Appl Math Optim 52:129–141 (2005)
2005 Springer Science+Business Media, Inc.
Supremal Representation of L
and Francesca Prinari
Laboratoire de Math´ematiques UMR 6205, Universit´e De Brest,
6 avenue Victor Le Gorgeu, CS 93837, F-29238 Brest Cedex 3, France
Dipartimento di Matematica “Ennio De Giorgi,” Universit´a di Lecce,
Via Prov. le Lecce-Arnesano, 73100 Lecce, Italy
Abstract. We characterize the maps F = F(u, A) deﬁned for u ∈ W
A open, which can be written as supremal functionals of the form F(u, A) =
f (x, u(x), Du(x)).
Key Words. Supremal functionals, Calculus of variations in L
AMS Classiﬁcation. 49J45, 28A20.
The classical problems in calculus of variations are formulated through the introduction
of an integral functional. This viewpoint had brought us to refer to as a “variational
functional” any functional F(u, A) deﬁned on a space of functions X and on a class A
of open sets such that F(u, ·) is a regular measure with respect to A (see  and ).
In this way, an integral functional belongs to this class and, thanks to the representa-
tion results shown in  and in , all the variational functionals which satisfy some
lower continuity properties with respect to u, can be written in the integral form
G(u, A) =
g(x, u(x), Du(x)) dx.
In several applications, an integral functional is not suitable to describe certain phe-
During the preparation of this paper the second author was partially supported by the European Research
Training Network “Homogenization and Multiple Scales.”