Formation of singularities for viscosity solutions of Hamilton-Jacobi equations in higher dimensions

Abstract: "In this work we study the generation of singularities (shock waves) of the solution of the Cauchy problem for Hamilton-Jacobi equations in several space variables, under no assumption on convexity or concavity of the hamiltonian. We study the problem in the class of viscosity solutions, which are the correct class of weak solutions. We first examine the way the characteristics cross by identifying the set of critical points of the characteristic manifold with the caustic set of the related lagrangian mapping. We construct the viscosity solution by selecting a single-valued branch of the multi-valued function given as a solution by the method of characteristics. We finally discuss how the shocks propagate and undergo catastrophe in the case of two space variables."