795-801. doi : 10.1051/cocv:2008010. http://www.numdam.org/articles/10.1051/cocv:2008010/

[1] 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. | MR | Zbl

[2] 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. | MR | Zbl

[3] B. Dacorogna, Direct Methods in the Calculus of Variations. Springer-Verlag (1989). | MR | Zbl

[4] D. Faraco and L. Székelyhidi, Tartar’s conjecture and localization of the quasiconvex hull in ℝ 2×2 . Max Planck Institute for Mathematics in the Sciences, Preprint N ∘ 60 (2006). | MR

[5] S. Gutiérrez, A necessary condition for the quasiconvexity of polynomials of degree four. J. Convex Anal. 13 (2006) 51-60. | MR | Zbl

[6] T. Iwaniec, Nonlinear Cauchy-Riemann operators in ℝ n . Trans. Amer. Math. Soc. 354 (2002) 1961-1995. | MR | Zbl

[7] T. Iwaniec and J. Kristensen, A construction of quasiconvex functions. Rivista di Matematica Università di Parma 4 (2005) 75-89. | MR | Zbl

[8] J. Kristensen, On the non-locality of quasiconvexity. Ann. Inst. H. Poincaré Anal. Non Linéaire 16 (1999) 1-13. | Numdam | MR | Zbl

[9] C.B. Morrey, Quasi-convexity and the lower semicontinuity of multiple integrals. Pacific J. Math. 2 (1952) 25-53. | MR | Zbl

[10] S. Müller, A sharp version of Zhang's theorem on truncating sequences of gradients. Trans. Amer. Math. Soc. 351 (1999) 4585-4597. | MR | Zbl

[11] S. Müller, Rank-one convexity implies quasiconvexity on diagonal matrices. Internat. Math. Res. Not. 20 (1999) 1087-1095. | MR | Zbl

[12] F. Sauvigny, Partial differential equations, Foundations and Integral Representations 1. Springer-Verlag (2006). | MR

[13] V. Šverák, Rank-one convexity does not imply quasiconvexity. Proc. Roy. Soc. Edinburgh 120A (1992) 185-189. | MR | Zbl

Cité par Sources :