On global discontinuous solutions of Hamilton-Jacobi equations

The uniqueness of classical semicontinuous viscosity solutions of the Cauchy problem for Hamlton-Jacobi equations is established for globally Lipschitz continuous and convex Hamiltonian H = H(Du), provided the discontinuous initial value function (x) is continuous outside a set Γ of measure zero and satisfies A formula is presented. We prove that the discontinuous solutions with almost everywhere continuous initial data satisfying (*) become Lipschitz continuous after finite time for locally strictly convex Hamiltonians. The L1-accessibility of initial data and a comparison principle for discontinuous solutions are shown for a general Hamiltonian. The equivalence of semicontinuous viscosity solutions, bi-lateral solutions, L-solutions, minimax solutions, and L∞-solutions is clarified. © 2002 Académie des sciences/Éditions scientifiques et médicales Elsevier SAS.