DOI: 10.68381/jca12019 ISSN: 0944-6532
Maximum Principle for Vector Valued Minimizers
Francesco Leonetti, Francesco Siepe
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.