Generalizing Hopf and Lax-Oleînik formulae via conjugate integral
Tóm tắt
Using the second Fenchel conjugate transform the conjugate integral sums and the conjugate integral are introduced. An estimate of speed of convergence of the sums to the integral is obtained. In the case of a convex integrant the conjugate integral reduces to the Riemannian one. It is proved that the Fenchel conjugate transform of the conjugate integral with variable upper limit provides a formula for the viscosity solution to a Hamilton-Jacobi equation in which the Hamiltonian depends both on time and the gradient of the unknown function. In the autonomous case the obtained formula coincides with Hopf's one. Two examples are considered in which an application of the conjugate integral allows to find viscosity solutions explicitly. It is shown how the extension of the Lax-Oleînik formula to the nonautonomous case may be obtained using the generalized Hopf formula.
Tài liệu tham khảo
Bardi H, Evans LC (1984) On Hopf's formulas for solutions of Hamilton Jacobi equations. Nonlinear Analysis8: 1373–1281
Brennier Y (1989) Un algorithme rapide pour le calcul de transformées de Legendre-Fenchel discrétes. CR Acad Sci Paris Serie I308: 587–589
Crandall MG, Evans LC, Lions PL (1984) Some properties of viscosity solutions of Hamilton-Jacobi equations. Trans Amer Math Soc283: 487–502
Crandall MG, Lions PL (1983) Viscosity solutions of Hamilton-Jacobi equations. Trans Amer Math Soc277: 1–42
Evans LC, Souganidis P (1984) Differential games and representation formulas for solutions of Hamilton-Jacobi-Isaacs equations. Indiana Univ Math33: 773–797
Hopf E (1965) Generalized solutions of non-linear equations of first order. J Math Mech14: 951–973
Ioffe AD, Tikhomirov VM (1979) Theory of Extremal Problems. Amsterdam: North-Holland
Kruzkov SN (1965) Generalized solutions of non-linear equations of first order. Math USSR Sbornik1: 93–116
Lax PD (1957) Hyperbolic systems of conservation laws. II Comm Pure Appl Math10: 537–566
Lions P-L (1982) Generalized Solutions of Hamilton-Jacobi Equations. Boston: Pitman
Lions PL, Rochet J-C (1986) Hopf formula and multitime Hamilton-Jacobi equation. Proc Amer Math Soc96: 79–84
Oleînik OA (1957) Discontinuous solutions of non-linear differential equations. Uspekhi Matem Nauk12: 3–73: (in Russian)
Pshenichnyi BN (1968) Linear differential games. Automat Remote Control10: 55–67
Rockafellar RT (1970) Convex Analysis. Princeton: Univ Press
Silin DB (1994) On approximation of generalized alternate integral. Moscow Univ Comput Math and Cybern 1, 35–43: (in Russian)
Silin DB (1995) Set-valued differentiation and integration with quasi-affine mappings. Russian Math Doklady340: 164–167: (in Russian)
Silin DB (1995) Set-valued integration and viscosity solutions to Hamilton-Jacobi equations. Differential Equations31: 129–137: (in Russian)
Silin DB, Viscosity solutions via unbounded set-valued integration. Nonlinear Analysis. (to appear)
Silin DB, Trinko NG (1994) A modification of Graham's algorithm for the convexification of a positive-uniform functions. Comput Math and Math Physics34: 545–548
Subbotin AI (1995) Generalized solution of First-Order PDEs. The Dynamical Optimization Perspective. Boston: Birkhäuser
Vasiliev FP (1982) Numerical Methods for solving Extremal Problems. Moscow: Nauka (in Russian)
Zhikov VV (1995) On Lavrentiev's effect. Russian Math Doklady345: 10–14 (in Russian)