Abstract
The Cauchy problem for the Hamilton - Jacobi equation with Hamiltonian independent of time and the phase variable is considered under the assumption that the Hamiltonian and the boundary function are piecewise linear. A finite algorithm for the construction of the exact piecewise linear minimax ( and/or viscosity) solution is developed in the case when the phase space is twodimensional. The fact that a minimax solution is a piecewise linear function is established also for one special case when the phase space is three-dimensional. This results can be used in the researches of bifurcations of piecewise smooth solutions of PDEs of first order, and in the development of numerical methods
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