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arXiv · 1906.05625

Discontinuous viscosity solutions of first order Hamilton-Jacobi equations

M. Bertsch·F. Smarrazzo·A. Terracina·A. Tesei
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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.

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M. Bertsch, F. Smarrazzo, A. Terracina, A. Tesei. 2019-06-13. Discontinuous viscosity solutions of first order Hamilton-Jacobi equations. https://arxiv.org/abs/1906.05625

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