Appl Math Optim 48:39–66 (2003)
2003 Springer-Verlag New York Inc.
Autonomous Integral Functionals with Discontinuous Nonconvex
Integrands: Lipschitz Regularity of Minimizers, DuBois–Reymond
Necessary Conditions, and Hamilton–Jacobi Equations
Gianni Dal Maso
and H´el`ene Frankowska
SISSA, via Beirut 2-4,
34014 Trieste, Italy
CNRS, CREA, Ecole Polytechnique,
1 Rue Descartes, 75005 Paris, France
Communicated by I. Lasiecka
Abstract. This paper is devoted to the autonomous Lagrange problem of the cal-
culus of variations with a discontinuous Lagrangian. We prove that every minimizer
is Lipschitz continuous if the Lagrangian is coercive and locally bounded. The main
difference with respect to the previous works in the literature is that we do not as-
sume that the Lagrangian is convex in the velocity. We also show that, under some
additional assumptions, the DuBois–Reymond necessary condition still holds in the
discontinuous case. Finally, we apply these results to deduce that the value function
of the Bolza problem is locally Lipschitz and satisﬁes (in a generalized sense) a
Key Words. Discontinuous Lagrangians, Nonconvex integrands, Lipschitz mini-
mizers, DuBois–Reymond necessary conditions, Hamilton–Jacobi equations.
AMS Classiﬁcation. Primary 49N60, Secondary 49K05, 49L25.
The work of Gianni Dal Maso is part of the European Research Training Network “Homogenization
and Multiple Scales” under Contract HPRN-2000-00109, and the Project “Calculus of Variations,” supported
by SISSA and by the Italian Ministry of Education, University, and Research. The work of H´el`ene Frankowska
was supported in part by the European Community’s Human Potential Programme under Contract HPRN-CT-
2002-00281, Evolution Equations.