Applications of the Hopf-Lax formula for ut + H(u, Du) = 0

SIAM Journal on Mathematical Analysis

Volumn 29, Issue 4, 1998, Pages 1022-1039

  Barron, E N ^a   Jensen, R ^a   Liu, W ^a  

^a LOYOLA UNIVERSITY CHICAGO   (United States)

Author keywords

Differential game; Hamilton Jacobi equation; Hopf Lax formula; Level sets

Indexed keywords

EID: 0032362359     PISSN: 00361410     EISSN: None     Source Type: Journal    
DOI: 10.1137/S0036141097319966     Document Type: Article

References (26)

1. M. BARDI AND L. C. EVANS, On Hopf formula for solutions of Hamilton-Jacobi equations, Nonlinear Anal., 8 (1984), pp. 1373-1381.
2. M. BARDI AND S. OSHER, The nonconvex multi-dimensional Riemann problem for Hamilton-Jacobi equations, SIAM J. Math. Anal., 22 (1991), pp. 344-351.
3. M. BARDI AND S. FAGGIAN, Hopf-type formulas for non-convex non-concave Hamilton-Jacobi equations, SIAM J. Math. Anal., to appear.
4. G. BARLES, Solutions de Viscosite des Equations de Hamilton-Jacobi, Mathematiques and Applications 17, Springer-Verlag, New York, 1994.
5. ∞, in Optimal Control of Differential Equations, N. H. Pavel, ed., Marcel Dekker, New York, 1994, pp. 39-47.
6. E. N. BARRON, Differential games with maximum cost, Nonlinear Anal., 14 (1990), pp. 971-989.
7. E. N. BARRON AND H. ISHII, The Bellman equation for minimizing the maximum cost, Nonlinear Anal., 13 (1989), pp. 1067-1090.
8. ∞, Appl. Math. Optim., 35 (1997), pp. 237-263.
9. t + H(u, Du) = 0, J. Differential Equations, 126 (1996), pp. 48-61.
10. t + H(u, Du) = 0: II, Comm. PDE's, 22 (1997), pp. 1141-1160.
11. E. N. BARRON, R. JENSEN, AND W. LIU, Explicit solutions for some first order PDEs, J. Dynam. Control Systems, 3 (1997), pp. 149-164.
12. M. G. CRANDALL AND P.-L. LIONS, Viscosity solutions of Hamilton Jacobi equations, Trans. Amer. Math. Soc., 277 (1983), pp. 1-42.
13. 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.
14. n, in Generalized Concavity in Optimization and Economics, S. Schaible and W. T. Ziemba, eds., Academic Press, New York, 1981, pp. 109-130.
15. J.-P. CROUZEIX, A duality framework in quasi-convex programming, in Generalized Concavity in Optimization and Economics, S. Schaible and W. T. Ziemba, eds., Academic Press, New York, 1981, pp. 207-226.
16. L. C. EVANS AND P. SOUGANIDIS, Differential games and representation formulas for solutions of Hamilton Jacobi Isaacs equations, Indiana Univ. Math. J., 33 (1984), pp. 773-797.
17. E. HOPF, Generalized solutions of nonlinear equations of first order, J. Math. Mech., 14 (1965), pp. 951-973.
18. J. E. MARTINEZ-LEGAZ, On lower subdifferentiable functions, in Trends in Mathematical Optimization, K. H. Hoffman, J. B. Hiriart-Urruty, C. Lemarchal, and J. Zowe, eds., International Series Numer. Math. 84, Birkhauser, Boston, MA, 1988, pp. 197-232.
19. J. E. MARTINEZ-LEGAZ, quasi-convex duality theory by generalized conjugation methods, Optimization, 19 (1988), pp. 603-652.
20. J. E. MARTINEZ-LEGAZ AND S. ROMANO-RODRIGUEZ, α-lower subdifferentiable functions, SIAM J. Optim., 3 (1993), pp. 800-825.
21. J.-P. PENOT AND M. VOLLE, On quasi-convex duality, Math. Oper. Res., 15 (1990), pp. 597-625.
22. R. T. ROCKAFELLAR, Convex Analysis, Princeton University Press, Princeton, NJ, 1970.
23. D. B. SILIN, Generalizing Hopf and Lax-Oleinik formulae via conjugate integral, preprint.
24. D. B. SILIN, Set valued integration and viscosity solutions to Hamilton Jacobi equations, Differential Equations, 31 (1995), pp. 129-137 (in Russian).
25. A. I. SUBBOTIN, Generalized Solutions of First Order PDEs, Birkhauser, Boston, MA, 1995.
26. A. I. SUBBOTIN, Existence and uniqueness results for Hamilton Jacobi equations, Nonlinear Anal., 16 (1991), pp. 683-699.