Opuscula Math. 27, no. 1 (2007), 89-111
On the geometric structure of characteristic vector fields related with nonlinear equations of the Hamilton-Jacobi type
Natalia K. Prykarpatska
Abstract. The Cartan-Monge geometric approach to the characteristics method for Hamilton-Jacobi type equations and nonlinear partial differential equations of higher orders is analyzed. The Hamiltonian structure of characteristic vector fields related with nonlinear partial differential equations of first order is analyzed, the tensor fields of special structure are constructed for defining characteristic vector fields naturally related with nonlinear partial differential equations of higher orders. The generalized characteristics method is developed in the framework of the symplectic theory within geometric Monge and Cartan pictures. The related characteristic vector fields are constructed making use of specially introduced tensor fields, carrying the symplectic structure. Based on their inherited geometric properties, the related functional-analytic Hopf-Lax type solutions to a wide class of boundary and Cauchy problems for nonlinear partial differential equations of Hamilton-Jacobi type are studied. For the non-canonical Hamilton-Jacobi equations there is stated a relationship between their solutions and a good specified functional-analytic fixed point problem, related with Hopf-Lax type solutions to specially constructed dual canonical Hamilton-Jacobi equations.
Keywords: Hamilton-Jacobi equations, the Cartan-Monge geometric approach, Hopf-Lax type representation.
Mathematics Subject Classification: 34A30, 34B05, 34B15.