Sufficient conditions for the existence of viscosity solutions for nonconvex Hamiltonians

SIAM Journal on Mathematical Analysis

Volumn 36, Issue 1, 2005, Pages 186-203

  Pisante, Giovanni ^a,b  

^a UNIVERSITY OF NAPLES FEDERICO II   (Italy)

^b EPFL   (Switzerland)

Author keywords

Hamilton Jacobi equations; Viscosity solutions

Indexed keywords

FUNCTION EVALUATION; GAME THEORY; PROBLEM SOLVING; SET THEORY; THEOREM PROVING; VISCOSITY;

DIFFERENTIAL GAME THEORY; HAMILTON-JACOBI EQUATIONS; OPTIMAL CONTROL THEORY; VISCOSITY SOLUTIONS;

HAMILTONIANS;

EID: 14544300078     PISSN: 00361410     EISSN: None     Source Type: Journal    
DOI: 10.1137/S0036141003426902     Document Type: Article

References (24)

1. M. BARDI AND I. CAPUZZO-DOLCETTA, Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations. Systems and Control: Foundations and Applications, Birkhäuser Boston, Boston, 1997.
2. G. BARLES, Solutions de viscosité des équations de Hamilton-Jacobi, Math. Appl. 17, Springer-Verlag, Paris, 1994.
3. E. N. BARRON AND R. JENSEN, Semicontinuous viscosity solutions for Hamilton-Jacobi equations with convex Hamiltonians, Comm. Partial Differential Equations, 15 (1990), pp. 1713-1742.
4. G. BOULIGAND, Sur la semi-continuité d'inclusions et quelques sujets connexes, Enseign. Math., 31 (1932), pp. 11-22.
5. M. BORN AND E. WOLF, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light, 3rd ed., Pergamon Press, Oxford, UK, 1965.
6. P. CARDALIAGUET, B. DACOROGNA, W. GANGBO, AND N. GEORGY, Geometric restrictions for the existence of viscosity solutions, Ann. Inst. H. Poincaré Anal. Non Linéaire, 16 (1999), pp. 189-220.
7. F. H. CLARKE, Generalized gradients and applications, Trans. Amer. Math. Soc., 205 (1975), pp. 247-262.
8. F. H. CLARKE, Optimization and Nonsmooth Analysis, Canadian Mathematical Society Series of Monographs and Advanced Texts, John Wiley and Sons, New York, 1983.
9. M. G. CRANDALL, L. C. EVANS, AND P.-L. LIONS, Some properties of viscosity solutions of Hamilton-Jacobi equations, Trans. Amer. Math. Soc., 282 (1984), pp. 487-502.
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. (N.S.), 27 (1992), pp. 1-67.
11. M. G. CRANDALL AND P.-L. LIONS, Viscosity solutions of Hamilton-Jacobi equations, Trans. Amer. Math. Soc., 277 (1983), pp. 1-42.
12. B. DACOROGNA AND P. MARCELLINI, Sur le problème de Cauchy-Dirichlet pour les systèmes d'équations non linéaires du premier ordre, C. R. Acad. Sci. Paris Sér. I Math., 323 (1996), pp. 599-602.
13. B. DACOROGNA AND P. MARCELLINI, Théorèmes d'existence dans les cas scalaire et vectoriel pour les équations de Hamilton-Jacobi, C. R. Acad. Sci. Paris Sér. I Math., 322 (1996), pp. 237-240.
14. B. DACOROGNA AND P. MARCELLINI, General existence theorems for Hamilton-Jacobi equations in the scalar and vectorial cases, Acta Math., 178 (1997), pp. 1-37.
15. B. DACOROGNA AND P. MARCELLINI, Implicit Partial Differential Equations, Progr. Nonlinear Differential Equations Appl. 37, Birkhäuser Boston, Boston, 1999.
16. L. C. EVANS AND R. F. GARIEPY, Measure Theory and Fine Properties of Functions, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1992.
17. H. FRANKOWSKA, Hamilton-Jacobi equations: Viscosity solutions and generalized gradients, J. Math. Anal. Appl., 141 (1989), pp. 21-26.
18. N. GEORGY, Equations de type implicite du premier ordre, Ph.D. thesis, EPFL, Lausanne, Switzerland, 1999.
19. M. R. HESTENES, Optimization Theory. The Finite Dimensional Case. Pure and Applied Mathematics, Wiley-Interscience (John Wiley and Sons), New York, 1975.
20. S. N. KRUŽKOV, Generalized solutions of Hamilton-Jacobi equations of eikonal type. I. Statement of the problems; existence, uniqueness and stability theorems; certain properties of the solutions, Mat. Sb. (N.S.), 98 (1975), pp. 450-493; 496. (In Russian.)
21. P.-L. LIONS, Generalized Solutions of Hamilton-Jacobi Equations, Res. Notes Math. 69, Pitman, Boston, 1982.
22. R. T. ROCKAFELLAR, La théorie des sous-gradients et ses applications à l'optimisation: Fonctions convexes et non convexes, Presses de l'Université de Montréal, Montreal, Quebec, Canada, 1979.
23. R. T. ROCKAFELLAR AND R. J.-B. WETS, Variational Analysis, Grundlehren Math. Wiss. 317, Springer-Verlag, Berlin, 1998.
24. S. SAKS, Theory of the Integral, 2nd ed., Dover, New York, 1964.