  The aim of this work is to deal with a discontinuous Hamilton-Jacobi equation
in the whole euclidian N-dimensional space, associated to a possibly unbounded
optimal control problem. Here, the discontinuities are located on a hyperplane
and the typical questions we address concern the existence and uniqueness of
solutions, and of course the definition itself of solution. We consider
viscosity solutions in the sense of Ishii. The convex Hamiltonians are
associated to a control problem with specific cost and dynamics given on each
side of the hyperplane. We assume that those are Lipschitz continuous but the
main difficulty we deal with is that they are potentially unbounded, as well as
the control spaces. Using Bellman's approach we construct two value functions
which turn out to be the minimal and maximal solutions in the sense of Ishii.
Moreover, we also build a whole family of value functions, which are still
solutions in the sense of Ishii and connect continuously the minimal solution
to the maximal one.
