Degenerate Eikonal equations with discontinuous refraction index

Issue		ESAIM: COCV Volume 12, Number 2, April 2006


Page(s)		216 - 230
DOI		10.1051/cocv:2005033
Published online		22 March 2006

ESAIM: COCV, April 2006, Vol. 12, pp. 216-230
DOI: 10.1051/cocv:2005033

Degenerate Eikonal equations with discontinuous refraction index

Pierpaolo Soravia

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

(Received June 28, 2004. Revised February 28 and March 7, 2005. / Published online: 22 March 2006)

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.

Mathematics Subject Classification. 35A05, 35F30, 49L20, 49L25

Key words: Geometric optics, viscosity solutions, eikonal equation, minimum time problem, discontinuous coefficients.

© EDP Sciences, SMAI 2006

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