  We study a differential game where two players separately control their own
dynamics, pay a running cost, and moreover pay an exit cost (quitting the game)
when they leave a fixed domain. In particular, each player has its own domain
and the exit cost consists of three different exit costs, depending whether
either the first player only leaves its domain, or the second player only
leaves its domain, or they both simultaneously leave their own domain. We prove
that, under suitable hypotheses, the lower and upper value are continuous and
are, respectively, the unique viscosity solution of a suitable Dirichlet
problem for a Hamilton-Jacobi-Isaacs equation. The continuity of the values
relies on the existence of suitable non-anticipating strategies respecting the
domain-constraint. This problem is also treated in this work.
