Abstract

We prove a maximum principle for vector valued minimizers u:Ω⊂Rn→RNu: \Omega \subset\R^n\to\R^N of some functionals F(u)=∫Ωf(x,Du(x))dx.\mathcal{F}(u) = \int_{\Omega} f(x,Du(x)) dx. The main assumption on the density f(x,z)f(x,z) is a kind of "monotonicity" with respect to the N×nN \times n matrix zz. A model density is f(z)=∣z∣4−(det⁡z)2f(z)=|z|^4 - (\det z)^2, where z∈R2×2z \in \R^{2 \times 2}. We also consider relaxed functionals RF(u)=inf⁡{lim inf⁡kF(uk):uk→u}\mathcal{RF}(u) = \inf \{ \liminf\limits_{k} \mathcal{F}(u_k): \quad u_k \to u \} and we prove maximum principle under suitable assumptions.

Author information

Contact details are reproduced from the original publication and may be historical.

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

Keywords

  • Calculus of variations
  • minimizers
  • rank-one convexity
  • maximum principle
  • relaxation

Mathematics Subject Classification

49N60; 35J60

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Suggested citation

F. Leonetti, F. Siepe. “Maximum Principle for Vector Valued Minimizers.” Journal of Convex Analysis 12 (2005), No. 2, 267–278. https://doi.org/10.68381/jca12019

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Published by Heldermann Verlag, 2005. Rights now held by Banach Press LLC.