Let
\Omega \subset {\mathbb R}^n
Ω
⊂
R
n
be a bounded domain with Lipschitz boundary, and assume that
f: \Omega \times {\mathbb R}^{m \times n} \to {\mathbb R}
f
:
Ω
×
R
m
×
n
→
R
is a Carathéodory integrand such that
f(x, \cdot)
f
(
x
,
⋅
)
is polyconvex for
{\mathcal L}^n
L
n
- a.e.
x \in \Omega
x
∈
Ω
. In this paper we consider integral functionals of the form
{\mathcal F}(u, \Omega):= \int_{\Omega} f(x, Du(x)) \, dx,
F
(
u
,
Ω
)
:
=
∫
Ω
f
(
x
,
D
u
(
x
)
)
d
x
,
where
f
f
satisfies a growth condition of the type
|f(x,A)| \le c (1 + |A|^p),
∣
f
(
x
,
A
)
∣
≤
c
(
1
+
∣
A
∣
p
)
,
for some
c>0
c
>
0
and
1 \le p < \infty
1
≤
p
<
∞
, and
u
u
lies in the Sobolev space of vector-valued functions
W^{1,p}(\Omega, {\mathbb R}^m)
W
1
,
p
(
Ω
,
R
m
)
. We study the implications of a function
u_0
u
0
being a critical point of
{\mathcal F}
F
. In this regard we show among other things that if
f
f
does not depend on the spatial variable
x
x
, then every piecewise affine critical point of
{\mathcal F}
F
is a global minimizer subject to its own boundary condition. Moreover for the general case, we construct an example exhibiting that the uniform positivity of the second variation at a critical point is not sufficient for it to be a strong local minimizer. In this example
f
f
is discontinuous in
x
x
but smooth in
A
A