On the range of some sets of homogenized potential operators

We condsider homogenized Nemitskii operators a=a(z),a:Rn×m→Rn×m, defined by a family M of potential elliptic operators div [F(x,∇u(x))] with F′(x,·)∈M where the set M of strictly convex and continuously differentiable functions F:Rn×m→R is fixed. We show that there exists a (curlm,divm)-quasiconvex function L:Rn×m×Rn×m→R defined by the set M such that for every homogenized operator a and every z∈Rn×m the pair (z,a(z)) belongs to the zero level set of L.