DOI: 10.68381/jca17005 ISSN: 0944-6532
Normality and Quasiconvex Integrands
Abdessamad Amir, Hocine Mokhtar-Kharroubi
Let
(T, \mathcal{A})
(
T
,
A
)
be an arbitrary measurable space and
f
f
an integrand defined on
T\times \mathbb{R}^n
T
×
R
n
such that
f(t, \cdot)
f
(
t
,
⋅
)
is quasiconvex and lower semicontinuous. Here, convexity is present by the level set mapping. We show that the normality property of the integrand in the sense of R. T. Rockafellar [Pacific Journal of Mathematics 24 (1968) 525–539; and in: Nonlinear Operators and the Calculus of Variations; Bruxelles 1975, Lecture Notes in Mathematics 543, 157–207, Springer, Berlin] can be characterized by the normality of the level set mapping, and that normality is preserved for quasiconvex conjugates. Finally we obtain for the integral
I_f (x(\cdot)) = \int_T f(t, x(t)) d\mu (t)
I
f
(
x
(
⋅
)
)
=
∫
T
f
(
t
,
x
(
t
)
)
d
μ
(
t
)
the equality (in appropriate topology) between the lower semicontinuous regularization and the second quasiconvex conjugate.