Journal Home Page Cumulative Index List of all Volumes Complete Contentsof this Volume Previous Article Journal of Convex Analysis 12 (2005), No. 2, 267--278Copyright Heldermann Verlag 2005 Maximum Principle for Vector Valued Minimizers Francesco Leonetti Dip. di Matematica, Università d'Aquila, 67100 L'Aquila, Italy leonetti@univaq.it Francesco Siepe Dip. di Matematica, Università di Firenze, Piazza Ghiberti 27, 50122 Firenze, Italy siepe@math.unifi.it [Abstract-pdf] \newcommand{\R}{\mathbb R} 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. Keywords: Calculus of variations, minimizers, rank-one convexity, maximum principle, relaxation. MSC: 49N60; 35J60 [ Fulltext-pdf  (329  KB)] for subscribers only.