Abstract
A level set formulation is presented to characterize a maximal solution of the Cauchy problem for the Hamilton-Jacobi equation with semicontinuous initial data in an explicit way. No convexity assumptions on Hamiltonians are imposed. The solution proposed in the present paper is interpreted as the level set of an auxiliary problem and called an L-solution. It turns out that our L-solution is consistent with a classical discontinuous viscosity solution and a bitateral viscosity solution. Moreover, our L-solution is unique and enjoy the comparison principle. The condition that initial data is really attained is also discussed.
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