Yakuninagalina@yandex.ru UDC 517.9The modern theory of fully nonlinear second order partial differential equations is based on some algebraic facts and, in particular, on the theory of a-hyperbolic polynomials created by L. Gårding in 1959. The goal of this paper is to describe the Gårding cones in this context. Bibliography: 30 titles.
We prove that the Dirichlet problem is solvable in a generalized sense ]or a class of nonlinear elliptic equations and related equations of Monge-Ampdre type. Bibliography: 1~ titles. v u 1 [ Jm-l( )= n-m+i jv(u~);'~-l[u]dx' m=2,...,n, f~ where/~ = 1, and the operators #m-l," m = 1, .. . , n, are defined by recursion as the variational derivatives for the functionals J,~,_2(u); u U ~ u 1= It turns out [9, 11], that #V~[u] --~:m[A, u]with A(p) = vp-~ 1, where vpp is the Hessian matrix for v(p). In particular for the function v = X/1 + p2 of the matrix A = X/1 + p2(~hj. + pipj) we Obtain the curvature operator of order m with a vector k defining the principal curvatures of the graph of the function u(x). If v = p2/2, then A = E, the vector k is the vector of eigenvalues of the matrix u~, and we are dealing with operators of Monge-Amp&re type. It might be more correct to call them operators of Laplacian type.
В статье обсуждается влияние новых геометрических инвариантов
области на гессиановские интегральные неравенства и
дано новое доказательство известных неравенств Трудингера-Ванга.
Проведен сравнительный анализ неравенств Трудингера-Ванга
с классическим неравенством Пуанкаре-Фридрихса,
который показывает, что они качественно различны. Показано,
что гессиановские интегральные неравенства содержат
информацию нового типа и не имеют аналогов
в классическом функциональном анализе.
Библиография: 19 названий.
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.