On the asymptotics and stability of the point spectrum of a waveguide with thin shielding obstacle

Some explicit conditions are found for the existence and absence of an eigenvalue in the interval (0, π ²) of the continuous spectrum of the Neumann problem for the Laplace operator in the unit strip with a thin (of width O(ε)) symmetric screen, which, as ε → +0, shrinks into a line segment perpendicular to the sides of the strip. An asymptotics of this eigenvalue is constructed, as well as the asymptotics of the reflection coefficient, which describes Wood’s anomalies, namely, quick changes of the diffraction pattern near a frequency threshold in the continuous spectrum. Bibliography: 32 titles.     

Homogenization with corrector for a multidimensional periodic elliptic operator near an edge of an inner gap

For a second order differential operator A \mathcal{A} _ε  = −div g(x/ε)∇ + ε ⁻²p(x/ε) in L ₂(ℝ ^d ) with periodic coefficients and small parameter ε > 0 we study an approximation of the resolvent of A \mathcal{A} _ε at a point close to an edge of an inner gap in the spectrum of A \mathcal{A} _ε . Under certain regularity conditions, we construct an approximation (with a first order corrector taken into account) for the resolvent with error estimate of order O(ε ²). We show that a proper effective operator and a proper corrector are associated to each (regular) edge of the gap. Bibliography: 14 titles.     

On the structure of the solution of singularly perturbed initial boundary value problems with an unbounded spectrum of the limit operator

Singularly perturbed initial boundary value problems are studied for some classes of linear systems of ordinary differential equations on the semiaxis with an unbounded spectrum of the limit operator. We give a new version of the proof of the existence of a unique and bounded (as ε→+0) solution for which with the help of the splitting method we construct a uniform asymptotic expansion on the entire semiaxis and describe all singularities (reflecting the structure of the corresponding boundary layers) in closed analytic form, including the critical case in which the points of the spectrum of the limit operator can touch the imaginary axis; this supplements previous results. Translated fromMatematicheskie Zametki, Vol. 65, No. 6, pp. 831–835, June, 1999.     

Approximations of the exponential of an operator with periodic coefficients

We consider the operator exponential e ^−tA , t > 0, where A is a selfadjoint positive definite operator corresponding to the diffusion equation in \mathbbRⁿ {\mathbb{R}^n} with measurable 1-periodic coefficients, and approximate it in the operator norm ||   ·   ||_{L²( \mathbbRⁿ ) ? L²( \mathbbRⁿ )} {\left\| {\; \cdot \;} \right\|_{{{L^2}\left( {{\mathbb{R}^n}} \right) \to {L^2}\left( {{\mathbb{R}^n}} \right)}}} with order O( t ^{- \fracm2} ) O\left( {{t^{{ - \frac{m}{2}}}}} \right) as t → ∞, where m is an arbitrary natural number. To construct approximations we use the homogenized parabolic equation with constant coefficients, the order of which depends on m and is greater than 2 if m > 2. We also use a collection of 1-periodic functions N _α (x), x ? \mathbbRⁿ x \in {\mathbb{R}^n} , with multi-indices α of length | a| \leqslant m \left| \alpha \right| \leqslant m , that are solutions to certain elliptic problems on the periodicity cell. These results are used to homogenize the diffusion equation with ε-periodic coefficients, where ε is a small parameter. In particular, under minimal regularity conditions, we construct approximations of order O(ε ^m ) in the L ²-norm as ε → 0. Bibliography: 14 titles.     

Stefan problem

We prove the existence of a global classical solution of the multidimensional two-phase Stefan problem. The problem is reduced to a quasilinear parabolic equation with discontinuous coefficients in a fixed domain. With the help of a small parameter ε, we smooth coefficients and investigate the resulting approximate solution. An analytical method that enables one to obtain the uniform estimates of an approximate solution in the cross-sections t = const is developed. Given the uniform estimates, we make the limiting transition as ε → 0. The limit of the approximate solution is a classical solution of the Stefan problem, and the free boundary is a surface of the class H ^2+α,1+α/2.     

Random evolutions with locally independent increments on increasing time intervals

Three main schemes of limit theorems for random evolutions are discussed: averaging, diffusion approximation, and the asymptotics of large deviations. Markov stochastic evolutions with locally independent increments on increasing time intervals T _ε  = t/ε → ∞, ε → 0, are considered. The asymptotic behavior of random evolutions is investigated with the use of solutions of the singular perturbation problems for reducibly invertible operators.     

Homogenization of a parabolic signorini boundary value problem in a thick plane junction

We consider a parabolic Signorini boundary value problem in a thick plane junction Ω _ε which is the union of a domain Ω₀ and a large number of ε−periodically situated thin rods. The Signorini conditions are given on the vertical sides of the thin rods. The asymptotic analysis of this problem is done as ε → 0, i.e., when the number of the thin rods infinitely increases and their thickness tends to zero. With the help of the integral identity method we prove a convergence theorem and show that the Signorini conditions are transformed (as ε → 0) in differential inequalities in the region that is filled up by the thin rods in the limit passage. Bibliography: 31 titles. Illustrations: 1 figure.     

Homogenization of solutions to problems for the Laplace operator in unbounded domains with many concentrated masses on the boundary

A problem for the Laplace operator is considered in a three-dimensional unbounded domain with singular density. The density, depending on a small positive parameter ε, is equal to 1 outside small inclusions, and is equal to (δε)^−m in these inclusions. These domains, concentrated masses of diameter εδ, are located along the plane part of the boundary at the distance of order O(δ), where δ = δ(ε). The Dirichlet condition is imposed on the boundary parts tangent to the concentrated masses. We construct the limit (averaged) operator and study the asymptotic behavior of solutions to the original problem with m < 1. __________ Translated from Problemy Matematicheskogo Analiza, No. 33, 2006, pp. 103–111.     

The order of convergence in the stefan problem with vanishing specific heat

The paper is concerned with the two-phase Stefan problem with a small parameter ϵ, which coresponds to the specific heat of the material. It is assumed that the initial condition does not coincide with the solution for t = 0 of the limit problem related to ε = 0. To remove this discrepancy, an auxiliary boundary layer type function is introduced. It is proved that the solution to the two-phase Stefan problem with parameter ϵ differs from the sum of the solution to the limit Hele–Shaw problem and a boundary layer type function by quantities of order O(ϵ). The estimates are obtained in H?lder norms. Bibliography: 13 titles.     

On eigenfunctions localized in a neighborhood of the lateral surface of a thin domain

We study the spectrum of the boundary-value problem for the Laplace operator in a thin domain Ω(ε) obtained by small perturbation of the cylinder Ω(ε)=ω×(-ε/2.ε/2) ⊂ ℝ³in a neighborhood of the lateral surface. The Dirichlet condition is imposed on the bases of the cylinder, and the Dirichlet condition or the Neumann condition is imposed on the remaining part of ∂Ω(ε). We construct and justify asymptotic formulas (as ε→+0) for eigenvalues and eigenfunctions. In view of a special form of the lateral surface, there are eigenfunctions of boundary-layer type that exponentially decrease far from the lateral surface. For the mixed boundary-value problem such a localization is possible in neighborhoods of local maxima of the curvature of the contour ∂ω. This property of eigenfunctions is a characteristic feature of the first points of the spectrum (in particular, the first eigenvalue) and, under the passage from Ω(h)₍₎ to Ω(h), the spectrum itself has perturbation O(h⁻²). Bibliography: 29 titles. Translated fromProblemy Matematicheskogo Analiza, No. 19, 1999, pp. 105–149.     

Stopping games with randomized strategies

We study stopping games in the setup of Neveu. We prove the existence of a uniform value (in a sense defined below), by allowing the players to use randomized strategies. In constrast with previous work, we make no comparison assumption on the payoff processes. Moreover, we prove that the value is the limit of discounted values, and we construct ε-optimal strategies. Received: 10 May 1999 / Revised version: 18 May 2000 / Published online: 15 February 2001     

Homogenization of the diffusion equation with nonlinear flux condition on the interior boundary of a perforated domain — the influence of the scaling on the nonlinearity in the effective sink-source term

In this paper, we study the asymptotic behavior of solutions u _ε of the initial boundary value problem for parabolic equations in domains W_e ì \mathbbRⁿ {\Omega_\varepsilon } \subset {\mathbb{R}^n} , n ≥ 3, perforated periodically by balls with radius of critical size ε ^α , α = n/(n − 2), and distributed with period ε. On the boundary of the balls a nonlinear third boundary condition is imposed. The weak convergence of the solutions u _ε to the solution of an effective equation is given. Furthermore, an improved approximation for the gradient of the microscopic solutions is constructed, and a corrector result with respect to the energy norm is proved.     

Homogenization of a model spectral problem for the Laplace operator in a domain with many closely located “ heavy” and “intermediate heavy” concentrated masses

We study the asymptotic behavior of eigenelements of boundary value problems in a domain Ω ⊂ ℝ^d, d ⩾ 3, with rapidly alternating type of boundary conditions. The density is equal to 1 outside tiny domains and is equal to ε^−m inside them, where ε is a small parameter. These domains (concentrated masses) of diameter εa are located on the boundary at a positive distance of order O(ε) from each other, where a = const. The Dirichlet boundary condition is on parts of ∂Ω that are tangent to concentrated masses, and the Neumann boundary condition is stated outside concentrated masses. We construct the limit (homogenized) operator, prove the convergence of eigenelements of the original problem to the eigenelements of the limit (homogenized) problem in the case m ⩾ 2, and estimate the difference between the eigenelements. Bibliography: 79 titles. Illustrations: 4 figures. __________ Translated from Problemy Matematicheskogo Analiza, No. 32, 2006, pp. 45–75.     

Limit behavior of thin heterogeneous domain with rapidly oscillating boundary

Abstract We give a generalization of the work presented in [6] where the asymptotic behaviour, as ε→0, of a monotone nonlinear problem in a bounded multidomain of R^N depending on ε was addressed. We extend the previous results to the case where the nonlinear operator depends both on the slow and rapid variable and we prove that, due to the presence of the rapid variable, the algebraic equation contained in the limit problem obtained in [6] must be replaced by a partial differential equation with respect to the microscopic variable y′. Keywords: Homogenization, Dimension reduction, Multidomain, Rapid variable, Limit problem Mathematics Subject Classification (2000): 35B27, 35J60     

Singular perturbations of some nonlinear problems

We deal with singular perturbations of nonlinear problems depending on a small parameter ε > 0. First we consider the abstract theory of singular perturbations of variational inequalities involving some nonlinear operators, defined in Banach spaces, and describe the asymptotic behavior of these solutions as ε → 0. Then these abstract results are applied to some boundary value problems. Bibliography: 15 titles.     

On the negative Pell equation

Let ε be the fundamental unit of a field Q( ?d ) Q\left( {\sqrt {d} } \right) . In the paper, it is proved that ε > d ^3/2/ log² d for almost all d such that N(ε) = −1. Bibliography: 6 titles.     

Two-phase Stefan problem with vanishing specific heat

The unique solvability of the two-phase Stefan problem with a small parameter ε ∈ [0; ε ₀] at the time derivative in the heat equations is proved. The solution is obtained on a certain time interval [0; t ₀] independent of ε. The solution of the Stefan problem is compared with the solution to the Hele–Shaw problem corresponding to the case ε = 0. The solutions of both problems are not assumed to coincide at the initial moment of time. Bibliography: 18 titles. Dedicated to Vsevolod Alekseevich Solonnikov on the occasion of his jubilee Published in Zapiski Nauchnykh Seminarov POMI, Vol. 362, 2008, pp. 337–363.     

Analytical and numerical investigation of the spectra of three-point difference operators

The properties of the spectra of discrete spatially one-dimensional problems of convection — diffusion type with constant coefficients and nonstandard boundary conditions are examined in the framework of stability of explicit algorithms for time-dependent problems of mathematical physics. An analytical method is proposed for finding isolated limit points of the operator spectrum. Limit points are determined for the difference transport equation with different versions of nonreflecting boundary conditions and for an approximation of the heat conduction equation on a grid with condensation near the boundary. Stability and other properties of the spectrum are also established numerically. __________ Translated from Prikladnaya Matematika i Informatika, No. 27, pp. 25–45, 2007.     

A singularly perturbed spectral problem for a biharmonic operator with Neumann conditions

We study a mathematical model of a composite plate that consists of two components with similar elastic properties but different distributions of density. The area of the domain occupied by one of the components is infinitely small as ε → 0. We investigate the asymptotic behavior of the eigenvalues and eigenfunctions of the boundary-value problem for a biharmonic operator with Neumann conditions as ε → 0. We describe four different cases of the limiting behavior of the spectrum, depending on the ratio of densities of the medium components. In particular, we describe the so-called Sanches-Palensia effect of local vibrations: A vibrating system has a countable series of proper frequencies infinitely small as ε → 0 and associated with natural forms of vibrations localized in the domain of perturbation of density. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 51, No. 11, pp. 1467–1475, November, 1999.     

Calculation of the SDE-contribution of the dissipation operator to the energy spectrum of developed turbulence

The nonanalytic correction to the energy spectrum of the developed turbulence in the one-loop approximation of ε-expansion is calculated. It has the form y⁴ ln u (u≡kL, L is the external scale of the turbulence), which is in agreement with Wilson's short distance expansion prediction. The amplitude of the contribution of the dissipation operator to the energy spectrum is determined. Bibliography: 7 titles. Dedicated to the memory of V. N. Popov Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 224, 1995, pp. 36–42. Translated by N. Yu. Netsvetaev.