Polyconvex functionals and maximum principle

Let us consider continuous minimizers $u:\overline{\mathrm{\Omega }}\subset {\mathbb{R}}^n\to {\mathbb{R}}^n$ of

$$\mathcal{F}(v)={\int }_{\mathrm{\Omega }}[|Dv{|}^p+|\mathrm{d}\mathrm{e}\mathrm{t}Dv{|}^r]dx,$$

with $p>1$ and $r>0$; then it is known that every component $u^{\alpha }$ of $u=(u^1,...,u^n)$ 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, ${\mathcal{L}}^n(\{x\in \mathrm{\Omega }:u^{\alpha }(x)>\underset{\mathrm{\partial }\mathrm{\Omega }}{sup}{u}^{\alpha }\})=0$. If we change the structure of the functional, it might happen that the maximum principle fails, as in the case

$$\mathcal{F}(v)={\int }_{\mathrm{\Omega }}[max\{(|Dv{|}^p-1);0\}+|\mathrm{d}\mathrm{e}\mathrm{t}Dv{|}^r]dx,$$

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 ${\mathcal{L}}^n(\{x\in \mathrm{\Omega }:u^{\alpha }(x)>\underset{\mathrm{\partial }\mathrm{\Omega }}{sup}{u}^{\alpha }\})>0$. In this paper we show that the measure of the image of these bad points is zero, that is ${\mathcal{L}}^n(u(\{x\in \mathrm{\Omega }:u^{\alpha }(x)>\underset{\mathrm{\partial }\mathrm{\Omega }}{sup}{u}^{\alpha }\}))=0$, provided $p>n$. This is a particular case of a more general theorem.