Geodesic
Maximum Principle for Vector Valued Minimizers
Journal of convex analysis, Tome 12 (2005) no. 2, pp. 267-278.

Voir la notice de l'article provenant de la source Heldermann Verlag

\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.
Classification : 49N60, 35J60
Mots-clés : Calculus of variations, minimizers, rank-one convexity, maximum principle, relaxation 
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     author = {F. Leonetti and F. Siepe},
     title = {Maximum {Principle} for {Vector} {Valued} {Minimizers}},
     journal = {Journal of convex analysis},
     pages = {267--278},
     publisher = {mathdoc},
     volume = {12},
     number = {2},
     year = {2005},
     url = {http://geodesic.mathdoc.fr/item/JCA_2005_12_2_JCA_2005_12_2_a1/}
}
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F. Leonetti; F. Siepe. Maximum Principle for Vector Valued Minimizers. Journal of convex analysis, Tome 12 (2005) no. 2, pp. 267-278. http://geodesic.mathdoc.fr/item/JCA_2005_12_2_JCA_2005_12_2_a1/