ESAIM: Control, Optimisation and Calculus of Variations

Table of Contents - April 2006 - Volume , Issue 02  

Research Article

Degenerate Eikonal equations with discontinuous refraction index

Soravia, Pierpaolo

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

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.

(Received June 28 2004)

(Revised February 28 2005)

(Revised March 7 2005)

(Online publication March 22 2006)

Key Words:

  • Geometric optics;
  • viscosity solutions;
  • eikonal equation;
  • minimum time problem;
  • discontinuous coefficients.

Mathematics Subject Classification:

  • 35A05;
  • 35F30;
  • 49L20;
  • 49L25