A general convergence result for viscosity solutions of Hamilton-Jacobi equations and non-linear semigroups

Abstract

We extend the Barles-Perthame procedure [4] (see also [22]) of semi-relaxed limits of viscosity solutions of Hamilton-Jacobi equations of the type f−λHf=h to the context of non-compact spaces. The convergence result allows for equations on a ‘converging sequence of spaces’ as well as Hamilton-equations written in terms of two equations in terms of operators H† and H‡ that serve as natural upper and lower bounds for the ‘true’ operator H. In the process, we establish a strong relation between non-linear pseudo-resolvents and viscosity solutions of Hamilton-Jacobi equations. As a consequence we derive a convergence result for non-linear semigroups.