The asymptotics of the solution to the Neumann spectral problem in a domain of the “dense-comb” type

Convergence theorems and asymptotic estimates (as ε → 0) are proved for the eigenvalues and the eigenfunctions of the Neumann problem in a dense singular junction Ω _ɛ of a domain Ω₀ and a large number N of thin cylinders with thickness of order ε=lN⁻¹, where l is the total length of common boundaries for Ω₀ and the cylinders in question. Bibliography: 27 titles. We dedicate the present paper to Olga Arsenievna Oleinik, as a symbol of our deep respect and gratitude Translated from Trudy Seminara imeni I G. Petrovskogo, No. 19. pp. 000-000. 0000.     

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.     

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.     

Asymptotic Analysis of an Optimal Control Problem Involving a Thick Two-Level Junction with Alternate Type of Controls

We study the asymptotic behavior (as ε→0) of an optimal control problem in a plane thick two-level junction, which is the union of some domain and a large number 2N of thin rods with variable thickness of order e = O(N^-1).\varepsilon =\mathcal{O}(N^{-1}). The thin rods are divided into two levels depending on the geometrical characteristics and on the controls given on their bases. In addition, the thin rods from each level are ε-periodically alternated and the inhomogeneous perturbed Fourier boundary conditions are given on the lateral sides of the rods. Using the direct method of the calculus of variations and the Buttazzo-Dal Maso abstract scheme for variational convergence of constrained minimization problems, the asymptotic analysis of this problem for different kinds of controls is made as ε→0.     

On the spectrum of the Dirichlet Laplacian in a narrow strip

We consider the Dirichlet Laplacian Δ_∈ in a family of bounded domains {−a < x < b, 0 < y < εh(x)}. The main assumption is that x = 0 is the only point of global maximum of the positive, continuous function h(x). We find the two-term asymptotics in ε → 0 of the eigenvalues and the one-term asymptotics of the corresponding eigenfunctions. The asymptotic formulas obtained involve the eigenvalues and eigenfunctions of an auxiliary ODE on ℝ that depends on the behavior of h(x) as x → 0. The proof is based on a detailed study of the resolvent of the operator Δ_∈.     

The central connection problem at turning points of linear differential equations

A system of linear differential equations of the vectorial form εdy/dx=A (x, ε) y is considered, where ε is a positive parameter, and the matrixA (x, ε) is holomorphic in |x|⩽x ₀, 0 < ε ⩽ ε₀ , with an asymptotic expansionsA (x, ε) ∼ ∑ _r=0 ^∞ A _r (x) ε ^r , as ε→0. The eigenvalues ofA ₀(x) are supposed to coalesce atx=0 so as to make this point a simple turning point. With the help of refinements of the representations for the inner and outer asymptotic solutions, as ε→0, that were introduced in the articles [9] and [10] by the author (see the references at the end of the paper), explicit connection formulas between these solutions are calculated. As part of this derivation it is shown that only the diagonal entries of the connection matrix are asymptotically relevant.     

Asymptotic analysis of a spectral problem in a periodic thick junction of type 3:2:1

Convergence theorems and asymptotic estimates (as ϵ→0) are proved for eigenvalues and eigenfunctions of a mixed boundary value problem for the Laplace operator in a junction Ω_ϵ of a domain Ω₀ and a large number N² of ϵ‐periodically situated thin cylinders with thickness of order ϵ=O(N⁻¹). We construct an extension operator that is only asymptotically bounded in ϵ on the eigenfunctions in the Sobolev space H¹. An approach based on the asymptotic theory of elliptic problem in singularly perturbed domains is used. Copyright © 2000 John Wiley & Sons, Ltd.     

Boundary Asymptotic Analysis for an Incompressible Viscous Flow: Navier Wall Laws

We consider a new way of establishing Navier wall laws. Considering a bounded domain Ω of R ^N , N=2,3, surrounded by a thin layer Σ _ε , along a part Γ₂ of its boundary ∂Ω, we consider a Navier-Stokes flow in Ω∪∂Ω∪Σ _ε with Reynolds’ number of order 1/ε in Σ _ε . Using Γ-convergence arguments, we describe the asymptotic behaviour of the solution of this problem and get a general Navier law involving a matrix of Borel measures having the same support contained in the interface Γ₂. We then consider two special cases where we characterize this matrix of measures. As a further application, we consider an optimal control problem within this context.     

Asymptotic analysis of a parabolic semilinear problem with nonlinear boundary multiphase interactions in a perforated domain

We consider a parabolic semilinear problem with rapidly oscillating coefficients in a domain Ω_ε that is ε-periodically perforated by small holes of size O\mathcal {O}(ε). The holes are divided into two ε-periodical sets depending on the boundary interaction at their surfaces, and two different nonlinear Robin boundary conditions σ_ε(u _ε) + εκ _m (u _ε) = εg ^(m) _ε, m = 1, 2, are imposed on the boundaries of holes. We study the asymptotics as ε → 0 and establish a convergence theorem without using extension operators. An asymptotic approximation of the solution and the corresponding error estimate are also obtained. Bibliography: 60 titles. Illustrations: 1 figure.     

Homogenization of eigenvalue problems in perforated domains

In this paper, we treat some eigenvalue problems in periodically perforated domains and study the asymptotic behaviour of the eigenvalues and the eigenvectors when the number of holes in the domain increases to infinity Using the method of asymptotic expansion, we give explicit formula for the homogenized coefficients and expansion for eigenvalues and eigenvectors. If we denote by ε the size of each hole in the domain, then we obtain the following aysmptotic expansion for the eigenvalues: Dirichlet: λ_ε = ε⁻² λ + λ₀ +O (ε), Stekloff: λ_ε = ελ₁ +O (ε²), Neumann: λ_ε = λ₀ + ελ₁ +O (ε²). Using the method of energy, we prove a theorem of convergence in each case considered here. We briefly study correctors in the case of Neumann eigenvalue problem.     

Homogenization of a Singularly Perturbed Parabolic Problem in a Thick Periodic Junction of the Type 3:2:1

We prove a convergence theorem and obtain asymptotic (as 0) estimates for a solution of a parabolic initial boundary-value problem in a junction that consists of a domain ₀ and a large number N ² of -periodically located thin cylinders whose thickness is of order = O(N ^–1).     

Perturbed boundary eigenvalue problem for the Schrödinger operator on an interval

A perturbed two-parameter boundary value problem is considered for a second-order differential operator on an interval with Dirichlet conditions. The perturbation is described by the potential μ⁻¹ V((x − x ₀)ɛ⁻¹), where 0 < ɛ ≪ 1 and μ is an arbitrary parameter such that there exists δ > 0 for which ɛ/μ = o(ɛ^δ). It is shown that the eigenvalues of this operator converge, as ɛ → 0, to the eigenvalues of the operator with no potential. Complete asymptotic expansions of the eigenvalues and eigenfunctions of the perturbed operator are constructed.     

Bounds on Supremum Norms for Hecke Eigenfunctions of Quantized Cat Maps

We study extreme values of desymmetrized eigenfunctions (so called Hecke eigenfunctions) for the quantized cat map, a quantization of a hyperbolic linear map of the torus. In a previous paper it was shown that for prime values of the inverse Planck’s constant N = 1/h, such that the map is diagonalizable (but not upper triangular) modulo N, the Hecke eigenfunctions are uniformly bounded. The purpose of this paper is to show that the same holds for any prime N provided that the map is not upper triangular modulo N. We also find that the supremum norms of Hecke eigenfunctions are ≪_ε N^ε for all ε > 0 in the case of N square free. Submitted: March 6, 2006; Accepted: April 30, 2006     

On a biharmonic equation involving nearly critical exponent

This paper is concerned with a biharmonic equation under the Navier boundary condition , u > 0 in Ω and u = Δu = 0 on ∂Ω, where Ω is a smooth bounded domain in , n ≥ 5, and ε > 0. We study the asymptotic behavior of solutions of (P _−ε) which are minimizing for the Sobolev quotient as ε goes to zero. We show that such solutions concentrate around a point x ₀ ∈Ω as ε → 0, moreover x ₀ is a critical point of the Robin’s function. Conversely, we show that for any nondegenerate critical point x ₀ of the Robin’s function, there exist solutions of (P _−ε) concentrating around x ₀ as ε → 0. Finally we prove that, in contrast with what happened in the subcritical equation (P _−ε), the supercritical problem (P _+ε) has no solutions which concentrate around a point of Ω as ε → 0. Work finished when the authors were visiting Mathematics Department of the University of Roma “La Sapienza”. They would like to thank the Mathematics Department for its warm hospitality. The authors also thank Professors Massimo Grossi and Filomena Pacella for their constant support.     

Asymptotic analysis of a boundary-value problem with nonlinear multiphase boundary interactions in a perforated domain

We consider a boundary-value problem for the second-order elliptic differential operator with rapidly oscillating coefficients in a domain Ω _ε that is ε-periodically perforated by small holes. The holes are split into two ε-periodic sets depending on the boundary interaction via their boundary surfaces. Therefore, two different nonlinear boundary conditions σ _ε (u _ε ) + εκ _m (u _ε ) = εg _ε ^(m) , m = 1, 2, are given on the corresponding boundaries of the small holes. The asymptotic analysis of this problem is performed as ε → 0, namely, the convergence theorem for both the solution and the energy integral is proved without using an extension operator, asymptotic approximations for the solution and the energy integral are constructed, and the corresponding approximation error estimates are obtained.     

On homogenization of the mixed boundary-value problem for the heat equation in a domain whose part contains channels arranged in a periodic way

In this paper, we study the asymptotic behavior of the solutionsu _ε (ε is a small parameter) of boundaryvalue problems for the heat equation in the domain Ω_ε=Ω⁻∪Ω _ε ⁺ ∪γ one part of which (Ω _ε ⁺ ) contains ε-periodically situated channels with diameters of order ε and the other part of which (Ω⁺) is a homogeneous medium; γ=∂Ω _ε ⁺ ∩∂Ω⁺. On the boundary of the channels the Neumann boundary condition is posed, and on ∂Ω_ε∩∂Ω the Dirichlet boundary condition is prescribed. The homogenized problem is the Dirichlet problem in Ω with the transmission condition on γ. The estimates for the difference betweenu _ε and the solution of the homogenized problem are obtained. Bibliography: 14 titles. Translated from Trudy Seminara imeni I. G. Petrovskogo, No. 20, pp. 27–47, 1997.     

Existence of Solutions to a Singular Elliptic Equation

We study the equation ${{-{\Delta}u = (-\frac{1}{u^{\beta}}+\lambda{u}^{p})\chi\{u >0 }\}}${{-{\Delta}u = (-\frac{1}{u^{\beta}}+\lambda{u}^{p})\chi\{u >0 }\}} in Ω with Dirichlet boundary condition, where 0 < p < 1 and 0 < β < 1. We regularize the term 1/u ^β near u ~ 0 by using a function g _ε (u) which pointwisely tends to 1/u ^β as ε → 0. When the parameter λ > 0 is large enough, the corresponding energy functional has critical points u _ε . Letting ε → 0, then u _ε converges to a solution of the original problem, which is nontrivial, nonnegative and vanishes at some portion of Ω. There are two nontrivial solutions.     

Estimates for the Second Order Derivatives of Eigenvectors in Thin Anisotropic Plates with Variable Thickness

For the second order derivatives of eigenvectors in a thin anisotropic heterogeneous plate Ω_h, we derive estimates of their weighted L₂-norms with majorants whose dependence on the plate thickness h and on the eigenvalue number is expressed explicitly. These estimates maintain the asymptotic sharpness throughout the entire spectrum, whereas inside its low-frequency band the majorants remain bounded as h → +0. The latter is a rather unexpected fact, because for the first eigenfunction u¹ of a similar boundary-value problem for a scalar second order differential operator with variable coefficients, the norm ‖∇ _x ² u⁰; L₂(Ω_h)‖ is of order h⁻¹ and grows as h tends to zero. Bibliography: 35 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 308, 2004, pp. 161–180.     

A capacitary method for the asymptotic analysis of dirichlet problems for monotone operators

Given a non-linear elliptic equation of monotone type in a bounded open set Ω ⊂ Rⁿ, we prove that the asymptotic behaviour, asj → ∞, of the solutions of the Dirichlet problems corresponding to a sequence (Ω_j) of open sets contained in Ω is uniquely determined by the asymptotic behaviour, asj → ∞, of suitable non-linear capacities of the setsKΩ _j, whereK runs in the family of all compact subsets of Ω.     

The asymptotic behavior of a solution to the boundary-value problem in a degenerate domain with rapidly oscillating boundary

We construct and justify the asymptotics (as ε → +0) of a solution of the mixed boundary-value problem for the Poisson equation in the domain obtained by joining two sets Ω₊ and Ω_- by a large number of thin (of width O (ε)) curvilinear strips (a hub and a rim with a large number of spokes). As a resulting limit problem describing the principal terms of exterior expansions (in Ω_± and in the set ω occupied by the strips) we take the problem of conjugating the partial differential equations and an ordinary differential equation depending on a parameter. Bibliography: 16 titles; Illustrations: 1 figure. Translated fromProblemy Matematicheskogo Analiza, No. 14, 1995, pp. 63–90.