Polyconvex functionals and maximum principle

Let us consider continuous minimizers u : , subset of Rn -> Rn of Z F (v) = [|Dv|p + |det Dv|r]dx, , with p > 1 and r > 0; then it is known that every component u alpha of u = (u1, ..., un) enjoys maximum principle: the set of interior points x, for which the value u alpha(x) is greater than the supremum on the boundary, has null measure, that is, Ln({x is an element of , : u alpha(x) > sup partial differential , u alpha}) = 0. If we change the structure of the functional, it might happen that the maximum principle fails, as in the case Z F(v) = [max{(|Dv|p - 1); 0} + |det Dv|r]dx, , with p > 1 and r > 0. Indeed, for a suitable boundary value, the set of the interior points x, for which the value u alpha(x) is greater than the supremum on the boundary, has a positive measure, that is Ln({x is an element of , : u alpha(x) > sup partial differential , u alpha}) > 0. In this paper we show that the measure of the image of these bad points is zero, that is Ln(u({x is an element of , : u alpha(x) > sup partial differential , u alpha})) = 0, provided p > n. This is a particular case of a more general theorem.