On the range of some sets of homogenized potential operators Original Research Article

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