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.
Keywords: geometric optics, viscosity solutions, eikonal equation, minimum time problem, discontinuous coefficients
@article{COCV_2006__12_2_216_0,
author = {Soravia, Pierpaolo},
title = {Degenerate {Eikonal} equations with discontinuous refraction index},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
pages = {216--230},
year = {2006},
publisher = {EDP Sciences},
volume = {12},
number = {2},
doi = {10.1051/cocv:2005033},
mrnumber = {2209351},
zbl = {1105.35026},
language = {en},
url = {https://www.numdam.org/articles/10.1051/cocv:2005033/}
}
TY - JOUR AU - Soravia, Pierpaolo TI - Degenerate Eikonal equations with discontinuous refraction index JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2006 SP - 216 EP - 230 VL - 12 IS - 2 PB - EDP Sciences UR - https://www.numdam.org/articles/10.1051/cocv:2005033/ DO - 10.1051/cocv:2005033 LA - en ID - COCV_2006__12_2_216_0 ER -
%0 Journal Article %A Soravia, Pierpaolo %T Degenerate Eikonal equations with discontinuous refraction index %J ESAIM: Control, Optimisation and Calculus of Variations %D 2006 %P 216-230 %V 12 %N 2 %I EDP Sciences %U https://www.numdam.org/articles/10.1051/cocv:2005033/ %R 10.1051/cocv:2005033 %G en %F COCV_2006__12_2_216_0
Soravia, Pierpaolo. Degenerate Eikonal equations with discontinuous refraction index. ESAIM: Control, Optimisation and Calculus of Variations, Tome 12 (2006) no. 2, pp. 216-230. doi: 10.1051/cocv:2005033
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