We study the Cauchy problem for the simplest first-order Hamilton-Jacobi equation in one space dimension, with a bounded and Lipschitz continuous Hamiltonian which only depends on the spatial derivative. Uniqueness of discontinuous viscosity solutions is proven, if the initial data function has a finite number of jump discontinuities. Main ingredients of the proof are the barrier effect of spatial discontinuities of a solution (which is linked to the boundedness of the Hamiltonian), and a comparison theorem for semicontinuous viscosity subsolution and supersolution. These are defined in the spirit of the paper [H. Ishii, Perron's method for Hamilton-Jacobi equations, Duke Math. J. 55 (1987) 368-384], yet using essential limits to introduce semicontinuous envelopes. The definition is shown to be compatible with Perron's method for existence and is crucial in the uniqueness proof. We also describe some properties of the time evolution of spatial jump discontinuities of the solution, and obtain several results about singular Neumann problems which arise in connection with the above referred barrier effect.
Bertsch, M., Smarrazzo, F., Terracina, A., Tesei, A. (2021). Discontinuous viscosity solutions of first-order Hamilton-Jacobi equations. JOURNAL OF HYPERBOLIC DIFFERENTIAL EQUATIONS, 18(4), 857-898 [10.1142/S0219891621500259].