A. The extremality problem representation of the Lax type [1] is studied in detail for some class of Hamilton-Jacobi equations in the many-dimensional case. The regularity properties of solutions of the Cauchy problem in the class of convex lower semicontinuous functions are established. A generalisation to a wider class of functions is obtained. The Hamilton-Jacobi equation on the sphere is considered, and its exact solutions are found in terms of a Lax type extremality problem. Some generalisation of the results for the general case of many-dimensional Hamilton-Jacobi equations is obtained by using the Fan-Brouwder fixed point techniques in a Banach space.
Abstract:The projection-algebraic approach of the Calogero type for discrete approximations of linear and nonlinear differential operator equations in Banach spaces is studied. The solution convergence and realizability properties of the related approximating schemes are analyzed. For the limiting-dense approximating scheme of linear differential operator equations a new convergence theorem is stated. In the case of nonlinear differential operator equations the effective convergence conditions for the approximated solution sets, based on a Leray-Schauder type fixed point theorem, are obtained. 1. The projection-algebraic method of discrete approximations for linear differential operator equations
MSC:
Preliminaries: a Lie-algebraic scheme of the Calogero typeA projection-algebraic approach to discrete approximations of differential operator equations has been proposed by F. Calogero in 1983 for calculating the eigenvalues of linear differential operators [5,7] in Hilbert spaces. There were *
517.9The generalized method of characteristics is developed within the framework of the geometric Monge picture. Hopf-Lax-type extremality solutions are obtained for a broad class of Cauchy problems for nonlinear partial differential equations of the first and higher orders. A special Hamilton-Jacobi-type case is analyzed separately. An exact extremality Hopf-Lax-type solution of the Cauchy problem for the nonlinear Burgers equation is obtained, and its linearization to the Hopf-Cole expression and to the corresponding Airy-type linear partial differential equation is found and discussed.
We develop the Cartan -Monge geometric approach to the characteristic method for nonlinear partial differential equations of the first and higher orders. The Hamiltonian structure of characteristic vector fields related with nonlinear partial differential equations of the 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.Розвинуто геометричний пiдхiд Картана -Монжа до методу характеристик для нелiнiйних диференцiальних рiвнянь з частинними похiдними першого та вищих порядкiв. Дослiджено га-мiльтонову структуру характеристичних векторних полiв, пов'язаних iз нелiнiйними диферен-цiальними рiвняннями з частинними похiдними першого порядку, та побудовано тензорнi поля зi спецiальною структурою для визначення характеристичних полiв, природно пов'язаних iз нелiнiйними диференцiальними рiвняннями з частинними похiдними вищих порядкiв.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.