On Hamilton Jacobi equations with time measurable Hamiltonians posed on a 1-dimensional junction

Abstract

In this paper, we study evolutive Hamilton Jacobi equations with Hamiltonians that are discontinuous in time, posed on a simple network consisting of two edges on the real line connected at a single junction. We introduce a notion of (flux-limited) viscosity solution for Hamiltonians H=H(t,x,p) that are assumed to be only measurable in time t. The flux limiter, A=A(t), acting at the junction, is not required to be continuous but only in L infinity. In the case of convex Hamiltonians, we prove a comparison principle and establish an existence result via the construction of an optimal control problem. Generalisations to the nonconvex case and to more general networks are also discussed.