On the System of Hamilton–Jacobi and Transport Equations Arising in Geometrical Optics

Abstract

Abstract In the present article, we study the system of eikonal and transport equations arising in geometrical optics. The mathematical analysis is performed by using the suitable notion of solution, i.e., the viscosity solution for the Hamilton–Jacobi equation and the measure solution for the transport equation defined via the generalized Filippov characteristics. We study the stability as well as the geometry of the solution to the system. Keywords: High frequency asymptoticsWKB approximationParaxial approximationEikonal equationTransport equationViscosity solutionsMeasure solutionsCausticsSingularitiesMultivalued solutionsBlow-up Acknowledgment This work has been partially supported by the European Union TMR grant No. ERBFMRX-CT98-0234 (Viscosity Solutions).