01073nas a2200121   4500008004300000245008500043210006900128260002100197520064400218100002800862700002500890856003600915       2007                        en_Ud  00aViscosity solutions of Hamilton-Jacobi equations with discontinuous coefficients0 aViscosity solutions of HamiltonJacobi equations with discontinuo  bWorld Scientific3 aWe 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.1 aCoclite, Giuseppe Maria1 aRisebro, Nils Henrik  uhttp://hdl.handle.net/1963/290700982nas a2200121   4500008004100000245006900041210006900110260001800179520057400197100002800771700002500799856003600824       2005                        en  d00aConservation laws with time dependent discontinuous coefficients0 aConservation laws with time dependent discontinuous coefficients  bSISSA Library3 aWe consider scalar conservation laws where the flux function depends discontinuously on both the spatial and temporal location. Our main results are the existence and well-posedness of an entropy solution to the Cauchy problem. The existence is established by showing that a sequence of front tracking approximations is compact in L1, and that the limits are entropy solutions. Then, using the definition of an entropy solution taken form [11], we show that the solution operator is L1 contractive. These results generalize the corresponding results from [16] and [11].1 aCoclite, Giuseppe Maria1 aRisebro, Nils Henrik  uhttp://hdl.handle.net/1963/1666