Maximum Principle for Vector Valued Minimizers

We prove a maximum principle for vector valued minimizers $u: \Omega \subset\R^n\to\R^N$ of some functionals $$\mathcal{F}(u) = \int_{\Omega} f(x,Du(x)) dx.$$ The main assumption on the density $f(x,z)$ is a kind of "monotonicity" with respect to the $N \times n$ matrix $z$. A model density is $f(z)=|z|^4 - (\det z)^2$, where $z \in \R^{2 \times 2}$. We also consider relaxed functionals $$\mathcal{RF}(u) = \inf \{ \liminf\limits_{k} \mathcal{F}(u_k): \quad u_k \to u \}$$ and we prove maximum principle under suitable assumptions.