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Degenerate Eikonal equations with discontinuous refraction index

Published online by Cambridge University Press:  22 March 2006

Pierpaolo Soravia
Pierpaolo Soravia*
Affiliation:
Università degli Studi di Padova, Dipartimento di Matematica Pura e Applicata, via Belzoni, 7, 35131 Padova, Italy; [email protected]
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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.

Keywords

Geometric opticsviscosity solutionseikonal equationminimum time problemdiscontinuous coefficients.
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 12 , Issue 2 , April 2006 , pp. 216 - 230
Copyright
© EDP Sciences, SMAI, 2006

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References

Bardi, M., A boundary value problem for the minimum-time function. SIAM J. Control Optim. 27 (1989) 776–785. CrossRef
M. Bardi and I. Capuzzo-Dolcetta, Optimal control and viscosity solutions of Hamilton-Jacobi-Bellman equations. Birkhäuser (1997).
Bardi, M. and Soravia, P., Hamilton-Jacobi equations with a singular boundary condition on a free boundary and applications to differential games. Trans. Amer. Math. Soc. 325 (1991) 205–229. CrossRef
G. Barles, Solutions de viscosité des équations de Hamilton-Jacobi. Springer-Verlag (1994).
Barles, G. and Perthame, B., Discontinuous solutions of deterministic optimal stopping time problems. RAIRO: M2AN 21 (1987) 557–579.
Caffarelli, L., Crandall, M.G., Kocan, M. and Swiech, A., On viscosity solutions of fully nonlinear equations with measurable ingredients. Comm. Pure Appl. Math. 49 (1996) 365–397. 3.0.CO;2-A>CrossRef
Camilli, F. and Siconolfi, A., Hamilton-Jacobi equations with measurable dependence on the state variable. Adv. Differential Equations 8 (2003) 733–768.
Capuzzo Dolcetta, I. and Lions, P.L., Hamilton-Jacobi equations with state constraints. Trans. Am. Math. Soc. 318 (1990) 643–683. CrossRef
R. Courant and D. Hilbert, Methods of mathematical physics Vol. II. John Wiley & Sons (1989).
Crandall, M.G., Ishii, H. and Lions, P.L., User's guide to viscosity solutions of second order partial differential equations. Bull. Amer. Math. Soc. 27 (1992) 1–67. CrossRef
Garavello, M. and Soravia, P., Optimality principles and uniqueness for Bellman equations of unbounded control problems with discontinuous running cost. Nonlin. Diff. Equations Appl. 11 (2004) 271–298. CrossRef
Haynes, G.W. and Hermes, H., Nonlinear controllability via Lie theory. SIAM J. Control 8 (1970) 450–460. CrossRef
Ishii, H., A boundary value problem of the Dirichlet type for Hamilton-Jacobi equations. Ann. Sc. Norm. Sup. Pisa (IV) 16 (1989) 105–135.
Katsoulakis, M.A., Viscosity solutions of second order fully nonlinear elliptic equations with state constraints. Indiana Univ. Math. J. 43 (1994) 493–519. CrossRef
P.L. Lions, Generalized solutions of Hamilton-Jacobi equations. Pitman (1982).
Newcomb II, R.T. and Eikonal, J. Su equations with discontinuities. Diff. Integral Equations 8 (1995) 1947–1960.
Ostrov, D.N., Extending viscosity solutions to eikonal equations with discontinuous spatial dependence. Nonlinear Anal. TMA 42 (2000) 709–736. CrossRef
F. Rampazzo and H. Sussmann, Set-valued differentials and a nonsmooth version of Chow's theorem, in Proc. of the 40th IEEE Conference on Decision and Control. Orlando, Florida (2001) 2613–2618.
Soner, H.M., Optimal control problems with state constraints I. SIAM J. Control Optim. 24 (1987) 551–561.
Soravia, P., Hölder continuity of the minimum time function with C 1-manifold targets. J. Optim. Theory Appl. 75 (1992) 401–421. CrossRef
Soravia, P., Discontinuous viscosity solutions to Dirichlet problems for Hamilton-Jacobi equations with convex hamiltonians. Commun. Partial Diff. Equations 18 (1993) 1493–1514. CrossRef
Soravia, P., Boundary value problems for Hamilton-Jacobi equations with discontinuous Lagrangian. Indiana Univ. Math. J. 51 (2002) 451–476. CrossRef
P. Soravia, Uniqueness results for viscosity solutions of fully nonlinear, degenerate elliptic equations with discontinuous coefficients. Commun. Pure Appl. Anal. (To appear).
Swiech, A., $W^{1,p}$ -interior estimates for solutions of fully nonlinear, uniformly elliptic equations. Adv. Differ. Equ. 2 (1997) 1005–1027.
Tourin, A., A comparison theorem for a piecewise Lipschitz continuous Hamiltonian and applications to shape-from-shading. Numer. Math. 62 (1992) 75–85. CrossRef