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Discontinuous Solutions of Hamilton-Jacobi Equations: Existence, Uniqueness, and Regularity

In: Hyperbolic Problems: Theory, Numerics, Applications

Author

Listed:
  • Gui-Qiang Chen

    (Northwestern University, Department of Mathematics)

  • Bo Su

    (University of Wisconsin at Madison, Department of Mathematics)

Abstract

The theory of continuous viscosity solutions for Hamilton-Jacobi equations and fully nonlinear second-order elliptic and parabolic equations has been established (see [8]) since Crandall-Lions introduced the viscosity solutions in [6]. In this Note, we are concerned with global discontinuous solutions of the Cauchy problem for Hamilton-Jacobi equations: (1) $$ {{u}_{t}} + H(t,x,u,Du) = 0,\quad x \in {{R}^{d}},t > 0,\quad u(0,x) = \varphi (x). $$ The discontinuous solutions of Hamilton-Jacobi equations arise in many important situations. The study of geometrically based motions demands deep understanding of discontinuous solutions of Hamilton-Jacobi equations. (e.g. [2]). Many examples in the control theory and the differential game theory do not have continuous solutions. Another motivation is that Hamiltonians arising in the differential game theory and other areas are discontinuous with respect to some or all t, x Du (e.g. [10, 13]). The conventional theories of viscosity solutions do not apply.

Suggested Citation

  • Gui-Qiang Chen & Bo Su, 2003. "Discontinuous Solutions of Hamilton-Jacobi Equations: Existence, Uniqueness, and Regularity," Springer Books, in: Thomas Y. Hou & Eitan Tadmor (ed.), Hyperbolic Problems: Theory, Numerics, Applications, pages 443-453, Springer.
    • Handle: RePEc:spr:sprchp:978-3-642-55711-8_40
    DOI: 10.1007/978-3-642-55711-8_40
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