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

top
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

top

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

top
  1. M. Bardi, A boundary value problem for the minimum-time function. SIAM J. Control Optim.27 (1989) 776–785.  
  2. M. Bardi and I. Capuzzo-Dolcetta, Optimal control and viscosity solutions of Hamilton-Jacobi-Bellman equations. Birkhäuser (1997).  
  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.  
  4. G. Barles, Solutions de viscosité des équations de Hamilton-Jacobi. Springer-Verlag (1994).  
  5. G. Barles and B. Perthame, Discontinuous solutions of deterministic optimal stopping time problems. RAIRO: M2AN21 (1987) 557–579.  
  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.  
  7. F. Camilli and A. Siconolfi, Hamilton-Jacobi equations with measurable dependence on the state variable. Adv. Differential Equations8 (2003) 733–768.  
  8. I. Capuzzo Dolcetta and P.L. Lions, Hamilton-Jacobi equations with state constraints. Trans. Am. Math. Soc.318 (1990) 643–683.  
  9. R. Courant and D. Hilbert, Methods of mathematical physics Vol. II. John Wiley & Sons (1989).  
  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.  
  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.  
  12. G.W. Haynes and H. Hermes, Nonlinear controllability via Lie theory. SIAM J. Control8 (1970) 450–460.  
  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.  
  14. M.A. Katsoulakis, Viscosity solutions of second order fully nonlinear elliptic equations with state constraints. Indiana Univ. Math. J.43 (1994) 493–519.  
  15. P.L. Lions, Generalized solutions of Hamilton-Jacobi equations. Pitman (1982).  
  16. R.T. Newcomb II and J. Su, Eikonal equations with discontinuities. Diff. Integral Equations8 (1995) 1947–1960.  
  17. D.N. Ostrov, Extending viscosity solutions to eikonal equations with discontinuous spatial dependence. Nonlinear Anal. TMA42 (2000) 709–736.  
  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.  
  20. P. Soravia, Hölder continuity of the minimum time function with C1-manifold targets. J. Optim. Theory Appl.75 (1992) 401–421.  
  21. P. Soravia, Discontinuous viscosity solutions to Dirichlet problems for Hamilton-Jacobi equations with convex hamiltonians. Commun. Partial Diff. Equations18 (1993) 1493–1514.  
  22. P. Soravia, Boundary value problems for Hamilton-Jacobi equations with discontinuous Lagrangian. Indiana Univ. Math. J.51 (2002) 451–476.  
  23. P. Soravia, Uniqueness results for viscosity solutions of fully nonlinear, degenerate elliptic equations with discontinuous coefficients. Commun. Pure Appl. Anal. (To appear).  
  24. A. Swiech, W 1 , p -interior estimates for solutions of fully nonlinear, uniformly elliptic equations. Adv. Differ. Equ.2 (1997) 1005–1027.  
  25. A. Tourin, A comparison theorem for a piecewise Lipschitz continuous Hamiltonian and applications to shape-from-shading. Numer. Math.62 (1992) 75–85.  

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.