Discontinuous viscosity solutions of first order Hamilton-Jacobi equations

Abstract

We consider the simplest example of a time-dependent first order Hamilton-Jacobi equation, in one space dimension and with a bounded and Lipschitz continuous Hamiltonian which only depends on the spatial derivative. We show that if the initial function has a finite number of jump discontinuities, the corresponding discontinuous viscosity solution of the corresponding Cauchy problem on the real line is unique. Uniqueness follows from a comparison theorem for semicontinuous viscosity sub-and supersolutions, using the barrier effect of spatial discontinuities of a solution. We also prove an existence theorem, as well as a comparison theorem for viscosity solutions with different initial data. In addition, we describe several properties of the evolution of the jump discontinuities.As a byproduct of our analysis, we obtain new existence and uniqueness results for the initial-boundary value problem on an interval with (possibly singular) Neumann boundary conditions.

Discontinuous solutions of Hamilton-Jacobi equations versus Radon measure-valued solutions of scalar conservation laws: Disappearance of singularities

“…(see (5.40)). Similarly, by the proof of [6, Theorem 3.4] (see also [6,Lemma 5.2]), the unique viscosity solution U of problem (N ) in Q τ1 with the same boundary conditions has the following features:…”

“…Let us point out that suitably defined discontinuous viscosity solutions of (HJ) are unique [6], but, as observed in [10], measure-valued entropy solutions of (CL) are not. Only recently an additional compatibility condition (see Definition 3.2 below) was identified which guarantees their uniqueness [4].…”

“…The positivity of the waiting time is in contrast with the case of nonlinearities H with superlinear growth, where the regularizing effect is instantaneous [16]. Similarly, in [6] we studied problem (HJ) in the context of discontinuous viscosity solutions, and showed that if H is bounded and…”

“…By (3.21) and the envelope properties we have that (U 0 ) * (x) = U * (x, 0) ≥ U * 1 (x, 0) for all x ≥ x j , thus inequality (7.3) gives (7.4) v(x, 0) ≥ U * 1 (x, 0) for all x ≥ x j . Since v is a viscosity supersolution of (7.1) (see [6,Definition 3.2]), by the comparison principle in [6, Theorem 3.1] and (7.4) we get (7.5) (U 1 ) * (x, t) ≤ v(x, t) for all (x, t) ∈ [x j , ∞) × [0, τ j ) .…”

Non-Equivalence of Stochastic Optimal Control Problems with Open and Closed Loop Controls

“…When v(t, x) is discontinuous, although there are some nice works on discontinuous viscosity solutions, see e.g. Barles-Perthame [2], Barron-Jensen [3], Bertsch-Dal Passo-Ughi [4], Bardi-Capuzzo-Dolcetta [1], Chen-Su [7], and Bertsch-Smarrazzo-Terracina-Tesei [5], the theory is far from complete. In particular, in this case we are not able to conclude (1.7) or (1.5) from the viscosity solution approach.…”