On the first eigenpair of singularly perturbed operators with oscillating coefficients

The paper deals with a Dirichlet spectral problem for a singularly perturbed second order elliptic operator with rapidly oscillating locally periodic coefficients. We study the limit behavior of the first eigenpair (ground state) of this problem. The main tool in deriving the limit \mbox(effective) problem is the viscosity solutions technique for Hamilton-Jacobi equations. The effective problem need not have a unique solution. We study the non-uniqueness issue in a particular case of zero potential and construct the higher order term of the ground state asymptotics.     

一类具有无穷边界值的二次奇摄动边值问题

研究一类具有无穷边界值的二次奇摄动Robin边值问题解的存在性与解的渐进行为,重点关注边界值的奇异程度对解的边界层行为的影响;同时将所得的结果与Chang及Howes的结果(带正常边界值)进行比较.研究表明:(1)当边界值大小为O(1/)时,得到的边界层大小为O( ln ),这比Chang及Howes带正常边界值的情形提高了O(ln )量级;(2)增大边界值的奇性至O(1/ r),这里r >1,边界层大小的量级不变,依然为O( ln );(3)若要使得边界层大小为O(1),则边界值的大小需为O(e?1/).最后给出一个算例验证得到的结果.     

Asymptotics of eigenvalues of the Dirichlet boundary value problem for the Lame operator in a three-dimensional domain with a small cavity

A boundary value problem for the Lame operator in a bounded three-dimensional domain with a small cavity is studied. The domain is filled with an elastic homogeneous isotropic medium that is clamped at the boundary, which corresponds to the Dirichlet boundary condition. The leading term of an asymptotic expansion for the eigenvalue is constructed in the case of the Dirichlet limit problem. The asymptotic expansion is constructed in powers of a small parameter ? that is the diameter of the cavity.     

The Cauchy problem for a singularly perturbed Volterra integro-differential equation

The Cauchy problem for a singularly perturbed Volterra integro-differential equation is examined. Two cases are considered: (1) the reduced equation has an isolated solution, and (2) the reduced equation has intersecting solutions (the so-called case of exchange of stabilities). An asymptotic expansion of the solution to the Cauchy problem is constructed by the method of boundary functions. The results are justified by using the asymptotic method of differential inequalities, which is extended to a new class of problems.     

Uniform grid approximation of nonsmooth solutions to the mixed boundary value problem for a singularly perturbed reaction-diffusion equation in a rectangle

A mixed boundary value problem for a singularly perturbed reaction-diffusion equation in a square is considered. A Neumann condition is specified on one side of the square, and a Dirichlet condition is set on the other three. It is assumed that the coefficient of the equation, its right-hand side, and the boundary values of the desired solution or its normal derivative on the sides of the square are smooth enough to ensure the required smoothness of the solution in a closed domain outside the neighborhoods of the corner points. No compatibility conditions are assumed to hold at the corner points. Under these assumptions, the desired solution in the entire closed domain is of limited smoothness: it belongs only to the Hölder class C ^μ, where μ ∈ (0, 1) is arbitrary. In the domain, a nonuniform rectangular mesh is introduced that is refined in the boundary domain and depends on a small parameter. The numerical solution to the problem is based on the classical five-point approximation of the equation and a four-point approximation of the Neumann boundary condition. A mesh refinement rule is described under which the approximate solution converges to the exact one uniformly with respect to the small parameter in the L _∞ ^h norm. The convergence rate is O(N ^?2ln² N), where N is the number of mesh nodes in each coordinate direction. The parameter-uniform convergence of difference schemes for mixed problems without compatibility conditions at corner points was not previously analyzed.     

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.     

On the principal eigenvalue of a Robin problem with a large parameter

We study the asymptotic behaviour of the principal eigenvalue of a Robin (or generalised Neumann) problem with a large parameter in the boundary condition for the Laplacian in a piecewise smooth domain. We show that the leading asymptotic term depends only on the singularities of the boundary of the domain, and give either explicit expressions or two‐sided estimates for this term in a variety of situations. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

The Cauchy problem for a singularly perturbed integro-differential Fredholm equation

An initial problem is considered for an ordinary singularly perturbed integro-differential equation with a nonlinear integral Fredholm operator. The case when the reduced equation has a smooth solution is investigated, and the solution to the reduced equation with a corner point is analyzed. The asymptotics of the solution to the Cauchy problem is constructed by the method of boundary functions. The asymptotics is validated by the asymptotic method of differential inequalities developed for a new class of problems.     

Approximation and eigenvalue extrapolation of biharmonic eigenvalue problem by nonconforming finite element methods

In this paper, we analyze the biharmonic eigenvalue problem by two nonconforming finite elements, Q and E Q. We obtain full order convergence rate of the eigenvalue approximations for the biharmonic eigenvalue problem based on asymptotic error expansions for these two nonconforming finite elements. Using the technique of eigenvalue error expansion, the technique of integral identities, and the extrapolation method, we can improve the accuracy of the eigenvalue approximations. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2008     

On the stability and domain of attraction of asymptotically nonsmooth stationary solutions to a singularly perturbed parabolic equation

A stationary solution to the singularly perturbed parabolic equation ?u _t + ε² u _xx ? f(u, x) = 0 with Neumann boundary conditions is considered. The limit of the solution as ε → 0 is a nonsmooth solution to the reduced equation f(u, x) = 0 that is composed of two intersecting roots of this equation. It is proved that the stationary solution is asymptotically stable, and its global domain of attraction is found.     

On a two-phase Stefan problem with convective boundary condition including a density jump at the free boundary

We consider a two-phase Stefan problem for a semi-infinite body with a convective boundary condition including a density jump at the free boundary with a time-dependent heat transfer coefficient of the type $h/\sqrt{t}$, whose solution was given in D. A. Tarzia, PAMM. Proc. Appl. Math. Mech. 7, 1040307–1040308 (2007). We demonstrate that the solution to this problem converges to the solution to the analogous one with a temperature boundary condition when the heat transfer coefficient $h\to +\infty$. Moreover, we analyze the dependence of the free boundary respecting to the jump density.     

Asymptotics of simple eigenvalues and eigenfunctions for the Laplace operator in a domain with an oscillating boundary

The asymptotic behavior of solutions to spectral problems for the Laplace operator in a domain with a rapidly oscillating boundary is analyzed. The leading terms of the asymptotic expansions for eigenelements are constructed, and the asymptotics are substantiated for simple eigenvalues. The text was submitted by the authors in English.     

Numerical approximation of two‐dimensional convection‐diffusion equations with boundary layers

Our aim in this article is to show how one can improve the numerical solution of singularity perturbed problems involving boundary layers. Incorporating the structures of boundary layers into finite element spaces can improve the accuracy of approximate solutions and result in significant simplifications. In this article we discuss convection‐diffusion equations in the two‐dimensional space with a homogeneous Dirichlet boundary condition and a mixed boundary condition. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2005     

Two-point boundary value problem for a singularly perturbed equation with a reduced equation having multiple roots

Contrast steplike structures for a singularly perturbed equation are considered in the case when the corresponding reduced equation has multiple roots.     

On Limit Results for a Class of Singularly Perturbed Switching Diffusions

This work is devoted to the weak convergence analysis of a class of aggregated processes resulting from singularly perturbed switching diffusions with fast and slow motions. The processes consist of diffusion components and pure jump components. The states of the pure jump component are naturally divisible into a number of classes. Aggregate the states in each weakly irreducible class by a single state leading to an aggregated process. Under suitable conditions, it is shown that the aggregated process converges weakly to a switching diffusion process whose generator is an average with respect to the quasi-stationary distribution of the jump process.     

ASYMPTOTIC BEHAVIOR FOR A CLASS OF ELLIPTIC EQUIVALUED SURFACE BOUNDARY VALUE PROBLEM WITH DISCONTINUOUS INTERFACE CONDITIONS

Spontaneous potential well-logging is one of the important techniques in petroleum exploitation. A spontaneous potential satisfies an elliptic equivalued surface boundary value problem with diseontinuous interface conditlons. In practice, the measuring electrode is so small that we can simplify the corresponding equivalued surface to a point. In this paper, we give a positive answer to this approximation process: when the equivalued surface shrinks to a point, the solution of the original equivalued surface boundary value problem converges to the solution of the corresponding limit boundary value problem.     

On the uniform convergence of the Fourier series for one spectral problem with a spectral parameter in a boundary condition

In this paper, we investigate the uniform convergence of the Fourier series expansions in terms of eigenfunctions for the spectral problem where λ is a spectral parameter, q(x) is a real‐valued continuous function on the interval [0,1], and a₁,b₀,b₁,c₁,d₀, and d₁ are real constants that satisfy the conditions Copyright © 2015 John Wiley & Sons, Ltd.     

On the index of the boundary value problem for a homogeneous Hamiltonian system of differential equations

The index of the homogeneous self-adjoint boundary value problem for the Hamiltonian systems of ordinary differential equations is introduced. It is assumed that the system has a nontrivial solution. The relationship between the index of an eigenvalue of the nonlinear eigenvalue problem and the index of the corresponding homogeneous problem is established. Properties of the index of the problem and those of the eigenvalue are examined.     

Characteristic frequencies of bodies with thin spikes III. Frequency splitting

The eigenvalue problem for the Laplace operator with the Neumann boundary conditions in a domain that has a thin spike of finite length is considered for the case in which the limit value is an eigenvalue both for the main body and the spike. The method of matched asymptotic expansions is used to construct total asymptotics of the eigenvalues of the perturbed problem and obtain closed formulas for the leading asymptotic terms. Translated fromMatematicheskie Zametki, Vol. 61, No. 4, pp. 494–502, April, 1997. Translated by V. E. Nazaikinskii