Viscosity solutions of discontinuous Hamilton–Jacobi equations

We define viscosity solutions for the Hamilton–Jacobi equation φt = v(x, t)H(∇φ) in RN × (0,∞) where v is positive and bounded measurable and H is non-negative and Lipschitz continuous. Under certain assumptions, we establish the existence and uniqueness of Lipschitz continuous viscosity solutions. The uniqueness result holds in particular for those v which are independent of t and piecewise continuous with discontinuity sets consisting of finitely many smooth lower dimensional surfaces not tangent to each other at any point of their intersection.