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1.
Jean Louis Woukeng 《Applications of Mathematics》2009,54(6):465-489
This paper is devoted to the homogenization beyond the periodic setting, of nonlinear monotone operators in a domain in ℝ
N
with isolated holes of size ɛ2 (ɛ > 0 a small parameter). The order of the size of the holes is twice that of the oscillations of the coefficients of the
operator, so that the problem under consideration is a reiterated homogenization problem in perforated domains. The usual
periodic perforation of the domain and the classical periodicity hypothesis on the coefficients of the operator are here replaced
by an abstract assumption covering a great variety of behaviors such as the periodicity, the almost periodicity and many more
besides. We illustrate this abstract setting by working out a few concrete homogenization problems. Our main tool is the recent
theory of homogenization structures. 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.
Jorge San Martín Loredana Smaranda 《Zeitschrift für Angewandte Mathematik und Physik (ZAMP)》2010,75(3):401-424
This paper considers the periodic spectral problem associated with the Laplace operator written in
\mathbbRN{\mathbb{R}^N} (N = 3, 4, 5) periodically perforated by balls, and with homogeneous Dirichlet condition on the boundary of holes. We give an
asymptotic expansion for all simple eigenvalues as the size of holes goes to zero. As an application of this result, we use
Bloch waves to find the classical strange term in homogenization theory, as the size of holes goes to zero faster than the
microstructure period. 相似文献
4.
In one space dimension we address the homogenization of the spectral problem for a singularly perturbed diffusion equation
in a periodic medium. Denoting by ε the period, the diffusion coefficient is scaled as ε2. The domain is made of two purely periodic media separated by an interface. Depending on the connection between the two cell
spectral equations, three different situations arise when ε goes to zero. First, there is a global homogenized problem as
in the case without an interface. Second, the limit is made of two homogenized problems with a Dirichlet boundary condition
on the interface. Third, there is an exponential localization near the interface of the first eigenfunction.
Received: January 10, 2001; in final form: July 9, 2001?Published online: June 11, 2002 相似文献
5.
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
6.
M. Vanninathan 《Proceedings Mathematical Sciences》1981,90(3):239-271
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:
λε = ε−2 λ + λ0 +O (ε), Stekloff: λε = ελ1 +O (ε2), Neumann: λε = 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. 相似文献
7.
V. V. Yablokov 《Journal of Mathematical Sciences》2006,135(1):2803-2811
The problem of homogenization is considered for the solutions of the Neumann problem for the Lamé system of plane elasticity
in two-dimensional domains with channels that have the form of rectilinear cylinders of length ε
q (ε is a small positive parameter, q = const > 0) and radius a
ɛ. The bases of the channels form an ε-periodic structure on the hyperplane {x ∈ ℝ2: x
1 = 0} and their number is equal to N
ɛ= O(ɛ−1) as ε → 0. Under the limit condition lim
on the parameters characterizing the geometry of the domain, the weak H
1-limit of the generalized solution of this problem is found.
__________
Translated from Trudy Seminara imeni I. G. Petrovskogo, No. 25, pp. 310–322, 2005. 相似文献
8.
G. Cardone C. D'Apice U. De Maio 《NoDEA : Nonlinear Differential Equations and Applications》2002,9(3):325-346
In this paper we study the asymptotic behaviour of the Laplace equation in a periodically perforated domain of R
n
, where we assume that the period is ε and the size of the holes is of the same order of greatness. An homogeneous Dirichlet
condition is given on the whole exterior boundary of the domain and on a flat portion of diameter if (, if n=2) of the boundary of every hole, while we take an homogeneous Neumann condition elsewhere. 相似文献
9.
A. N. Averyanov 《Journal of Mathematical Sciences》2006,135(1):2603-2615
This article deals with the asymptotic behavior as ε → 0 of the solution {u
ɛ, p
ɛ} of the plane Stokes problem in a perforated domain. The limit problem is constructed and estimates for the speed of convergence
are obtained. It is shown that the speed of convergence is of order O(ε
3/2).
__________
Translated from Trudy Seminara imeni I. G. Petrovskogo, No. 25, pp. 3–20, 2005. 相似文献
10.
We consider approximation of linear multivariate problems defined over
weighted tensor product Hilbert spaces with finite-order weights. This means we
consider functions of d variables that can be represented as sums of functions of at
most q* variables. Here, q* is fixed (and presumably small) and d may be arbitrarily
large. For the univariate problem, d = 1, we assume we know algorithms A1,ε that
use O(ε−p) function or linear functional evaluations to achieve an error ε in the
worst case setting. Based on these algorithms A1,ε, we provide a construction of polynomial-time algorithms Ad,ε for the general d-variate problem with the number of evaluations bounded roughly by ε−pdq* to achieve an error ε in the worst case setting. 相似文献
11.
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
12.
We consider quasilinear and linear parabolic problems with rapidly oscillating coefficients in a domain Ω
ε
that is ε-periodically perforated by small holes of order
相似文献
13.
Shmuel Friedland 《Israel Journal of Mathematics》1980,35(1-2):89-108
LetA(ε) andB(ε) be complex valued matrices analytic in ε at the origin.A(ε)≈
p
B(ε) ifA(ε) is similar toB(ε) for any |ε|<r,A(ε)≈a
B(ε) ifB(ε)=T(ε)A(ε)T
−1(ε) andT(ε) is analytic and |T(ε)|≠0 for |ε|<r! In this paper we find a necessary and sufficient conditions onA(ε) andB(ε) such thatA(ε)≈
a
B(ε) provided thatA(ε)≈
p
B(ε). This problem arises in study of certain ordinary differential equations singular with respect to a parameter ε in the
origin and was first stated by Wasow.
Sponsored by the United States Army under Contract No. DAAG29-75-C-0024 相似文献
14.
On the validity of the Ginzburg-Landau equation 总被引:1,自引:0,他引:1
A. van Harten 《Journal of Nonlinear Science》1991,1(4):397-422
Summary The famous Ginzburg-Landau equation describes nonlinear amplitude modulations of a wave perturbation of a basic pattern when
a control parameterR lies in the unstable regionO(ε
2) away from the critical valueR
c for which the system loses stability. Hereε>0 is a small parameter. G-L's equation is found for a general class of nonlinear evolution problems including several classical
problems from hydrodynamics and other fields of physics and chemistry. Up to now, the rigorous derivation of G-L's equation
for general situations is not yet completed. This was only demonstrated for special types of solutions (steady, time periodic)
or for special problems (the Swift-Hohenberg equation). Here a mathematically rigorous proof of the validity of G-L's equation
is given for a general situation of one space variable and a quadratic nonlinearity. Validity is meant in the following sense.
For each given initial condition in a suitable Banach space there exists a unique bounded solution of the initial value problem
for G-L's equation on a finite interval of theO(1/ε2)-long time scale intrinsic to the modulation. For such a finite time interval of the intrinsic modulation time scale on which
the initial value problem for G-L's equation has a bounded solution, the initial value problem for the original evolution
equation with corresponding initial conditions, has a unique solutionO(ε2) — close to the approximation induced by the solution of G-L's equation. This property guarantees that, for rather general
initial conditions on the intrinsic modulation time scale, the behavior of solutions of G-L's equation is really inherited
from solutions of the original problem, and the other way around: to a solution of G-L's equation corresponds a nearby exact
solution with a relatively small error. 相似文献
15.
Kersten Schmidt Sébastien Tordeux 《Zeitschrift für Angewandte Mathematik und Physik (ZAMP)》2010,42(1):603-626
We derive and analyse models which reduce conducting sheets of a small thickness ε in two dimensions to an interface and approximate their shielding behaviour by conditions on this interface. For this we
consider a model problem with a conductivity scaled reciprocal to the thickness ε, which leads to a nontrivial limit solution for ε → 0. The functions of the expansion are defined hierarchically, i.e. order by order. Our analysis shows that for smooth sheets
the models are well defined for any order and have optimal convergence meaning that the H
1-modelling error for an expansion with N terms is bounded by O(ε
N+1) in the exterior of the sheet and by O(ε
N+1/2) in its interior. We explicitly specify the models of order zero, one and two. Numerical experiments for sheets with varying
curvature validate the theoretical results. 相似文献
16.
In this article, we study the homogenization of the family of parabolic equations over periodically perforated domains
. Here, Ωɛ
= ΩS
ε
is a periodically perforated domain andd
ε
is a sequence of positive numbers which goes to zero. We obtain the homogenized equation. The homogenization of the equations
on a fixed domain and also the case of perforated domain with Neumann boundary condition was studied by the authors. The homogenization
for a fixed domain and
has been done by Jian. We also obtain certain corrector results to improve the weak convergence. 相似文献
17.
O. P. Filatov 《Mathematical Notes》2000,67(3):365-371
For the class II(ℝ
m
) of continuous almost periodic functionsf: ℝ
m
→ ℝ, we consider the problem of the existence of the limit
where the least upper bound is taken over all solutions (in the sense of Carathéodory) of the generalized differential equation
{ie365-1} εG, γ(0)=a
0. We establish that if the compact setG ⊂ ℝ
m
is not contained in a subspace of ℝ
m
of dimensionm−1 (i.e., if it is nondegenerate), then the limit exists uniformly in the initial vectora
0 ε ℝ
m
. Conversely, if for any functionf ε π(ℝ
m
), the limit exists uniformly in the initial vectora
0 ε ℝ
m
, then the compact setG is nondegenerate. We also prove that there exists an extremal solution for which a limit of the maximal mean uniform in the
initial conditions is realized.
Translated fromMatematicheskie Zametki, Vol. 67, No. 3, pp. 433–440, March, 2000. 相似文献
(1) |
18.
S. E. Pastukhova 《Journal of Mathematical Sciences》2006,132(3):313-330
We study elasticity problems in the plane (space) which is reinforced with a periodic thin network (box structure). This highly
contrasting medium depends on two small related parameters ε and h which control the size of a periodicity cell and thickness
of the reinforcement. For combined structures, we prove the classical homogenization principle, which is the same for any
relations between parameters ε and h; this is in a contrast with the case of thin structures. We use the method of two-scale
convergence with respect to a variable measure, which is natural for combined structures. Bibliography: 17 titles.
__________
Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 310, 2004, pp. 114–144. 相似文献
19.
T. Shaposhnikova 《Journal of Mathematical Sciences》1995,75(3):1631-1645
The behavior of the solution of a boundary value problem for a parabolic equation with rapidly oscillating coefficientsɛ
−1
x,ɛ
−2k
t), (k⋝0) in a perforated domain for ε→0 is studied. Some estimates of the deviation of the solution and energy for the original
boundary value problem from the solution and energy of the corresponding homogenized problem are found. In this investigation
methods developed by Oleinik, Zhikov, Kozlov, Bensoussan, Lions, Papanikolaou, Cioranescu, and Paulin are used. Bibliography:
15 titles.
Translated from Trudy Seminara imeni I. G. Petrovskogo, No. 17, pp. 27–50, 1994. 相似文献
20.
We use regularized semigroups to consider local linear and semilinear inhomogeneous abstract Cauchy problems on a Banach space
in a unified way. We show that the inhomogeneous abstract Cauchy problem {fx43-1} has a unique classical solution, for allf εC([0,T], [Im(C)]),x inC(D(A)), if and only ifA generates aC-regularized semigroup of bounded semivariation, and has a strong solution for allf εL
1 ([0,T], [Im(C)]),x εC(D(A)) if and only if theC-regularized semigroup is what we call of bounded super semivariation. This includes locally Lipschitz continuousC-regularized semigroups. We give similar simple sufficient conditions for the semilinear abstract Cauchy problem {fx43-2}
to have a unique solution.
Well-known results for generators of strongly continuous semigroups, as well as more recent results for Hille-Yosida operators,
originally due to Da Prato and Sinestrari, regarding (0.1), are immediate corollaries of our results. Results due to Desch,
Schappacher and Zhang, on (0.2), for generators of strongly continuous semigroups, are similarly generalized to Hille-Yosida
operators with our approach.
This article appeared in the last issue of the Forum. However, due to an error by the Journal Secetary, the Abstract was omitted,
and with it the equations which are the focus of the article. We therefore are reprinting the article in its entirety. The
Journal Secretary regrets the error. 相似文献