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Viscosity solutions of Hamilton-Jacobi equations with discontinuous coefficients. (English) Zbl 1144.35016

Summary: We consider Hamilton-Jacobi equations, where the Hamiltonian depends discontinuously on both the spatial and temporal location. Our main result is the existence of viscosity solution to the Cauchy problem, and that the front tracking algorithm yields an \(L^\infty\) contractive semigroup. We define a viscosity solution by treating the discontinuities in the coefficients analogously to “internal boundaries”. The existence of viscosity solutions is established constructively via a front tracking approximation, whose limits are viscosity solutions, where by “viscosity solution” we mean a viscosity solution that posses some additional regularity at the discontinuities in the coefficients. We then show a comparison result that is valid for these viscosity solutions.

MSC:

35F25 Initial value problems for nonlinear first-order PDEs
70H20 Hamilton-Jacobi equations in mechanics
49L25 Viscosity solutions to Hamilton-Jacobi equations in optimal control and differential games
35L60 First-order nonlinear hyperbolic equations
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