Appl Math Optim 41:9–24 (2000)
2000 Springer-Verlag New York Inc.
Variational Solutions of Coupled Hamilton–Jacobi Equations
and G. Vergara Caffarelli
Istituto per le Applicazioni del Calcolo “Mauro Picone”,
Consiglio Nazionale delle Ricerche,
Viale del Policlinico 137, 00161 Roma, Italy
Dipartimento di Metodi e Modelli Matematici per le Scienze Applicate,
Universit´a di Roma “La Sapienza”,
Via A. Scarpa 16, 00161 Roma, Italy
Communicated by D. Kinderlehrer
Abstract. In this note we study variationalsolutions of weakly coupled Hamilton–
Jacobi equations in the case where the Hamiltonians are convex. More precisely, we
build the variational solution by an approximation scheme.
Key Words. Hamilton–Jacobi equations, Variational solutions, Viscosity solu-
AMS Classiﬁcation. 35F, 35J50.
1. Introduction and Formulation of the Main Result
In this paper we consider the Cauchy problem for weakly coupled Hamilton–Jacobi
equations with convex Hamiltonians. Previously, Engler and Lenhart investigated the
problem of the existence of solutions for this type of system, in the context of viscosity
solutions . To get comparison results, of maximum principle type, Engler and Lenhart
make some assumptions which restrict the class of Hamilton–Jacobi systems. Other
authors have studied systems of partial differential equations via the viscosity solution
method: see, for instance, , –, and .
Current address: Dipartimento di Metodi e Modelli Matematici per le Scienze Applicate, Universit´adi
Roma “La Sapienza”, Via A. Scarpa 16, 00161 Roma, Italy. firstname.lastname@example.org.