Remarks on Algebraic Lagrangian Formalism

An axiomatic approach to the notion of adjoint operator is provided within the framework of the algebraic Lagrangian formalism.     

Kamenev-type Oscillation Criteria for Semilinear Elliptic Differential Equations

Oscillation criteria for semilinear elliptic differential equations are obtained. The results are extensions of integral averaging technique of Kamenev. General means are employed to establish our results.     

微分算子方法在积分中的应用

函数的微分运算化积分运算容易,把积分运算化为微分运算这是一大难题.利用微分算子方法可以把某些积分运算化为微分运算,且使形如∫f(x)eαxsinβxdx,∫f(x)eαxcosβxdx的积分运算简便、快捷.     

Asymptotics of Solutions for Nonlocal Elliptic Problems in Plane Angles

In the paper we investigate asymptotics of solutions for nonlocal elliptic problems in plane angles and in ² \ {0}. These problems arise in studying nonlocal problems in bounded domains in the case where support of nonlocal terms intersects with a boundary of a domain. We obtain explicit formulas for the asymptotic coefficients in terms of eigenvectors and associated vectors of both adjoint nonlocal operators acting in spaces of distributions and formally adjoint (with respect to the Green formula) nonlocal transmission problems.     

Unique solvability of pseudohyperbolic equations with singular right-hand sides

In this paper, we consider the solvability of the Cauchy problem for pseudohyperbolic equations (partial differential equations of third order). For the case in which the right-hand side is a generalized function (distribution) of finite order, we establish a theorem on the unique solvability for a sufficiently general pseudohyperbolic operator. The method of proof is based on a specially constructed “scale” of a priori inequalities for the direct and adjoint operators.     

B-evolutions and sobolev equations

We apply the results on B-evolutions as developed by Sauer to the Sobolev equation ?^t:(Mu)+Lu=0 with initial condition Mu∣_{t = 0} = y and homogeneous boundaryconditions. M and L are uniformly strongly elliptic differential operators. Results on admissible initial conditions for strong solutions are obtained and compared to those of Fichera and Showalter.     

含控制参数的抽象动力方程连值问题之积分算子求解法

讨论一类抽象Volterra型积分算子,利用此获得含控制参数的抽象动力方程边值问题的解。这种新的求解法我们称为积分算子求解法。     

利用微积分算子求幂级数的和函数

针对幂级数求和函数的问题,引入借助微分算子、积分算子和微分方程进行计算的方法,可作为逐项微分法和逐项积分法的一种补充．实例说明其应用．     

On Duality in Spaces of Solutions to Elliptic Systems

Suppose that D is a bounded domain in (n2) with connected real-analytic boundary, A is an elliptic system with real-analytic coefficients in a neighborhood of the closure of D, and sol(A,D) is the space of solutions to the system Au=0 in D furnished with the standard Frechet–Schwartz topology. Then the dual of sol(A,D) represents the space sol(A, ) of solutions to the system Au=0 in a neighborhood of furnished with the standard inductive limit topology over some decreasing net of neighborhoods of . The corresponding pairing is generated by the inner product in the Lebesgue space L ²(D).     

微分算子方法在积分中的应用注记

指出了微分算子方法的缺陷,结果表明:它不能真正解决不定积分的快捷计算问题,有时甚至不如经典算法方便.     

波动方程初值问题的解在两类积分中的应用

利用波动方程初值问题解的特点给出了圆域上一类反常二重积分和球面上第一类曲面积分的微分算子级数公式解和定积分公式解.通过举例说明了该方法相对于常规解法的简便实用性.     

On Compact Solvability of Differentials of an Elliptic Differential Complex

We find sufficient conditions for compact solvability of differentials of an elliptic differential complex on a noncompact Riemannian manifold. As the main example we consider the de Rham complex of differential forms on a manifold with cylindrical ends.     

Continuous rearrangement and symmetry of solutions of elliptic problems

This work presents new results and applications for the continuous Steiner symmetrization. There are proved some functional inequalities, e.g. for Dirichlet-type integrals and convolutions and also continuity properties in Sobolev spacesW ^1,p. Further it is shown that the local minimizers of some variational problems and the nonnegative solutions of some semilinear elliptic problems in symmetric domains satisfy a weak, ‘local’ kind of symmetry.     

A criterion of solvability of the elliptic Cauchy problem in a multi-dimensional cylindrical domain

In this paper, we consider the Cauchy problem for multidimensional elliptic equations in a cylindrical domain. The method of spectral expansion in eigenfunctions of the Cauchy problem for equations with deviating argument establishes a criterion of the strong solvability of the considered elliptic Cauchy problem. It is shown that the ill-posedness of the elliptic Cauchy problem is equivalent to the existence of an isolated point of the continuous spectrum for a self-adjoint operator with deviating argument.     

Singular Perturbation Elliptic Boundary-Value Problems in the Case of a Nonisolated Root of the Degenerate Equation

For a singularly perturbed elliptic equation (the Neumann boundary-value problem), we prove a theorem on the passage to the limit for the case in which the degenerate equation has a nonisolated root.__________Translated from Matematicheskie Zametki, vol. 78, no. 1, 2005, pp. 26–36.Original Russian Text Copyright © 2005 by V. F. Butuzov, M. A. Terent’ev.     

一般形式的一阶椭圆组的非线性Riemann问题

讨论一般形式的一阶线性和拟线性椭圆型方程的非线性Riemann问题,通过把这些问题转化为奇异积分方程,利用压缩原理和广义压缩原理来证明在某些假设条件下所讨论问题的解的存在性．