Archive for Rational Mechanics and Analysis

, Volume 200, Issue 3, pp 1051–1073 | Cite as

Hamilton Jacobi Equations with Obstacles

  • Camillo De LellisEmail author
  • Roger Robyr


We consider a problem in the theory of optimal control proposed for the first time by Bressan. We characterize the associated minimum time function using tools from geometric measure theory and we obtain, as a corollary, an existence theorem for a related variational problem.


Viscosity Solution Maximal Element Differential Inclusion Hamilton Jacobi Equation Geometric Measure Theory 
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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  1. 1.
    Ambrosio, L., Fusco, N., Pallara, D.: Functions of bounded variation and free discontinuity problems. Oxford Mathematical Monographs, 2000Google Scholar
  2. 2.
    Aubin J.P., Cellina A.: Differential Inclusions. Springer-Verlag, Berlin (1984)zbMATHCrossRefGoogle Scholar
  3. 3.
    Bardi M., Capuzzo-Dolcetta I.: Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations. Birkhuser, Boston (1997)zbMATHCrossRefGoogle Scholar
  4. 4.
    Bardi, M., Crandall, M.G., Evans, L.C., Soner, H.M., Souganidis, E.: Viscosity Solutions and Applications. Lecture Notes in Mathematics, Vol. 1660. Springer-Verlag, Berlin, 1997Google Scholar
  5. 5.
    Bressan A.: Differential inclusions and the control of forest fires. J. Differ. Equ. 243, 179–207 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Bressan A., Burago M., Friend A., Jou J.: Blocking strategies for a fire control problem. Anal. Appl. 6, 229–246 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Bressan A., De Lellis C.: Existence of optimal strategies for a fire confinement problem. Commun. Pure Appl. Math. 62, 789–830 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Bressan, A., Wang, T.: Equivalent formulation and numerical analysis of fire confinement problem. Preprint, 2008Google Scholar
  9. 9.
    Bressan A., Wang T.: The minimum speed for a blocking problem on a half plane. J. Math. Anal. Appl. 356, 133–144 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Evans, L.C.: Partial Differential Equations. Graduate Studies in Mathematics. AMS, 1991Google Scholar
  11. 11.
    Evans L.C., Souganidis P.E.: Differential games and representation formulas for solutions of Hamilton-Jacobi-Isaacs equations. Indiana Univ. Math. J. 33, 773–797 (1984)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Federer, H.: Geometric Measure Theory. Die Grundlehren der mathematischen Wissenschaften, Band 153. Springer-Verlag, New York, 1969Google Scholar

Copyright information

© Springer-Verlag 2010

Authors and Affiliations

  1. 1.Institut für MathematikUniversität ZürichZurichSwitzerland

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