Viscosity solutions for junctions: well posedness and stability

We introduce a notion of state-constraint viscosity solutions for one dimensional junction''-type problems for Hamilton-Jacobi equations with non convex coercive Hamiltonians and study its well-posedness and stability properties. We show that viscosity approximations either select the state-constraint solution or have a unique limit, and we introduce another type of approximation by fattening the domain. We also make connections with existing results for convex equations and discuss extensions to time and/or multi-dimensional problems.