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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, UniversitÓ d'Aquila, 67100 L'Aquila, Italy

Francesco Siepe
Dip. di Matematica, UniversitÓ di Firenze, Piazza Ghiberti 27, 50122 Firenze, Italy


\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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