DOI: 10.68381/jca1025 ISSN: 0944-6532
On Lambda-Convexity Conditions in the Theory of Lower Semicontinuous Functionals
Agnieszka Kałamajska
Consider the functional
I_f(u)=\int_\Omega f(u(x))\, dx
I
f
(
u
)
=
∫
Ω
f
(
u
(
x
)
)
d
x
, where
u=(u_1,\dots,u_m)
u
=
(
u
1
,
…
,
u
m
)
. Assume additionally that each
u_j
u
j
is constant along
W_j
W
j
, some subspace of
{\bf R}^n
R
n
. We find the family of cones
\Lambda
Λ
in
{\bf R}^m
R
m
such that every
\Lambda
Λ
-convex function
f
f
defines a functional
I_f
I
f
which is lower semicontinuous under the sequential weak
*
∗
convergence in
L^\infty (\Omega,{\bf R}^m )
L
∞
(
Ω
,
R
m
)
. Then we apply our result to functionals acting on distributional kernels of differential operators. We also discuss the relations of our problem to the rank–one conjecture of Morrey.