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