Viscosity solutions of Hamilton--Jacobi equations with discontinuous coefficients

Giuseppe Maria Coclite; Nils Henrik Risebro

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Viscosity solutions of Hamilton--Jacobi equations with discontinuous coefficients

Giuseppe Maria Coclite·Nils Henrik Risebro

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Abstract

We consider Hamilton--Jacobi equations, where the Hamiltonian depends discontinuously on both the spatial and temporal location. Our main results are the existence and well--posedness of a viscosity solution to the Cauchy problem. We define a viscosity solution by treating the discontinuities in the coefficients analogously to ``internal boundaries''. By defining an appropriate penalization function, we prove that viscosity solutions are unique. The existence of viscosity solutions is established by showing that a sequence of front tracking approximations is compact in $L^\infty$, and that the limits are viscosity solutions.

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Giuseppe Maria Coclite, Nils Henrik Risebro. 2003-03-24. Viscosity solutions of Hamilton--Jacobi equations with discontinuous coefficients. https://arxiv.org/abs/math/0303288

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