Quasiconvexity is not invariant under transposition

Abstract

An example is given of a quasiconvex f : M^2×3 → R such that the transposed function f̃ : M^3×2 → R given by f̃(F) = f(F^T) is not quasiconvex. For f̃ one can take Sverák's quartic polynomial that is rank-one convex but not quasiconvex. The proof is closely related to the observation that the map v ↦ v¹v²v³ is weakly continuous from L³(R³; R³) into distributions provided that A(Dv) = (∂₂v¹, ∂₃v¹, ∂₁v², ∂₃v², ∂₁v³, ∂₂v³) is compact in W^−1,3(R³; R⁶).