Coerciveness of functional-differential equations

Abstract

ABSTRACT. We consider functional-differential equations with the Dirichlet conditions and with contraction and dilatation of the arguments. Necessary and sufficient conditions are obtained under which a Gs type inequality holds. These results allow us to verify coerciveness by using a special "symbol" of the equation considered.The coerciveness problem for quadratic forms generated by differential operators was studied by M. Figueiredo [4], and J. Necas in the book [5]. In particular, it was proved that the strong ellipticity of a system of differential operators with Dirichlet conditions on the boundary is a necessary and sufficient condition for this system to satisfy a Gs type inequality. A. Skubachevskii [6] obtained necessary as well as sufficient conditions in algebraic form for differencedifferential operators to satisfy a Gs type inequality. He also proved that these conditions coincide for almost all domains Q c R".The present paper treats functional-differential equations with Dirichlet conditions and with contracted and expanded arguments. We present necessary and sufficient conditions for a Gs type inequality. With the equations under consideration we associate an algebraic expression (the symbol of the equation) whose positiveness is the desired condition. Theorems 1 and 2 contain the main results. Let us begin with some preliminary considerations.w Let S =-1 be the unit sphere in R n. To each function g G C (S "-1) we assign a function G G L~(R ") by setting c(=)=g (= e R"). (i)The multiplication by G(x) is a bounded operator in L~(R ") k. It will be also denoted by G. Thus,Obviously, the mapping g ~-. G is a homomorphism of the algebra C(S n-1 ) into the algebra of bounded operators B(L2(R")). Let us show that this is an isometry. On the one hand, we have

The first boundary-value problem for strongly elliptic functional-differential equations with orthotropic contractions

“…We can show in a similar way to [4] that σ(P ) coincides with this circle. If R = a 0 I + a 1 P + a −1 P −1 is an operator with complex coefficients a 0 , a 1 , a −1 , then by R we denote the operator with complex conjugate coefficients,…”

“…To obtain sufficient conditions for the positive definiteness of the operators (2.13), we use the approach proposed in [4]. These conditions are given below.…”

“…Functional differential equations containing contractions and dilations of the arguments of the unknown function in the leading part were considered in [4]- [7], where it was assumed that the contraction (dilation) coefficient is the same for all variables:…”

Analysis of functional-differential equation with orthotropic contractions

“…First examples of boundary value problems for FDEs with contractions of independent variables in the principal part were considered in [3,15]. In the multidimensional case, equations with isotropic contractions were first considered, i.e.…”

Модельная Задача В Конусе Для Функционально-Дифференциального Оператора С Преобразованиями Поворота

“…Необходимые и достаточные условия сильной эллиптичности в алгебраической форме для дифференциальных операторов были получены М. И. Вишиком и Л. Гордингом [1], [2]. Позднее А. Л. Скубачевский [3] получил необходимые и достаточные условия сильной эллиптичности для дифференциально-разностных операторов в виде положительной определенности некоторых матриц.…”