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An Introduction to the Theory of Viscosity Solutions for First-Order Hamilton–Jacobi Equations and Applications

Part of the Lecture Notes in Mathematics book series (LNMCIME,volume 2074)

Abstract

In this course, we first present an elementary introduction to the concept of viscosity solutions for first-order Hamilton–Jacobi Equations: definition, stability and comparison results (in the continuous and discontinuous frameworks), boundary conditions in the viscosity sense, Perron’s method, Barron–Jensen solutions etc. We use a running example on exit time control problems to illustrate the different notions and results. In a second part, we consider the large time behavior of periodic solutions of Hamilton–Jacobi Equations: we describe recents results obtained by using partial differential equations type arguments. This part is complementary of the course of H. Ishii which presents the dynamical system approach (“weak KAM approach”).

Keywords

  • Viscosity Solution
  • Jacobi Equation
  • Large Time Behavior
  • Viscosity Subsolution
  • Viscosity Sense

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Notes

  1. 1.

    Here we use the notation Du for the full gradient of u in space and time but, in general, we will use it for the gradient in space of u.

  2. 2.

    Here we have also used a less important (but simplifying) property, namely the commutation with constants: for any \(c\,\in \,\mathbb{R}\), S, x, t and for any function u( ⋅), G(S, x, t, u( ⋅) + c) = G(S, x, t, u( ⋅)) + c.

  3. 3.

    In Biton [19], a non-trivial counterexample to the uniqueness for (17) is given in a situation where the Cauchy–Lipschitz Theorem cannot be applied to (18).

  4. 4.

    This is a key point: the compactness of the domain (periodicity) plays a crucial role here since local uniform convergence is the same as global uniform convergence.

References

  1. L. Alvarez, F. Guichard, P.L. Lions, J.M. Morel, Axioms and fundamental equations of image processing. Arch. Ration. Mech. Anal. 123(3), 199–257 (1993)

    CrossRef  MathSciNet  MATH  Google Scholar 

  2. M. Bardi, I. Capuzzo Dolcetta, Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations (Birkhäuser, Boston, 1997)

    CrossRef  MATH  Google Scholar 

  3. M. Bardi, M.G. Crandall, L.C. Evans, H.M. Soner, P.E. Souganidis, Viscosity Solutions and Applications, ed. by I. Capuzzo Dolcetta, P.L. Lions. Lecture Notes in Mathematics, vol. 1660 (Springer, Berlin, 1997), x + 259 pp

    Google Scholar 

  4. G. Barles, Discontinuous viscosity solutions of first order Hamilton-Jacobi equations: a guided visit. Nonlinear Anal. TMA 20(9), 1123–1134 (1993)

    CrossRef  MathSciNet  MATH  Google Scholar 

  5. G. Barles, Solutions de Viscosité des Équations de Hamilton-Jacobi Mathématiques & Applications (Berlin), vol. 17 (Springer, Paris, 1994)

    Google Scholar 

  6. G. Barles, Nonlinear Neumann boundary conditions for quasilinear degenerate elliptic equations and applications. J. Differ. Equ. 154, 191–224 (1999)

    CrossRef  MathSciNet  MATH  Google Scholar 

  7. G. Barles, H. Mitake, A PDE approach to large-time asymptotics for boundary-value problems for nonconvex Hamilton-Jacobi equations. Commun. Partial Differ. Equ. 37(1), 136–168 (2012). doi:10.1080/03605302.2011.553645

    CrossRef  MathSciNet  MATH  Google Scholar 

  8. G. Barles, B. Perthame, Discontinuous solutions of deterministic optimal stopping time problems. Model. Math. Anal. Numer. 21(4), 557–579 (1987)

    MathSciNet  MATH  Google Scholar 

  9. G. Barles, B. Perthame, Exit time problems in optimal control and vanishing viscosity method. SIAM J. Control Optim. 26, 1133–1148 (1988)

    CrossRef  MathSciNet  MATH  Google Scholar 

  10. G. Barles, B. Perthame, Comparison principle for Dirichlet type Hamilton-Jacobi Equations and singular perturbations of degenerated elliptic equations. Appl. Math. Optim. 21, 21–44 (1990)

    CrossRef  MathSciNet  MATH  Google Scholar 

  11. G. Barles, J.-M. Roquejoffre, Ergodic type problems and large time behaviour of unbounded solutions of Hamilton-Jacobi equations. Commun. Partial Differ. Equ. 31(7–9), 1209–1225 (2006)

    CrossRef  MathSciNet  MATH  Google Scholar 

  12. G. Barles, E. Rouy, A strong comparison result for the Bellman equation arising in stochastic exit time control problems and its applications. Commun. Partial Differ. Equ. 23(11 & 12), 1995–2033 (1998)

    MathSciNet  MATH  Google Scholar 

  13. G. Barles, P.E. Souganidis, Convergence of approximation schemes for fully nonlinear second order equations. Asymptot. Anal. 4, 271–283 (1991)

    MathSciNet  MATH  Google Scholar 

  14. G. Barles, P.E. Souganidis, A new approach to front propagation problems: theory and applications. Arch. Ration. Mech. Anal. 141, 237–296 (1998)

    CrossRef  MathSciNet  MATH  Google Scholar 

  15. G. Barles, P.E. Souganidis, On the large time behavior of solutions of Hamilton-Jacobi equations. SIAM J. Math. Anal. 31(4), 925–939 (2000)

    CrossRef  MathSciNet  MATH  Google Scholar 

  16. G. Barles, A. Tourin, Commutation properties of semigroups for first-order Hamilton-Jacobi equations and application to multi-time equations. Indiana Univ. Math. J. 50(4), 1523–1544 (2001)

    CrossRef  MathSciNet  MATH  Google Scholar 

  17. E.N. Barron, R. Jensen, Semicontinuous viscosity solutions of Hamilton-Jacobi Equations with convex hamiltonians. Commun. Partial Differ. Equ. 15(12), 1713–1740 (1990)

    CrossRef  MathSciNet  MATH  Google Scholar 

  18. E.N. Barron, R. Jensen, Optimal control and semicontinuous viscosity solutions. Proc. Am. Math. Soc. 113, 49–79 (1991)

    CrossRef  MathSciNet  Google Scholar 

  19. S. Biton, Nonlinear monotone semigroups and viscosity solutions. Ann. Inst. Henri Poincaré Anal. Non Linéaire 18(3), 383–402 (2001)

    CrossRef  MathSciNet  MATH  Google Scholar 

  20. F. Cardin, C. Viterbo, Commuting Hamiltonians and Hamilton-Jacobi multi-time equations. Duke Math. J. 144(2), 235–284 (2008)

    CrossRef  MathSciNet  MATH  Google Scholar 

  21. M.G. Crandall, L.C. Evans, P.L, Lions, Some properties of viscosity solutions of Hamilton-Jacobi equations. Trans. Am. Math. Soc. 282, 487–502 (1984)

    Google Scholar 

  22. M.G. Crandall, P.L. Lions, Viscosity solutions of Hamilton-Jacobi equations. Trans. Am. Math. Soc. 277, 1–42 (1983)

    CrossRef  MathSciNet  MATH  Google Scholar 

  23. M.G. Crandall, H. Ishii, P.-L. Lions, User’s guide to viscosity solutions of second order partial differential equations. Bull. AMS 27, 1–67 (1992)

    CrossRef  MathSciNet  MATH  Google Scholar 

  24. A. Davini, A. Siconolfi, A generalized dynamical approach to the large time behavior of solutions of Hamilton-Jacobi equations. SIAM J. Math. Anal. 38(2), 478–502 (2006)

    CrossRef  MathSciNet  Google Scholar 

  25. A. Fathi, Sur la convergence du semi-groupe de Lax-Oleinik. C. R. Acad. Sci. Paris Sér. I Math. 327(3), 267–270 (1998)

    CrossRef  MathSciNet  MATH  Google Scholar 

  26. W.H Fleming, H.M. Soner, Controlled Markov Processes and Viscosity Solutions. Applications of Mathematics (Springer, New-York, 1993)

    Google Scholar 

  27. N. Ichihara, H. Ishii, Asymptotic solutions of Hamilton-Jacobi equations with semi-periodic Hamiltonians. Commun. Partial Differ. Equ. 33(4–6), 784–807 (2008)

    CrossRef  MathSciNet  MATH  Google Scholar 

  28. N. Ichihara, H. Ishii, The large-time behavior of solutions of Hamilton-Jacobi equations on the real line. Methods Appl. Anal. 15(2), 223–242 (2008)

    MathSciNet  MATH  Google Scholar 

  29. N. Ichihara, H. Ishii, Long-time behavior of solutions of Hamilton-Jacobi equations with convex and coercive Hamiltonians. Arch. Ration. Mech. Anal. 194(2), 383–419 (2009)

    CrossRef  MathSciNet  MATH  Google Scholar 

  30. H. Ishii, Hamilton-Jacobi equations with discontinuous Hamiltonians on arbitrary open sets. Bull. Faculty Sci. Eng. Chuo Univ. 28, 33–77 (1985)

    Google Scholar 

  31. H. Ishii, Perron’s method for Hamilton-Jacobi equations. Duke Math. J. 55, 369–384 (1987)

    CrossRef  MathSciNet  MATH  Google Scholar 

  32. H. Ishii, A simple, direct proof of uniqueness for solutions of Hamilton-Jacobi equations of eikonal type. Proc. Am. Math. Soc. 100, 247–251 (1987)

    CrossRef  MATH  Google Scholar 

  33. H. Ishii, Asymptotic solutions for large time of Hamilton-Jacobi equations in Euclidean n space. Ann. Inst. Henri Poincaré Anal. Non Linéaire 25(2), 231–266 (2008)

    CrossRef  MATH  Google Scholar 

  34. H. Ishii, Fully nonlinear oblique derivative problems for nonlinear second-order elliptic PDE’s. Duke Math. J. 62, 663–691 (1991)

    CrossRef  MathSciNet  Google Scholar 

  35. J.M. Lasry, P.L. Lions, A remark on regularization in Hilbert spaces. Isr. J. Math. 55, 257–266 (1986)

    CrossRef  MathSciNet  MATH  Google Scholar 

  36. P.L. Lions, Generalized Solutions of Hamilton-Jacobi Equations. Research Notes in Mathematics, vol. 69 (Pitman Advanced Publishing Program, Boston, 1982)

    Google Scholar 

  37. P.-L. Lions, Neumann type boundary conditions for Hamilton-Jacobi equations. Duke Math. J. 52(4), 793–820 (1985)

    CrossRef  MathSciNet  MATH  Google Scholar 

  38. H. Mitake, Asymptotic solutions of Hamilton-Jacobi equations with state constraints. Appl. Math. Optim. 58(3), 393–410 (2008)

    CrossRef  MathSciNet  MATH  Google Scholar 

  39. H. Mitake, The large-time behavior of solutions of the Cauchy-Dirichlet problem for Hamilton-Jacobi equations. Nonlinear Differ. Equ. Appl. 15(3), 347–362 (2008)

    CrossRef  MathSciNet  MATH  Google Scholar 

  40. H. Mitake, Large time behavior of solutions of Hamilton-Jacobi equations with periodic boundary data. Nonlinear Anal. 71(11), 5392–5405 (2009)

    CrossRef  MathSciNet  MATH  Google Scholar 

  41. M. Motta, F. Rampazzo, Nonsmooth multi-time Hamilton-Jacobi systems. Indiana Univ. Math. J. 55(5), 1573–1614 (2006)

    CrossRef  MathSciNet  MATH  Google Scholar 

  42. G. Namah, J.-M. Roquejoffre, Remarks on the long time behaviour of the solutions of Hamilton-Jacobi equations. Commun. Partial Differ. Equ. 24(5–6), 883–893 (1999)

    CrossRef  MathSciNet  MATH  Google Scholar 

  43. J.-M. Roquejoffre, Convergence to steady states or periodic solutions in a class of Hamilton-Jacobi equations. J. Math. Pures Appl. (9) 80(1), 85–104 (2001)

    Google Scholar 

  44. H.M. Soner, Optimal control problems with state-space constraints. SIAM J. Control Optim. 24, 552–562, 1110–1122 (1986)

    CrossRef  MathSciNet  MATH  Google Scholar 

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Barles, G. (2013). An Introduction to the Theory of Viscosity Solutions for First-Order Hamilton–Jacobi Equations and Applications. In: Hamilton-Jacobi Equations: Approximations, Numerical Analysis and Applications. Lecture Notes in Mathematics(), vol 2074. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-36433-4_2

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