Degenerate Eikonal equations with discontinuous refraction index

Pierpaolo Soravia

ESAIM: Control, Optimisation and Calculus of Variations (2006)

  • Volume: 12, Issue: 2, page 216-230
  • ISSN: 1292-8119

Abstract

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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.

How to cite

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Soravia, Pierpaolo. "Degenerate Eikonal equations with discontinuous refraction index." ESAIM: Control, Optimisation and Calculus of Variations 12.2 (2006): 216-230. <http://eudml.org/doc/249679>.

@article{Soravia2006,
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. },
author = {Soravia, Pierpaolo},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {Geometric optics; viscosity solutions; eikonal equation; minimum time problem; discontinuous coefficients.; discontinuous coefficients; Dirichlet boundary value problem; ray light propagation; comparison principle},
language = {eng},
month = {3},
number = {2},
pages = {216-230},
publisher = {EDP Sciences},
title = {Degenerate Eikonal equations with discontinuous refraction index},
url = {http://eudml.org/doc/249679},
volume = {12},
year = {2006},
}

TY - JOUR
AU - Soravia, Pierpaolo
TI - Degenerate Eikonal equations with discontinuous refraction index
JO - ESAIM: Control, Optimisation and Calculus of Variations
DA - 2006/3//
PB - EDP Sciences
VL - 12
IS - 2
SP - 216
EP - 230
AB - 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.
LA - eng
KW - Geometric optics; viscosity solutions; eikonal equation; minimum time problem; discontinuous coefficients.; discontinuous coefficients; Dirichlet boundary value problem; ray light propagation; comparison principle
UR - http://eudml.org/doc/249679
ER -

References

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