Appl Math Optim 47:27–44 (2003)
2002 Springer-Verlag New York Inc.
From a Nonlinear, Nonconvex Variational Problem to a Linear,
Juan Jos´e Egozcue,
and Pablo Pedregal
ETS d’Enginyers de Camins, Canals i Ports,
Gran Capit`a, s/n, 08034 Barcelona, Spain
ETI Telecomunicaciones, Universidad de Castilla-La Mancha,
16071 Cuenca, Spain
Departamento de Matem´aticas, Universidad de los Andes,
Carrera 1, N 18A-10 Bogot´a, Colombia
ETSI Industriales, Universidad de Castilla-La Mancha,
13071 Ciudad Real, Spain
Abstract. We propose a general approach to deal with nonlinear, nonconvex varia-
tional problems based on a reformulation of the problem resulting in an optimization
problem with linear cost functional and convex constraints. As a ﬁrst step we explic-
itly explore these ideas to some one-dimensional variational problems and obtain
speciﬁc conclusions of an analytical and numerical nature.
Key Words. Convexiﬁcation, Young measures, Moments.
AMS Classiﬁcation. 49J05, 49J45, 49M37.
Nonlinear, nonconvex variational problems have lately received much attention be-
cause the behavior of such problems seems apparently connected to the explanation of
many situations in science and technology. We are referring to the oscillatory behavior
This work was partially supported by BFM2001-0738 of the MCyT (Spain) and by GC-02-001 of