A polyconvexity condition in dimension two

Abstract

We state a necessary and sufficient polyconvexity condition inR2X2for functions of classC1. This condition is applied tof(X) = |X|2(|X|2−2 detX) for obtaining a convex representation inR2X2xR.

Some numerical methods for the study of the convexity notions arising in the calculus of variations

“…The case a = 1 has been studied by Dacorogna-Marcellini [16] and Alibert-Dacorogna [1] (see also Hartwig [19], Iwaniec-Lutoborski [20]). They proved that Let y > 0 and a > 0, and consider the function This is essentially the counterexample of Sverak [30].…”

“…Since it is obvious from the définition (19) that $ h (f) ^ 0 for every h > 0, the following resuit follows from (20). It is possible to replace the space V h by the space V h of continuous functions u : Q -> IR that satisfy ut are periodic on the boundary dQ (see [30] for example), and to use the space in place of U h .…”