Degenerate Eikonal equations with discontinuous refraction index

Università degli Studi di Padova, Dipartimento di Matematica Pura e Applicata, via Belzoni, 7, 35131 Padova, Italy; soravia@math.unipd.it

Received: 28 June 2004
Revised: 28 February 2005
Revised: 7 March 2005

Abstract

We study the Dirichlet boundary value problem for eikonal type equations of ray light propagation in an inhomogeneous medium with discontinuous refraction index. We prove a comparison principle that allows us to obtain existence and uniqueness of a continuous viscosity solution when the Lie algebra generated by the coefficients satisfies a Hörmander type condition. We require the refraction index to be piecewise continuous across Lipschitz hypersurfaces. The results characterize the value function of the generalized minimum time problem with discontinuous running cost.