Discontinuous solutions for Hamilton-Jacobi equations: Uniqueness and regularity

The uniqueness of classical semicontinuous viscosity solutions of the Cauchy problem for Hamilton-Jacobi equations with convex Hamiltonians H = H (Du) is established, provided the discontinuous initial value function φ(x) is continuous outside a set Γ of measure zero and satisfies φ(x) ≥ φ**(x) ≡ liminfy→x,y∈ℝd\Γ φ (y). The regularity of discontinuous solutions to Hamilton-Jacobi equations with locally strictly convex Hamiltonians is proved: The discontinuous solutions with almost everywhere continuous initial data satisfying (*) become Lipschitz continuous after finite time. The L1-accessibility of initial data and a comparison principle for discontinuous solutions are shown. The equivalence of semicontinuous viscosity solutions, bi-lateral solutions, L-solutions, minimax solutions, and L∞-solutions is also clarified