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1.
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. 相似文献
2.
In this paper, we study the asymptotic behavior of solutions u
ε
of the initial boundary value problem for parabolic equations in domains
We 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. 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.
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
5.
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
6.
T. A. Mel’nyk 《Ukrainian Mathematical Journal》2006,58(2):220-243
A spectral boundary-value problem is considered in a plane thick two-level junction Ωε formed as the union of a domain Ω0 and a large number 2N of thin rods with thickness of order ε = O(N
−1). The thin rods are split into two levels depending on their length. In addition, the thin rods from the indicated levels
are ε-periodically alternating. The Fourier conditions are given on the lateral boundaries of the thin rods. The asymptotic
behavior of the eigenvalues and eigenfunctions is investigated as ε → 0, i.e., when the number of thin rods infinitely increases
and their thickness approaches zero. The Hausdorff convergence of the spectrum is proved as ε → 0, the leading terms of asymptotics
are constructed, and the corresponding asymptotic estimates are justified for the eigenvalues and eigenfunctions.
Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 2, pp. 195–216, February, 2006. 相似文献
7.
Summary This paper is concerned with second order differential systems involving two parameters with boundary conditions specified
at three points. In particular, we consider the system y' = k(x, λ, μ)z, z' = -g(x, λ, μ)y, where k and g are real-valued
junctions defined on X: a ≤ x ≤ c, L: L1 < λ < L2, and M: M1 < μ < M2. This system is studied together with the boundary conditions α(λ, μ)y(a) - β(λ, μ)z(a)=0, γ(λ, μ)y(b) - δ(λ, μ)z(b)=0, ε1(μ)y(b) - φ1(μ)z(b)=ε2(μ)y(c) - φ2(μ)z(c), where α, β, δ, γ, εi, φi, i=1, 2, are continuous functions of the parameters. This work establishes the existence of eigenvalue pairs for the boundary
problem and the oscillatory behavior of the associated solutions. These results complement those previously obtained by the
authors and B. D. Sleeman, where boundary conditions of the ? Sturm-Liouville ? type were studied.
Entrata in Redazione il 5 dicembre 1977.
The research for this paper was supported by a University College Reasearch Grant, University of Alabama in Birmingham. 相似文献
8.
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
9.
The asymptotic behavior of solutions to boundary value problems for the Poisson equation is studied in a thick two-level junction
of type 3:2:2 with alternating boundary conditions. The thick junction consists of a cylinder with ε-periodically stringed
thin disks of variable thickness. The disks are divided into two classes depending on their geometric structure and boundary
conditions. We consider problems with alternating Dirichlet and Neumann boundary conditions and also problems with different
alternating Fourier (Neumann) conditions. We study the influence of the boundary conditions on the asymptotic behavior of
solutions as ε → 0. Convergence theorems, in particular, convergence of energy integrals, are proved. Bibliography: 31 titles.
Illustrations: 1 figure. 相似文献
10.
This paper deals with the blow-up properties of solutions to a system of heat equations u
t=Δu, v
t=Δv in B
R×(0, T) with the Neumann boundary conditions εu/εη=e
v, εv/εη=e
u on S
R×[0, T). The exact blow-up rates are established. It is also proved that the blow-up will occur only on the boundary.
This work is supported by the National Natural Science Foundation of China 相似文献
11.
The collocation method by spline in tension for the problem: −εy"+p(x)y=f(x), y(0)=α0,y(1)=α1, p(x)>0, 0<ε<<1, is derived. The method has the second order of the global uniform convergence. For the corresponding difference
scheme the optimal estimate: O (himin(hi, ε) is obtained.
This research was supported partly by NSF and SIZ for Science of SAP Vojvodina through funds made available to the U.S.—Yugoalav
Joint Board on Scientific and Tchnological Cooperation (grants JF554, JF799). 相似文献
12.
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
13.
Mikhail A. Borodin 《Journal of Mathematical Sciences》2011,178(1):13-40
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. 相似文献
14.
On positivity of solutions of degenerate boundary value problems for second-order elliptic equations
In this paper we study thedegenerate mixed boundary value problem:Pu=f in Ω,B
u
=gon Ω∂Г where ω is a domain in ℝ
n
,P is a second order linear elliptic operator with real coefficients, Γ⊆∂Ω is a relatively closed set, andB is an oblique boundary operator defined only on ∂Ω/Γ which is assumed to be a smooth part of the boundary.
The aim of this research is to establish some basic results concerning positive solutions. In particular, we study the solvability
of the above boundary value problem in the class of nonnegative functions, and properties of the generalized principal eigenvalue,
the ground state, and the Green function associated with this problem. The notion of criticality and subcriticality for this
problem is introduced, and a criticality theory for this problem is established. The analogs for the generalized Dirichlet
boundary value problem, where Γ=∂Ω, were examined intensively by many authors. 相似文献
15.
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
16.
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
17.
S. P. Yadav 《Acta Mathematica Hungarica》2003,98(1-2):21-30
Let X represent either the space C[-1,1] L
p
(α,β) (w), 1 ≦ p < ∞ on [-1, 1]. Then Xare Banach spaces under the sup or the p norms, respectively. We prove that there exists a normalized Banach subspace X
1
αβ of Xsuch that every f ∈ X
1
αβ can be represented by a linear combination of Jacobi polynomials to any degree of accuracy. Our method to prove such an approximation
problem is Fourier–Jacobi analysis based on the convergence of Fourier–Jacobi expansions.
This revised version was published online in June 2006 with corrections to the Cover Date. 相似文献
18.
We consider the problem −Δu=|u|
p−1u+λu in Ω with
on δΩ, where Ω is a bounded domain inR
N
,p=(N+2)/(N−2) is the critical Sobolev exponent,n the outward pointing normal and λ a constant. Our main result is that if Ω is a ball inR
N
, then for every λ∈R the problem admits infinitely many solutions. Next we prove that for every bounded domain Ω inR
3, symmetric with respect to a plane, there exists a constant μ>0 such that for every λ<μ this problem has at least one non-trivial
solution.
This work was supported by the Paris VI-Leiden exchange program
Supported by the Netherlands organisation for scientific research NWO, under number 611-306-016. 相似文献
19.
Giuseppina Vannella 《Annali di Matematica Pura ed Applicata》2002,180(4):429-440
We consider a Neumann problem of the type -εΔu+F
′(u(x))=0 in an open bounded subset Ω of R
n
, where F is a real function which has exactly k maximum points.
Using Morse theory we find that, for ε suitably small, there are at least 2k nontrivial solutions of the problem and we give some qualitative information about them.
Received: October 30, 1999 Published online: December 19, 2001 相似文献
20.
An a priori estimate for the H?lder constant with exponent 0 < α < 1 is obtained for smooth solutions to m-Hessian equations in a closed domain with (m − 1)-convex boundary of class C
2. The H?lder constant depends on the L
p
-norm, p > [n(n + 1)]/2, of the right-hand side of the equation, and the estimate remains valid for approximate solutions. Bibliography: 4 titles.
Dedicated to Nina Nikolaevna Uraltseva
Translated from Problemy Matematicheskogo Analiza, 40, May 2009, pp. 69–76. 相似文献