Computational Mathematics and Modeling

, Volume 11, Issue 4, pp 391–400

Generalized Hopf formulas for the nonautonomous Hamilton-Jacobi equation

• I. V. Rublev
Mathematical Modeling

Abstract

Generalized Hopf formulas are provided for minimax (viscosity) solutions of Hamilton-Jacobi equations of the formVt+H(t, DxV)=0 andVt+H(t, V, DxV)=0 with the boundary conditionV (T, x)=ϕ(x), where ϕ is a convex function. The bounds within which these formulas apply are elucidated.

Keywords

Mathematical Modeling Computational Mathematic Industrial Mathematic Convex Function Hopf Formula
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.

References

1. 1.
A. I. Subbotin,Minimax Solutions and the Hamilton-Jacobi Equation [in Russian], Nauka, Moscow (1991).Google Scholar
2. 2.
A. I. Subbotin, “Minimax solutions of first-order partial differential equations,”Usp. Mat. Nauk,51, No. 2, 105–138 (1996).
3. 3.
D. B. Silin, “Set-valued integration and viscosity solutions of the Hamilton-Jacobi equation,”Differents. Uravn.,31, 129–137 (1995).
4. 4.
D. B. Silin, “Viscosity solutions via unbounded set-valued integration,”Nonlinear Anal. T. M. A.,31, 55–90 (1998).
5. 5.
E. Hopf, “Generalized solutions of nonlinear equations of first order,J. Math. Mech.,14, 951–973 (1965).
6. 6.
R. T. Rockafellar,Convex Analysis [Russian translation], Mir, Moscow (1973).Google Scholar
7. 7.
E. N. Barron and H. Ishii, “The Bellman equation for minimizing the maximum cost,”Nonlinear Anal. T. M. A.,13, 1067–1090 (1989).
8. 8.
E. N. Barron, R. Jensen, and W. Liu, “Hopf-Lax formula foru t+H(u, Du)=0, II,”Comm. PDE,22, 1141–1160 (1997).
9. 9.
M. G. Crandall and L. C. Evans, “Viscosity solutions of Hamilton-Jacobi equations,Trans. AMS,277, 1–42 (1983).Google Scholar
10. 10.
M. G. Crandall, L. C. Evans, and P.-L. Lions, “Some properties of viscosity solutions of Hamilton-Jacobi equations,”Trans. AMS,282, 487–502 (1984).