The Cartan-Monge geometric approach to the characteristic method for Hamilton-Jacobi type equations and its generalization for nonlinear partial differential equations of higher orders

Анотація

Th Cartan-Monge geometric approach to the characteristic 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 fields naturally related with nonlinear partial differential equations of higher orders. The generalizes characteristic method is developed in the framework of the symplectic theory within geometric Monge and Cartan pictures. 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 equatios of Hamilton-Jacobi type are studied. For the non-canonical Hamilton-Jacobi equations there is stated a relationship between their solutions and a good posed function-analytic fixed point problem, related with Hopf-Lax type solutions to specially constructed dual canonical Hamilton-Jacobi equations. Functional-analytic solutions to a Hamilton-Jacobi equation of Ricatti type are obtained and investigated reducing them to the classical Brouwer-Banach type fixed point theory.