Viscosity solutions of discontinuous Hamilton–Jacobi equations

Abstract

We define viscosity solutions for the Hamilton-Jacobi equation ϕ t = v(x, t)H (∇ϕ) in R N × (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.

Hamilton-Jacobi Equations with Discontinuous Source Terms

“…Deckelnick and Elliott [11] obtained the unique existence of continuous viscosity solutions for the one space dimensional case when f (x, t) = a(x) and h(x, p) = √ 1 + p 2 , where a is assumed to be bounded, of bounded variation and one-sided Lipschitz continuous. Afterwards Chen and Hu [8] studied a more general case when f depends on t but h depends only on p. They assumed that f is positive, bounded and measurable and h is non-negative and Lipschitz continuous. More recently, with the optimal control theory involved De Zan and Soravia [12] discussed the unique existence of solutions when h depends also on x while f is independent of t and piecewise Lipschitz continuous across Lipschitz hypersurfaces.…”

A comparison principle for Hamilton-Jacobi equations with discontinuous Hamiltonians

“…Let us mention that while the notion of semicontinuous super-and subsolutions for discontinuous Hamiltonians is well-defined (see [1], [9] [10]), the authors frequently assume in the statements of their Comparison Principles that either the supersolution of the subsolution is at least Lipschitz continuous [2], [3], [4], [5], [6], [11].…”

“…One stems from image analysis, like the 'shape-from-shading' problem [4], [11], and another is from flame propagation or etching [3] or from game theory [6]. In those papers (1.1) is a general form of the eikonal equation and H does not depend upon u.…”

“…There is a spectrum of assumptions on admissible discontinuities with respect to x and t. Jump discontinuities across Lipschitz hypersurfaces are quite common [3], [5], [6], [11]. We may add that sometimes the authors admit triple junctions of the discontinuity set; see [5].…”

Cauchy problems for noncoercive Hamilton–Jacobi–Isaacs equations with discontinuous coefficients

“…We recall the work of one of the present authors on the characterization of uniqueness of viscosity solutions [20], existence and uniqueness results for the stationary eikonal equation [21], and general uniqueness results for degenerate elliptic equations [22]; see however also the references therein for additional work on the subject. More recently, uniqueness results for Lipschitz continuous solutions of (1.1) with methods different than ours have been obtained by Chen-Hu [7] for nonconvex but radial and x-independent H , with new sets of assumptions on the discontinuous coefficient.…”