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

Gui-Qiang Chen and Bo Su
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Gui-Qiang Chen: Northwestern University, Department of Mathematics
Bo Su: University of Wisconsin at Madison, Department of Mathematics

A chapter in Hyperbolic Problems: Theory, Numerics, Applications, 2003, pp 443-453 from Springer

Abstract: 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.

Date: 2003
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DOI: 10.1007/978-3-642-55711-8_40

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