Asymptotic Expansions of Eigenelements of the Laplace Operator in a Domain with Many “Light” Concentrated Masses Closely Located on the Boundary. Multi-Dimensional Case

A spectral problem for the Laplace operator is considered in a domain with singular density near the boundary (with concentrated masses). The boundary condition is of rapidly variable type. We construct asymptotic expansions for the eigenvalues and eigenfunctions of the problem with respect to a small parameter characterizing the diameter and density of masses, as well as distance between the masses. The expansions are justified. Bibliography: 59 titles. Illustrations: 3 figures.__________Translated from Problemy Matematicheskogo Analiza, No. 30, 2005, pp. 87–119.     

On Stability Conditions for Stationary Solutions of Boundary-Value Problems with Free Boundary

In this paper, the free boundary problem for an elliptic equation is considered. The stability conditions for its stationary solutions are found. __________ Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 16, Partial Differential Equations, 2004.     

On Homogenization of Some Boundary-Value Problems in Domains with Thin Channels

Homogenization theorems are proved for some boundary-value problems for the Laplace operator and the biharmonic operator in plane domains consisting of two or more parts joined by thin strips of small length.__________Translated from Trudy Seminara imeni I. G. Petrovskogo, No. 24, pp. 324–340, 2004.     

Homogenization of Boundary Value Problems in Plane Domains with Frequently Alternating Type of Nonlinear Boundary Conditions: Critical Case

In the present paper we consider a boundary homogenization problem for the Poisson’s equation in a bounded domain and with a part of the boundary conditions of highly oscillating type (alternating between homogeneous Neumman condition and a nonlinear Robin type condition involving a small parameter). Our main goal in this paper is to investigate the asymptotic behavior as ε → 0 of the solution to such a problem in the case when the length of the boundary part, on which the Robin condition is specified, and the coefficient, contained in this condition, take so-called critical values. We show that in this case the character of the nonlinearity changes in the limit problem. The boundary homogenization problems were investigate for example in [1, 2, 4]. For the first time the effect of the nonlinearity character change via homogenization was noted for the first time in [5]. In that paper an effective model was constructed for the boundary value problem for the Poisson’s equation in the bounded domain that is perforated by the balls of critical radius, when the space dimension equals to 3. In the last decade a lot of works appeared, e.g., [6–10], in which this effect was studied for different geometries of perforated domains and for different differential operators. We note that in [6–10] only perforations by balls were considered. In papers [11, 12] the case of domains perforated by an arbitrary shape sets in the critical case was studied.     

On the Relation between Weak Solutions of Elliptic Dirichlet and Neumann Boundary-Value Problems for Plane Simply Connected Domains

Mathematical Notes - For any (possibly unbounded) simply connected domain Ω ? ?2 whose complement has nonempty interior, we establish an explicit relation between the solving...     

环形域上超线性边值问题的无穷多径向对称解

本文揭示了一类超线性边值问题在环域上有无限多的正对称解，此结果对非线性项的增长性除超线性外无其它限制．本文主要方法是相关常微分方程解的能量分析和相平面分析．     

On Boundary-Value Problems for the Laplacian in Bounded Domains with Micro Inhomogeneous Structure of the Boundaries

We consider boundary-value problems with rapidly alternating types of boundary conditions. We present the classification of homogenized （limit） problems depending on the ratio of small parameters, which characterize the diameter of parts of the boundary with different types of boundary conditions. Also we study the respective spectral problem of this type.     

On Solvability of Boundary-Value Problems for the Wave Equation with a Nonlinear Dissipation in Noncylindrical Domains

Solvability in noncylindrical domains is studied for some analogs of the first initial-boundary value problem for the wave equation with nonlinear increasing lower-order terms. Considered are the cases of domains that expand or contract with the growth of time. Alongside the existence theorems for regular solutions of boundary value problems some results are presented on the behavior of the energy norm of a solution as t+.     

The Minimax Estimation of Solutions of Boundary-Value Problems of Filtration in Multicomponent Schistous Media by Incomplete Data

We consider systems described by boundary-value problems for elliptic second-order partial differential equations with discontinuous coefficients appearing in the study of steady-state processes of filtration of a liquid in multicomponent media under nonhomogeneous conditions of a nonideal contact. Minimax estimates for functionals of solutions of these equations are found by using observations of states of the system. We assume that the right hand sides of equations, boundary conditions, and junction conditions on borders of media as well as errors in measurements are not known precisely, but we know only the sets to which they belong. We prove that the finding of minimax estimates can be reduced to the solving of some systems of integro-differential equations.     

Aggregate-Iterative Methods for the Approximation of Solutions of Boundary-Value Problems

We construct a special aggregate-iterative algorithm (a two-parameter method) for the iterative integration of a differential equation with two-point boundary conditions. We establish conditions for the convergence of this method and present partial cases of the two-parameter aggregate-iterative algorithm.     

On Coefficients of Eigenfunction Expansions for Boundary-Value Problems with Parameter in Boundary Conditions

In this paper, we consider boundary-value problems for an nth-order ordinary differential equation with polynomials of the spectral parameter in the equation and in the boundary conditions. For a wide class of such problems, we find coefficients of the eigen and associated function expansions for the Shkalikov linearizer.     

一类无穷区间上分数阶耦合系统边值问题解的存在性

讨论了在无穷区间上一类分数阶耦合系统边值问题解的存在性的问题.通过建立相应的相对紧的判定准则,构造相应的全连续算子,借助不动点理论及Leray-Schauder非线性抉择,得到解的存在性.     

Dirichlet Problems in Polyhedral Domains I: Regularity of the Solutions

The solution of the Dirichlet problem relative to an elliptic operator in a polyhedron has a complex singular behaviour near edges and vertices. Here we show that this solution and its conormal derivative have a global regularity in appropriate weighted Sobolev spaces. We also investigate some compact embeddings of these spaces. The present results will be applied in a forthcoming work to the constructive treatment of the problem by optimal convergent finite clement method and boundary element method.     

Homogenization of Elliptic Problems with Neumann Boundary Conditions in Non-smooth Domains

We consider a family of second-order elliptic operators {L_ε} in divergence form with rapidly oscillating and periodic coefficients in Lipschitz and convex domains in R~n. We are able to show that the uniform W~(1,p) estimate of second order elliptic systems holds for 2n/(n+1)-δ p 2n/(n-1)+ δ where δ 0 is independent of ε and the ranges are sharp for n = 2, 3. And for elliptic equations in Lipschitz domains, the W~(1,p) estimate is true for 3/2-δ p 3 + δ if n ≥ 4, similar estimate was extended to convex domains for 1 p ∞.     

Uniqueness of Solutions to Initial Boundary Value Problems for Parabolic Systems in Plane Bounded Domains with Nonsmooth Lateral Boundaries

Doklady Mathematics - We consider initial boundary value problems with boundary conditions of the first or second kind for one-dimensional (with respect to a spatial variable) Petrovskii parabolic...     

Boundary Properties of Solutions to Second-Order Parabolic Equations in Domains with Special Symmetry

We consider the first boundary-value problem for second-order nondivergent parabolic equations with, in general, discontinuous coefficients. We study the regularity of a boundary point assuming that in a neighborhood of this point the boundary of the domain is a surface of revolution. We prove a necessary and sufficient regularity condition in terms of parabolic capacities; for the heat equation this condition coincides with Wiener's criterion.