A level set approach to semicontinuous viscosity solutions for Cauchy problems

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