Maximum Principle for Vector Valued Minimizers
Francesco Leonetti, Francesco Siepe · Journal of convex analysis · 2005
We prove a maximum principle for vector valued minimizers u: \Omega \subset{\mathbb R}^n\to{\mathbb R}^N u : Ω ⊂ R n → R N of some functionals \mathcal{F}(u) = \int_{\Omega} f(x,Du(x)) dx. F ( u ) = ∫ Ω f ( x , D u ( x ) ) d x . The main assumption on the density f(x,z) f ( x , z ) is a kind of "monotonicity" with respect to the N \times n N × n matrix z z . A model density is f(z)=|z|^4 - (\det z)^2 f ( z ) = ∣ z ∣ 4 − ( det z ) 2 , where z \in {\mathbb R}^{2 \times 2} z ∈ R 2 × 2 . We also consider relaxed functionals \mathcal{RF}(u) = \inf \{ \liminf\limits_{k} \mathcal{F}(u_k): \quad u_k \to u \} R F ( u ) = inf { lim inf k F ( u k ) : u k → u } and we prove maximum principle under suitable assumptions.