Degenerate Eikonal equations with discontinuous refraction index
ESAIM: Control, Optimisation and Calculus of Variations, Volume 12 (2006) no. 2, pp. 216-230.

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.

DOI: 10.1051/cocv:2005033
Classification: 35A05,  35F30,  49L20,  49L25
Keywords: geometric optics, viscosity solutions, eikonal equation, minimum time problem, discontinuous coefficients
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author = {Soravia, Pierpaolo},
title = {Degenerate {Eikonal} equations with discontinuous refraction index},
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Soravia, Pierpaolo. Degenerate Eikonal equations with discontinuous refraction index. ESAIM: Control, Optimisation and Calculus of Variations, Volume 12 (2006) no. 2, pp. 216-230. doi : 10.1051/cocv:2005033. http://archive.numdam.org/articles/10.1051/cocv:2005033/

[1] M. Bardi, A boundary value problem for the minimum-time function. SIAM J. Control Optim. 27 (1989) 776-785. | Zbl

[2] M. Bardi and I. Capuzzo-Dolcetta, Optimal control and viscosity solutions of Hamilton-Jacobi-Bellman equations. Birkhäuser (1997). | MR | Zbl

[3] M. Bardi and P. Soravia, 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. | Zbl

[4] G. Barles, Solutions de viscosité des équations de Hamilton-Jacobi. Springer-Verlag (1994). | MR | Zbl

[5] G. Barles and B. Perthame, Discontinuous solutions of deterministic optimal stopping time problems. RAIRO: M2AN 21 (1987) 557-579. | Numdam | Zbl

[6] L. Caffarelli, M.G. Crandall, M. Kocan and A. Swiech, On viscosity solutions of fully nonlinear equations with measurable ingredients. Comm. Pure Appl. Math. 49 (1996) 365-397. | Zbl

[7] F. Camilli and A. Siconolfi, Hamilton-Jacobi equations with measurable dependence on the state variable. Adv. Differential Equations 8 (2003) 733-768. | Zbl

[8] I. Capuzzo Dolcetta and P.L. Lions, Hamilton-Jacobi equations with state constraints. Trans. Am. Math. Soc. 318 (1990) 643-683. | Zbl

[9] R. Courant and D. Hilbert, Methods of mathematical physics Vol. II. John Wiley & Sons (1989). | MR | Zbl

[10] M.G. Crandall, H. Ishii and P.L. Lions, User's guide to viscosity solutions of second order partial differential equations. Bull. Amer. Math. Soc. 27 (1992) 1-67. | Zbl

[11] M. Garavello and P. Soravia, Optimality principles and uniqueness for Bellman equations of unbounded control problems with discontinuous running cost. Nonlin. Diff. Equations Appl. 11 (2004) 271-298. | Zbl

[12] G.W. Haynes and H. Hermes, Nonlinear controllability via Lie theory. SIAM J. Control 8 (1970) 450-460. | Zbl

[13] H. Ishii, A boundary value problem of the Dirichlet type for Hamilton-Jacobi equations. Ann. Sc. Norm. Sup. Pisa (IV) 16 (1989) 105-135. | Numdam | Zbl

[14] M.A. Katsoulakis, Viscosity solutions of second order fully nonlinear elliptic equations with state constraints. Indiana Univ. Math. J. 43 (1994) 493-519. | Zbl

[15] P.L. Lions, Generalized solutions of Hamilton-Jacobi equations. Pitman (1982). | MR | Zbl

[16] R.T. Newcomb Ii and J. Su, Eikonal equations with discontinuities. Diff. Integral Equations 8 (1995) 1947-1960. | Zbl

[17] D.N. Ostrov, Extending viscosity solutions to eikonal equations with discontinuous spatial dependence. Nonlinear Anal. TMA 42 (2000) 709-736. | Zbl

[18] 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.

[19] H.M. Soner, Optimal control problems with state constraints I. SIAM J. Control Optim. 24 (1987) 551-561. | Zbl

[20] P. Soravia, Hölder continuity of the minimum time function with ${C}^{1}$-manifold targets. J. Optim. Theory Appl. 75 (1992) 401-421. | Zbl

[21] P. Soravia, Discontinuous viscosity solutions to Dirichlet problems for Hamilton-Jacobi equations with convex hamiltonians. Commun. Partial Diff. Equations 18 (1993) 1493-1514. | Zbl

[22] P. Soravia, Boundary value problems for Hamilton-Jacobi equations with discontinuous Lagrangian. Indiana Univ. Math. J. 51 (2002) 451-476. | Zbl

[23] P. Soravia, Uniqueness results for viscosity solutions of fully nonlinear, degenerate elliptic equations with discontinuous coefficients. Commun. Pure Appl. Anal. (To appear). | MR

[24] A. Swiech, ${W}^{1,p}$-interior estimates for solutions of fully nonlinear, uniformly elliptic equations. Adv. Differ. Equ. 2 (1997) 1005-1027. | Zbl

[25] A. Tourin, A comparison theorem for a piecewise Lipschitz continuous Hamiltonian and applications to shape-from-shading. Numer. Math. 62 (1992) 75-85. | Zbl

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