Appl Math Optim 42:35–50 (2000)
2000 Springer-Verlag New York Inc.
A Characterization of the Existence of Solutions for Hamilton–Jacobi
Equations in Ergodic Control Problems with Applications
and P.-L. Lions
Department of Computer and Mathematical Sciences,
GSIS, Tohoku University,
Aoba-ku, Sendai 980-8577, Japan
Department of Mathematics, Tokyo Metropolitan University,
Hachioji-shi, Tokyo 192-0397, Japan
CEREMADE, Universit´e de Paris-Dauphine,
Place de Lattre de Tassigny, 75775 Paris, France
Abstract. We give a characterization of the existence of bounded solutions for
Hamilton–Jacobi equations in ergodic control problems with state-constraint. This
result is applied to the reexamination of the counterexample given in  concern-
ing the existence of solutions for ergodic control problems in inﬁnite-dimensional
Hilbert spaces and also establishing results on effective Hamiltonians in periodic
homogenization of Hamilton–Jacobi equations.
Key Words. Hamilton–Jacobi equations, Ergodic control problems, Viscosity so-
lutions, Homogenization, Effective Hamiltonians.
AMS Classiﬁcation. 35F20, 49J15, 49L20, 49L25.
In this paper we consider the ﬁrst-order Hamilton–Jacobi equation
H(x,u, Du) = 0in. (1.1)
The second author was supported in part by Grant-in-Aid for Scientiﬁc Research No. 09440067, No.
09874034, No. 08640236, and (International Scientiﬁc Program) No. 07044094, the Ministry of Education,
Science, Sports, and Culture.