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

Abstract

We study the Cauchy problem for a homogeneous and not necessarily coercive Hamilton-JacobiIsaacs equation with an x-dependent, piecewise continuous coefficient. We prove that under suitable assumptions there exists a unique and continuous viscosity solution. The result applies in particular to the Carnot-Carathéodory eikonal equation with discontinuous refraction index of a family of vector fields satisfying the Hörmander condition. Our results are also of interest in connection with geometric flows with discontinuous velocity in anisotropic media with a non-euclidian ambient space.

Hamilton-Jacobi Equations with Discontinuous Source Terms

“…It turns out that the resulting unique envelope solution with zero initial data is 12) which tells us that the envelope solution is the maximum of solutions for each step source. As an another example we have r(x) = cχ S (x) (c > 0 and S is a nonempty closed subset of R n ).…”

“…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. Our results are therefore different from these above.…”

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].…”