An example of a quasiconvex function that is not polyconvex in two dimensions
Alibert, Jean-Jacques ; Dacorogna, Bernard
HAL, hal-00993881 / Harvested from HAL
Text on HAL
We study the different notions of convexity for the function f γ(ξ) = |ξ|2 (|ξ|2 − 2γ det ξ) where ξ ε ℝ2×2, introduced by Dacorogna & Marcellini. We show that f γ is convex, polyconvex, quasiconvex, rank-one convex, if and only if ¦γ¦≦ 2/3 √2, 1, 1+ɛ (for some ɛ>0), 2/√3, respectively.
Publié le : 1992-07-04
Classification:  [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP] 
@article{hal-00993881,
     author = {Alibert, Jean-Jacques and Dacorogna, Bernard},
     title = {An example of a quasiconvex function that is not polyconvex in two dimensions},
     journal = {HAL},
     volume = {1992},
     number = {0},
     year = {1992},
     language = {en},
     url = {http://dml.mathdoc.fr/item/hal-00993881}
}

Alibert, Jean-Jacques; Dacorogna, Bernard. An example of a quasiconvex function that is not polyconvex in two dimensions. HAL, Tome 1992 (1992) no. 0, . http://gdmltest.u-ga.fr/item/hal-00993881/