Journal of Convex Analysis 12 (2005), No. 2, 267--278
Copyright 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
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