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1.
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. 相似文献
2.
We consider a parabolic semilinear problem with rapidly oscillating coefficients in a domain Ωε 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 σε(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. 相似文献
3.
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. 相似文献
4.
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 Ωε=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 ∂Ωε∩∂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. 相似文献
5.
G. A. Chechkin 《Journal of Mathematical Sciences》2006,135(6):3485-3521
We study the asymptotic behavior of eigenelements of boundary value problems in a domain Ω ⊂ ℝd, 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. 相似文献
6.
G. A. Chechkin 《Journal of Mathematical Sciences》2006,139(1):6351-6362
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. 相似文献
7.
Kazuhiro Takimoto 《Calculus of Variations and Partial Differential Equations》2006,26(3):357-377
We consider the boundary blowup problem for k-curvature equation, i.e., H
k
[u] = f(u) g(|Du|) in an n-dimensional domain Ω, with the boundary condition u(x) → ∞ as dist (x,∂Ω) → 0. We prove the existence result under some hypotheses. We also establish the asymptotic behavior of a solution near the boundary ∂Ω.
Mathematics Subject Classification (2000) 35J65, 35B40, 53C21 相似文献
8.
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. 相似文献
9.
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. 相似文献
10.
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. 相似文献
11.
V. A. Solonnikov 《Journal of Mathematical Sciences》1998,92(6):4364-4385
The main result is an asymptotic formula for a solution to the conjugation problem for the Navier-Stokes equations describing
the slow motion of two immiscible liquids such that one of them occupies a bounded domain Ω1 ⊂ ℝ3, whereas the other occupies the exterior domain Ω2=ℝ4∖Ω. Such a formula was obtained for a solution to the exterior problem with sticking conditions on the boundary in the works
of Fischer, Hsiao, and Wendland. The result obtained is applied to the proof of the solvability of a free-boundary problem
describing a uniform drop in an infinite liquid. Bibliography: 10 titles.
Translated fromProblemy Matematicheskogo Analiza, No. 16. 1997, pp. 208–238. 相似文献
12.
A. A. Amosov 《Journal of Mathematical Sciences》2011,176(3):361-408
We consider semidiscrete and asymptotic approximations to a solution to the nonstationary nonlinear initial-boundary-value
problem governing the radiative–conductive heat transfer in a periodic system consisting of n grey parallel plate heat shields
of width ε = 1/n, separated by vacuum interlayers. We study properties of special semidiscrete and homogenized problems whose solutions approximate
the solution to the problem under consideration. We establish the unique solvability of the problem and deduce a priori estimates
for the solutions. We obtain error estimates of order O( ?{e} ) O\left( {\sqrt {\varepsilon } } \right) and O(ε) for semidiscrete approximations and error estimates of order O( ?{e} ) O\left( {\sqrt {\varepsilon } } \right) and O(ε
3/4) for asymptotic approximations. Bibliography: 9 titles. 相似文献
13.
Stephen Dias Barreto 《Proceedings Mathematical Sciences》2000,110(4):347-356
We study a quantum spin glass as a quantum spin system with random interactions and establish the existence of a family of
evolution groups {τt(ω)}ω∈/Ω of the spin system. The notion of ergodicity of a measure preserving group of automorphisms of the probability space Ω, is
used to prove the almost sure independence of the Arveson spectrum Sp(τ(ω)) of τt(ε). As a consequence, for any family of (τ(ω),β) — KMS states {ρ(ω)}, the spectrum of the generator of the group of unitaries
which implement τ(ω) in the GNS representation is also almost surely independent of ω. 相似文献
14.
Andreas Fr?hlich 《Annali dell'Universita di Ferrara》2000,46(1):11-19
We consider weights of Muckenhoupt classA
q, 1<q<∞. For a bounded Lipschitz domain Ω⊂ℝn we prove a compact embedding and a Poincaré inequality in weighted Sobolev spaces. These technical tools allow us to solve
the weak Neumann problem for the Laplace equation in weighted spaces on ℝn, ℝn
+, on bounded and on exterior domains Ω with boundary of classC
1, which will yield the Helmholtz decomposition ofL
ω
q(Ω)n for general ω∈A
q. This is done by transferring the method of Simader and Sohr [4] to the weighted case. Our result generalizes a result of
Farwig and Sohr [2] where the Helmholtz decomposition ofL
ω
p(Ω)n is proved for an exterior domain and weights of Muckenhoupt class without singularities or degeneracies in a neighbourhood
of ϖΩ.
Sunto In questo lavoro consideriamo dei pesi della classe di MuckenhouptA q, 1<q<∞. Per un dominio limitato lipschitziano Ω⊂ℝn, dimostriamo una immersione compatta ed una disuguaglianza di Poincaré in spazi di Sobolev con peso. Questa tecnica ci consente di risolvere il problema debole di Neumann per l’equazione di Laplace in spazi pesati in ℝn, ℝn + in domini limitati ed in domini esterni con frontiera di classeC 1, che conduce alla decomposizione di Helmholtz diL ω q(Ω)n per un qualsiasi ω∈A q. Il risultato è ottenuto trasferendo il metodo di Simader e Sohr [4] al caso pesato. Quello qui presente estende un risultato di Farwig e Sohr [2] dove la decomposizione di Helmholtz diL ω q(Ω)n è dimostrata per domini esterni e pesi della classe di Muckenhoupt privi di singolarità in un intorno di ϖΩ.相似文献
15.
Let Ω be a domain with piecewise smooth boundary. In general, it is impossible to obtain a generalized solution u ∈ W
2
2
(Ω) of the equation Δ
x
2
u = f with the boundary conditions u = Δxu = 0 by solving iteratively a system of two Poisson equations under homogeneous Dirichlet conditions. Such a system is obtained
by setting v = −Δu. In the two-dimensional case, this fact is known as the Sapongyan paradox in the theory of simply supported
polygonal plates. In the present paper, the three-dimensional problem is investigated for a domain with a smooth edge Γ. If
the variable opening angle α ∈ C∞(Γ) is less than π everywhere on the edge, then the boundary-value problem for the biharmonic equation is equivalent to the
iterated Dirichlet problem, and its solution u inherits the positivity preserving property from these problems. In the case
α ∈ (π 2π), the procedure of solving the two Dirichlet problems must be modified by permitting infinite-dimensional kernel
and co-kernel of the operators and determining the solution u ∈ W
2
2
(Ω) by inverting a certain integral operator on the contour Γ. If α(s) ∈ (3π/2,2π) for a point s ∈ Γ, then there exists a
nonnegative function f ∈ L2(Ω) for which the solution u changes sign inside the domain Ω. In the case of crack (α = 2π everywhere on Γ), one needs to
introduce a special scale of weighted function spaces. In this case, the positivity preserving property fails. In some geometrical
situations, the problems on well-posedness for the boundary-value problem for the biharmonic equation and the positivity property
remain open. Bibliography: 46 titles.
__________
Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 336, 2006, pp. 153–198. 相似文献
16.
N. A. Shirokov 《Journal of Mathematical Sciences》1997,87(5):3925-3940
Denote by Kω(z, ζ) the Bergman kernel of a pseudoconvex domain Ω. For some classes of domains Ω, a relationship is found between the rate
of increase of Kω(z, z) as z tends to ∂Ω, and a purely geometric property of Ω. Bibliography: 5 titles.
Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 222, 1995, pp. 222–245. 相似文献
17.
Dian K. Palagachev 《Journal of Global Optimization》2008,40(1-3):305-318
We derive W
2,p
(Ω)-a priori estimates with arbitrary
p ∈(1, ∞), for the solutions of a degenerate oblique derivative problem for linear uniformly elliptic operators with low regular
coefficients. The boundary operator is given in terms of directional derivative with respect to a vector field ℓ that is tangent
to ∂Ω at the points of a non-empty set ε ⊂ ∂Ω and is of emergent type on ∂Ω.
相似文献
18.
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. 相似文献
19.
E. Yu. Romanenko 《Ukrainian Mathematical Journal》2005,57(11):1792-1808
Let {I, f, Z
+} be a dynamical system induced by a continuous mapping f of a closed bounded interval I into itself. To describe the dynamics of neighborhoods of points unstable under the mapping f, we propose the concept of the εω-set ω
f, ε(x) of a point x as the ω-limit set of the ε-neighborhood of the point x. We investigate the relationship between the εω-set and the domain of influence of a point. It is also shown that the domain
of influence of an unstable point is always a cycle of intervals. The results obtained can be directly used in the theory
of difference equations with continuous time and similar equations.
__________
Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 57, No. 11, pp. 1534–1547, November, 2005. 相似文献
20.
A. A. Arkhipova 《Journal of Mathematical Sciences》1996,80(6):2208-2225
The partial regularity up to the boundary of a domain is established for a solution u ∈ H1 (Ω) ∩ L∞ (Ω) to the boundary-value problem for a second-order elliptic system with strong nonlinearity in the case of dimension n≥3.
Bibliography: 12 titles.
Translated fromProblemy Matematicheskogo Analiza, No. 15, 1995, pp. 47–69. 相似文献