Estimates for minimizers of some relaxed polyconvex functionals

We consider the relaxed functional RF(u)=inflim infkF(uk):uk→uwhere F is the polyconvex integral F(u)=∫Ω[|Du|p+h(detDu)]dx,with u:Ω⊂Rn→Rn and h≥0 is convex. We prove bounds for minimizers of RF(u). Similar results are already known when p≥2. In the present paper we use a different technique that allows us to get also the subquadratic case 1<2. The model case is h(t)=|t|s with s≥1: with such an h, we get maximum modulus inequality supΩ|u|≤sup∂Ω|u|