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1.
We consider the problem
where Ω is a bounded smooth domain in , 1 < p< + ∞ if N = 2, if N ≥ 3 and ε is a parameter. We show that if the mean curvature of ∂Ω is not constant then, for ε small enough, such a problem
has always a nodal solution u
ε with one positive peak and one negative peak on the boundary. Moreover, and converge to and , respectively, as ε goes to zero. Here, H denotes the mean curvature of ∂Ω.
Moreover, if Ω is a ball and , we prove that for ε small enough the problem has nodal solutions with two positive peaks on the boundary and arbitrarily
many negative peaks on the boundary.
The authors are supported by the M.I.U.R. National Project “Metodi variazionali e topologici nello studio di fenomeni non
lineari”. 相似文献
2.
T. V. Malovichko 《Ukrainian Mathematical Journal》2008,60(11):1789-1802
We consider the solution x
ε of the equation
where W is a Wiener sheet on . In the case where φε
2 converges to pδ(⋅ −a
1) + qδ(⋅ −a
2), i.e., the limit function describing the influence of a random medium is singular at more than one point, we establish the
weak convergence of (x
ε (u
1,⋅), …, x
ε (u
d
, ⋅)) as ε → 0+ to (X(u
1,⋅), …, X(u
d
, ⋅)), where X is the Arratia flow.
Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 60, No. 11, pp. 1529–1538, November, 2008. 相似文献
3.
Aleksandar Ivić 《Archiv der Mathematik》2008,90(5):412-419
If
denotes the error term in the classical Rankin-Selberg problem, then it is proved that
where Δ1(x) = ∫
x
0 Δ(u)du. The latter bound is, up to ‘ɛ’, best possible.
Received: 8 February 2007 相似文献
4.
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
5.
A general result on precise asymptotics for linear processes of positively associated sequences 总被引:2,自引:0,他引:2
Let {εt; t ∈ Z^+} be a strictly stationary sequence of associated random variables with mean zeros, let 0〈Eε1^2〈∞ and σ^2=Eε1^2+1∑j=2^∞ Eε1εj with 0〈σ^2〈∞.{aj;j∈Z^+} is a sequence of real numbers satisfying ∑j=0^∞|aj|〈∞.Define a linear process Xt=∑j=0^∞ ajεt-j,t≥1,and Sn=∑t=1^n Xt,n≥1.Assume that E|ε1|^2+δ′〈 for some δ′〉0 and μ(n)=O(n^-ρ) for some ρ〉0.This paper achieves a general law of precise asymptotics for {Sn}. 相似文献
6.
M. Rudelson 《Israel Journal of Mathematics》1999,111(1):143-155
Lett≥1 and letn, M be natural numbers,n<M. Leta=(a
i,j
) be ann xM matrix whose rows are orthonormal. Suppose that the ℓ2-norms of the columns ofA are uniformly bounded. Namely, for allj
Using majorizing measure estimates we prove that for every ε>0 there exists, a setI ⊃ {1,…,M} of cardinality at most
such that the matrix
, whereA
I
=(a
i,j
)
j∈I
, acts as a (1+ε)-isomorphism from ℓ
2
n
into
.
Research supported in part by a grant of the US-Israel BSF. Part of this research was performed when the author held a postdoctoral
position at MSRI. Research at MSRI was supported in part by NSF grant DMS-9022140. 相似文献
7.
The main result of this work is a Dancer-type bifurcation result for the quasilinear elliptic problem
Here, Ω is a bounded domain in denotes the Dirichlet p-Laplacian on , and is a spectral parameter. Let μ1 denote the first (smallest) eigenvalue of −Δ
p
. Under some natural hypotheses on the perturbation function , we show that the trivial solution is a bifurcation point for problem (P) and, moreover, there are two distinct continua, and , consisting of nontrivial solutions to problem (P) which bifurcate from the set of trivial solutions at the bifurcation point (0, μ1). The continua and are either both unbounded in E, or else their intersection contains also a point other than (0, μ1). For the semilinear problem (P) (i.e., for p = 2) this is a classical result due to E. N. Dancer from 1974. We also provide an example of how the union looks like (for p > 2) in an interesting particular case.
Our proofs are based on very precise, local asymptotic analysis for λ near μ1 (for any 1 < p < ∞) which is combined with standard topological degree arguments from global bifurcation theory used in Dancer’s original
work.
Submitted: July 28, 2007. Accepted: November 8, 2007. 相似文献
((P)) |
8.
V. G. Zhuravlev 《Journal of Mathematical Sciences》2008,150(3):2056-2083
The dynamics of contraction of the unit circle C under iterated two-color circle rotation Sɛ, dependent on a continuous parameter ε ∈ C, is studied. The dynamic distance from the deep hole Dh(ε) of the attraction domain
Spir
ɛ=C\Att
ɛ to the attractor Att
ɛ of the rotation Sɛ and the measure of the deep hole |Dh(ε)| are computed. It is proved that as ε ↑ 1, the phenomenon of localization of the
deep holes Dh(ε) occurs. It is shown that the process of contraction of the circle,
, goes on in three linear modes if the parameter ε coincides with an eigenvalue εm of a certain B-process, and in four modes in the general case. Bibliography: 7 titles.
__________
Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 350, 2007, pp. 89–138. 相似文献
9.
Yuexu Zhao 《Bulletin of the Brazilian Mathematical Society》2006,37(3):377-391
Let X1, X2, ... be i.i.d. random variables with EX1 = 0 and positive, finite variance σ2, and set Sn = X1 + ... + Xn. For any α > −1, β > −1/2 and for κn(ε) a function of ε and n such that κn(ε) log log n → λ as n ↑ ∞ and
, we prove that
*Supported by the Natural Science Foundation of Department of Education of Zhejiang Province (Grant No. 20060237 and 20050494). 相似文献
10.
LetH
ibe a finite dimensional complex Hilbert space of dimensiond
i associated with a finite level quantum system Ai for i = 1, 2, ...,k. A subspaceS ⊂
is said to becompletely entangled if it has no non-zero product vector of the formu
1⊗u
2 ⊗ ... ⊗u
k with ui inH
i for each i. Using the methods of elementary linear algebra and the intersection theorem for projective varieties in basic
algebraic geometry we prove that
where ε is the collection of all completely entangled subspaces.
When
andk = 2 an explicit orthonormal basis of a maximal completely entangled subspace of
is given.
We also introduce a more delicate notion of aperfectly entangled subspace for a multipartite quantum system, construct an example using the theory of stabilizer quantum codes and pose a
problem. 相似文献
11.
Unique solvability of the one-phase Stefan problem with a small multiplier ε at the time derivative in the equation is proved
on a certain time interval independent of ε for ε ∈ (0, ε0). The solution to the Stefan problem is compared with the solution to the Hele-Show problem, which describes the process
of melting materials with zero specific heat ε and can be regarded as a quasistationary approximation for the Stefan problem.
It is shown that the difference of the solutions has order
. This provides a justification of the quasistationary approximation. Bibliography: 23 titles.
__________
Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 348, 2007, pp. 209–253. 相似文献
12.
We consider one-dimensional difference Schr?dinger equations with real analytic function V(x). Suppose V(x) is a small perturbation of a trigonometric polynomial V
0(x) of degree k
0, and assume positive Lyapunov exponents and Diophantine ω. We prove that the integrated density of states is H?lder continuous for any k > 0. Moreover, we show that is absolutely continuous for a.e. ω. Our approach is via finite volume bounds. I.e., we study the eigenvalues of the problem
on a finite interval [1, N] with Dirichlet boundary conditions. Then the averaged number of these Dirichlet eigenvalues which fall into an interval
, does not exceed , k > 0. Moreover, for , this averaged number does not exceed exp , for any . For the integrated density of states of the problem this implies that for any . To investigate the distribution of the Dirichlet eigenvalues of on a finite interval [1, N] we study the distribution of the zeros of the characteristic determinants with complexified phase x, and frozen ω, E. We prove equidistribution of these zeros in some annulus and show also that no more than 2k
0 of them fall into any disk of radius exp. In addition, we obtain the lower bound (with δ > 0 arbitrary) for the separation of the eigenvalues of the Dirichlet eigenvalues over the interval [0, N]. This necessarily requires the removal of a small set of energies.
Received: February 2006, Accepted: December 2007 相似文献
13.
Alexander V. Kolesnikov 《Probability Theory and Related Fields》2008,140(1-2):1-17
Let γ be a Gaussian measure on a Suslin space X, H be the corresponding Cameron–Martin space and {e
i
} ⊂ H be an orthonormal basis of H. Suppose that μ
n
= ρ
n
· γ is a sequence of probability measures which converges weakly to a probability measure μ = ρ · γ Consider a sequence of Dirichlet forms , where and . We prove some sufficient conditions for Mosco convergence where . In particular, if X is a Hilbert space, and can be uniformly approximated by finite dimensional conditional expectations for every fixed e
i
, then under broad assumptions Mosco and the distributions of the associated stochastic processes converge weakly. 相似文献
14.
An Inverse Problem for Maxwell’s Equations in Anisotropic Media 总被引:3,自引:0,他引:3
The authors consider Maxwell's equations for an isomagnctic anisotropic and inhomogeneous medium in two dimensions, and discuss an inverse problem of determining the permittivity tensor (ε1ε2ε2ε3) and the permeabilityμin the constitutive relations from a finite number of lateral boundary measurements. Applying a Carlcman estimate, the authors prove an estimate of the Lipschitz type for stability, provided thatε1,ε2,ε3,μsatisfy some a priori conditions. 相似文献
15.
A. A. Kotsiolis 《Journal of Mathematical Sciences》1997,83(2):233-243
In this paper, the existence “in the large” of time-periodic classical solutions (with period T) is proved for the following
two dissipative ε-approximations for the Navier-Stokes equations modified in the sense of O. A. Ladyzhenskaya:
and the following two dissipative ε-approximations for the equations of motion of the Kelvin-Voight fluids:
satisfying the free surface conditions on the boundary ϖΩ of a domain Ω⊂R3:
.
The free term f(x, t) in systems (1)–(4) is assumed to be t-periodic with period T. It is shown that as ε→0, the classical
t-periodic solutions (with period T) of Eqs. (1)–(4) satisfying the free surface conditions (5) converge to the classicat
t-periodic solutions (with period T) of the Navier-Stokes equations modified in the sense of O. A. Ladyzhenskaya and to the
equations of motion of the Kelvin-Voight fruids, respectively, satisfying the boundary condition (5).
Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 210, 1994, pp. 109–124.
Translated by N. S. Zabavnikova. 相似文献
(1) |
(1) |
16.
Mohamed Ben Ayed Khalil El Mehdi 《NoDEA : Nonlinear Differential Equations and Applications》2006,13(4):485-509
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. 相似文献
17.
The asymptotic expansions are studied for the vorticity
to 2D incompressible Euler equations with-initial vorticity
, where ϕ0(x) satisfies |d ϕ0(x)|≠0 on the support of
and
is sufficiently smooth and with compact support in ℝ2 (resp. ℝ2×T) The limit,v(t,x), of the corresponding velocity fields {v
ɛ(t,x)} is obtained, which is the unique solution of (E) with initial vorticity ω0(x). Moreover,
(ℤ2)) for all 1≽p∞, where
and ϕ(t,x) satisfy some modulation equation and eikonal equation, respectively. 相似文献
18.
An estimate of
, Ω′⊃⊃Ω, for solutions uε of the family of equations
with a nondifferentiable lower term a is given. The majorant in the estimate depends on
and the distance between Ω′ and ∂Ω, and does not depend on ε. This publication is related to [2, 3]. Bibliography: 4 titles.
Dedicated to V. A. Solonnikov on his sixtieth anniversary
Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 213, 1994, pp. 75–92.
Translated by O. A. Ladyzhenskaya and N. N. Uraltseva. 相似文献
19.
Konstantin M. Dyakonov 《Constructive Approximation》2009,30(1):17-31
We study the action of Kolmogorov-type nonlinear averaging operators of the form V
−1
AV on smooth functions. Here, A runs through a family of convolution operators A
ε
[K], ε>0, generated by a single kernel K∈L
1(ℝ
n
) in the usual way and forming an “approximate identity” as ε→0, while V is a superposition map given by Vf=v○f, with a monotone continuous function v. The main result characterizes the kernels K with the property that the natural estimate
holds for all admissible functions f in the Lipschitz space Λ
ω
, associated with a majorant ω. Namely, it is shown that for fairly general (locally unbounded) functions v, the kernels in question must have compact support. Moreover, the same conclusion is already implied by various weak versions
of the above estimate (by infinitely weak ones, in a sense), even though the phenomenon has its limits.
Supported in part by grants MTM2005-08984-C02-02, MTM2006-26627-E and HF2006-0211 from El Ministerio de Educación y Ciencia
(Spain), and by grant 2005-SGR-00611 from DURSI (Generalitat de Catalunya). 相似文献
20.
Shuang-jie Peng 《应用数学学报(英文版)》2006,22(1):137-162
Abstract Let Ω be the unit ball centered at the origin in
. We study the following problem
By a constructive argument, we prove that for any k = 1, 2, • • •, if ε is small enough, then the above problem has positive a solution uε concentrating at k distinct points which tending to the boundary of Ω as ε goes to 0+. 相似文献