共查询到20条相似文献,搜索用时 46 毫秒
1.
Tetsutaro Shibata 《Journal d'Analyse Mathématique》2001,83(1):109-120
We consider the perturbed elliptic Sine-Gordon equation on an interval-u″t+γsinu(t)=μf(u(t)),t ∈I := (-T, T),u(t) > 0,t ∈I,u(±T)=0 where λ, μ>0 are parameters andT>0 is a constant. By applying variational methods subject to the constraint depending on λ, we obtain eigenpairs (μ,u)=(μ(λ),u
λ) which solve this eigenvalue problem for a given λ>0. Then we study the asymptotic behavior ofu
λ and μ(λ) as λ→∞. Especially, we study the location of interior transition layers ofu
λ as λ→∞.
This research has been supported by the Japan Society for the Promotion of Science. 相似文献
2.
Evangelos A. Latos Dimitrios E. Tzanetis 《NoDEA : Nonlinear Differential Equations and Applications》2010,17(2):137-151
We investigate the behaviour of solution u = u(x, t; λ) at λ = λ* for the non-local porous medium equation ${u_t = (u^n)_{xx} + {\lambda}f(u)/({\int_{-1}^1} f(u){\rm d}x)^2}We investigate the behaviour of solution u = u(x, t; λ) at λ = λ* for the non-local porous medium equation ut = (un)xx + lf(u)/(ò-11 f(u)dx)2{u_t = (u^n)_{xx} + {\lambda}f(u)/({\int_{-1}^1} f(u){\rm d}x)^2} with Dirichlet boundary conditions and positive initial data. The function f satisfies: f(s),−f ′ (s) > 0 for s ≥ 0 and s
n-1
f(s) is integrable at infinity. Due to the conditions on f, there exists a critical value of parameter λ, say λ*, such that for λ > λ* the solution u = u(x, t; λ) blows up globally in finite time, while for λ ≥ λ* the corresponding steady-state problem does not have any solution.
For 0 < λ < λ* there exists a unique steady-state solution w = w(x; λ) while u = u(x, t; λ) is global in time and converges to w as t → ∞. Here we show the global grow-up of critical solution u* = u(x, t; λ*) (u* (x, t) → ∞, as t → ∞ for all x ? (-1,1){x\in(-1,1)}. 相似文献
3.
Tetsutaro Shibata 《Annali di Matematica Pura ed Applicata》2003,182(2):211-229
We consider the two-parameter nonlinear eigenvalue problem?−Δu = μu − λ(u + u
p
+ f(u)), u > 0 in Ω, u = 0 on ∂Ω,?where p>1 is a constant and μ,λ>0 are parameters. We establish the asymptotic formulas for the variational eigencurves λ=λ(μ,α) as
μ→∞, where α>0 is a normalizing parameter. We emphasize that the critical case from a viewpoint of the two-term asymptotics
of the eigencurve is p=3. Moreover, it is shown that p=5/3 is also a critical exponent from a view point of the three-term asymptotics when Ω is a ball or an annulus. This sort
of criticality for two-parameter problems seems to be new.
Received: February 9, 2002; in final form: April 3, 2002?Published online: April 14, 2003 相似文献
4.
In this paper, the authors consider the behaviors of a class of parametric Marcinkiewicz integrals μ
Ω
ρ
, μ
Ω,λ
*,ρ
and μ
Ω,S
ρ
on BMO(ℝ
n
) and Campanato spaces with complex parameter ρ and the kernel Ω in Llog+
L(S
n−1). Here μ
Ω,λ
*,ρ
and μ
Ω,S
ρ
are parametric Marcinkiewicz functions corresponding to the Littlewood-Paley g
λ
*-function and the Lusin area function S, respectively. Under certain weak regularity condition on Ω, the authors prove that if f belongs to BMO(ℝ
n
) or to a certain Campanato space, then [μ
Ω,λ
*,ρ
(f)]2, [μ
Ω,S
ρ
(f)]2 and [μ
Ω
ρ
(f)]2 are either infinite everywhere or finite almost everywhere, and in the latter case, some kind of boundedness are also established. 相似文献
5.
Robert S. Strichartz 《Journal d'Analyse Mathématique》2006,99(1):333-353
For certain Cantor measures μ on ℝn, it was shown by Jorgensen and Pedersen that there exists an orthonormal basis of exponentialse
2πiγ·x for λεΛ. a discrete subset of ℝn called aspectrum for μ. For anyL
1 functionf, we define coefficientsc
γ(f)=∝f(y)e
−2πiγiy
dμ(y) and form the Mock Fourier series ∑λ∈Λcλ(f)e
2πiλ·x
. There is a natural sequence of finite subsets Λn increasing to Λ asn→∞, and we define the partial sums of the Mock Fourier series by
We prove, under mild technical assumptions on μ and Λ, thats
n(f) converges uniformly tof for any continuous functionf and obtain the rate of convergence in terms of the modulus of continuity off. We also show, under somewhat stronger hypotheses, almost everywhere convergence forfεL
1.
Research supported in part by the National Science Foundation, Grant DMS-0140194. 相似文献
6.
The existence of positive radial solutions of the equation -din( |Du|p-2Du)=f(u) is studied in annular domains in Rn,n≥2. It is proved that if f(0)≥0, f is somewherenegative in (0,∞), limu→0^ f‘ (u)=0 and limu→∞ (f(u)/u^p-1)=∞, then there is alarge positive radial solution on all annuli. If f(0)≤0 and satisfies certain conditions, then the equation has no radial solution if the annuli are too wide. 相似文献
7.
We consider the nonlinear eigenvalue problem −Δu=λ f(u) in Ω u=0 on ∂Ω, where Ω is a ball or an annulus in RN (N ≥ 2) and λ > 0 is a parameter. It is known that if λ >> 1, then the corresponding positive solution uλ develops boundary layers under some conditions on f. We establish the asymptotic formulas for the slope of the boundary layers of uλ with the exact second term and the ‘optimal’ estimate of the third term. 相似文献
8.
D. S. Lubinsky 《Constructive Approximation》2011,34(3):357-369
Let μ be a compactly supported positive measure on the real line, with associated Christoffel functions λ
n
(d
μ,⋅). Let g be a measurable function that is bounded above and below on supp[μ] by positive constants. We show that λ
n
(g
d
μ,⋅)/λ
n
(d
μ,⋅)→g in measure in {x:μ′(x)>0} and consequently in all L
p
norms, p<∞. The novelty is that there are no local or global restrictions on μ. The main idea is a new maximal function estimate for the “tail” in Nevai’s operators. 相似文献
9.
Yehoram Gordon 《Israel Journal of Mathematics》1966,4(3):177-188
GivenF(z),f
1(z), ..,f
n(z) defined on a finite point setE, and givenB — the set of generalised polynomials Σ
k
=1/n
a
kfk(z) — the definition of a juxtapolynomial is extended in the following manner: for a fixedλ(0<λ≦1),f(z) εB is called a generalizedλ-weak juxtapolynomial toF(z) onE if and only if there exists nog(z) εB for whichg(z)=F(z) wheneverf(z)=F(z) and |g(z)−F(z) |<λ|f(z)−F(z)| wheneverf(z)≠F(z). The properties of suchf(z) are investigated with particular attention given to the real case.
This note is an extension of a part of the author’s M.Sc. Thesis under the supervision of Prof. B. Grünbaum to whom the author
wishes to express his sincerest appreciation. The author also wishes to thank Dr. J. Lindenstrauss for his valuable remarks
in the preparation of this paper. 相似文献
10.
This is a continuation of our previous work. We classify all the simple ℋq(D
n
)-modules via an automorphismh defined on the set { λ | Dλ ≠ 0}. Whenf
n(q) ≠ 0, this yields a classification of all the simple ℋ
q
(D
n)- modules for arbitrary n. In general ( i. e., q arbitrary), if λ(1) = λ(2),wegivea necessary and sufficient condition ( in terms of some polynomials ) to ensure that the irreducible ℋq,1(B
n
)- module Dλ remains irreducible on restriction to ℋq(D
n
). 相似文献
11.
Sudhasree Gadam 《Rendiconti del Circolo Matematico di Palermo》1992,41(2):209-220
We study the behaviour of the positive solutions to the Dirichlet problem IR
n
in the unit ball in IR
R
wherep<(N+2)/(N−2) ifN≥3 and λ varies over IR. For a special class of functionsg viz.,g(x)=u
0
p
(x) whereu
0 is the unique positive solution at λ=0, we prove that for certain λ’s nonradial solutions bifurcate from radially symmetric
positive solutions. WhenN=1, we obtain the complete bifurcation diagram for the positive solution curve. 相似文献
12.
Ming Ju LIU Shan Zhen LU 《数学学报(英文版)》2007,23(1):7-16
In this paper, the authors study some properties of Littlewood-Paley g-functions gψ(f),Lusin area functions Sψ,α(f) and Littlewood-Paley gψ^*,λ(f) functions defined on H^n, where α,λ 〉 0 and ψ, f are suitable functions. They are the generalization of the corresponding operators on R^n. 相似文献
13.
LetK be a field, charK=0 andM
n
(K) the algebra ofn×n matrices overK. If λ=(λ1,…,λ
m
) andμ=(μ
1,…,μ
m
) are partitions ofn
2 let
wherex
1,…,x
n
2,y
1,…,y
n
2 are noncommuting indeterminates andS
n
2 is the symmetric group of degreen
2.
The polynomialsF
λ, μ
, when evaluated inM
n
(K), take central values and we study the problem of classifying those partitions λ,μ for whichF
λ, μ
is a central polynomial (not a polynomial identity) forM
n
(K).
We give a formula that allows us to evaluateF
λ, μ
inM(K) in general and we prove that if λ andμ are not both derived in a suitable way from the partition δ=(1, 3,…, 2n−3, 2n−1), thenF
λ, μ
is a polynomial identity forM
n
(K). As an application, we exhibit a new class of central polynomials forM
n
(K).
In memory of Shimshon Amitsur
Research supported by a grant from MURST of Italy. 相似文献
14.
Liang Hanying Zhang Dongxia Lu Baoxian Dept.of Appl.Math. Tongji Univ. Shanghai China. 《高校应用数学学报(英文版)》2004,19(3):302-310
§1 IntroductionConsider the following heteroscedastic regression model:Yi =g(xi) +σiei, 1≤i≤n,(1.1)whereσ2i=f(ui) ,(xi,ui) are nonrandom design points,0≤x0 ≤x1 ≤...≤xn=1and0≤u0≤u1 ≤...≤un=1,Yi are the response variables,ei are random errors,and f(·) andg(·) are unknown functions defined on closed interval[0 ,1] .It is well known thatregression model has many applications in practical problems,sothe model (1.1) and its special cases have been studied extensively. For instance,… 相似文献
15.
On any compact Riemannian manifold (M,g) of dimension n, the L
2-normalized eigenfunctions φ
λ
satisfy
||fl||¥ £ Cl\fracn-12\|\phi_{\lambda}\|_{\infty}\leq C\lambda^{\frac{n-1}{2}} where −Δφ
λ
=λ
2
φ
λ
. The bound is sharp in the class of all (M,g) since it is obtained by zonal spherical harmonics on the standard n-sphere S
n
. But of course, it is not sharp for many Riemannian manifolds, e.g., flat tori ℝ
n
/Γ. We say that S
n
, but not ℝ
n
/Γ, is a Riemannian manifold with maximal eigenfunction growth. The problem which motivates this paper is to determine the
(M,g) with maximal eigenfunction growth. In an earlier work, two of us showed that such an (M,g) must have a point x where the set ℒ
x
of geodesic loops at x has positive measure in S*xMS^{*}_{x}M. We strengthen this result here by showing that such a manifold must have a point where the set ℛ
x
of recurrent directions for the geodesic flow through x satisfies |{ℛ}
x
|>0. We also show that if there are no such points, L
2-normalized quasimodes have sup-norms that are o(λ
(n−1)/2), and, in the other extreme, we show that if there is a point blow-down x at which the first return map for the flow is the identity, then there is a sequence of quasimodes with L
∞-norms that are Ω(λ
(n−1)/2). 相似文献
16.
P. V. Chunaev 《Proceedings of the Steklov Institute of Mathematics》2010,270(1):278-284
We study the approximation of functions f(z) that are analytic in a neighborhood of zero by finite sums of the form H
n
(z) = H
n
(h, f, {λ
k
}; z) = Σ
k=1
n
λ
k
h(λ
k
z), where h is a fixed function that is analytic in the unit disk |z| < 1 and the numbers λ
k
(which depend on h, f, and n) are calculated by a certain algorithm. An exact value of the radius of the convergence H
n
(z) → f(z), n → ∞, and an order-sharp estimate for the rate of this convergence are obtained; an application to numerical analysis is given. 相似文献
17.
A. Kopotowski M G. Nadkarni K. P. S. Bhaskara Rao 《Proceedings Mathematical Sciences》2003,113(1):77-86
We discuss subsetsS of ℝn such that every real valued functionf onS is of the formf(x1, x2, ..., xn) =u
1(x1) +u
2(x2) +...+u
n(xn), and the related concepts and situations in analysis. 相似文献
18.
V. V. Shchigolev 《Journal of Mathematical Sciences》2007,142(2):2015-2019
In this paper, we calculate the space Ext
GL(n
1
)(L
n
(λ), L
n
(μ)), where GL(n) is the general linear group of degree n over an algebraically closed field of positive characteristic, L
n
(λ) and L
n
(μ) are rational irreducible GL(n)-modules with highest weights λ and μ, respectively, the restriction of L
n
(λ) to any Levi subgroup of GL(n) is semisimple, λ is a p-restricted weight, and μ does not strictly dominate λ.
__________
Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 11, No. 2, pp. 219–226, 2005. 相似文献
19.
We consider the algebra C
u
= C
u
(ℝ) of uniformly continuous bounded complex functions on the real line ℝ with pointwise operations and sup-norm. Let I be a closed ideal in C
u
invariant with respect to translations, and let ah
I
(f) denote the minimal real number (if it exists) satisfying the following condition. If λ > ah
I
(f), then ( [^(f)] - [^(g)] ) |V = 0\left. {\left( {\hat f - \hat g} \right)} \right|_V = 0 for some g ∈ I, where V is a neighborhood of the point λ. The classical Titchmarsh convolution theorem is equivalent to the equality ah
I
(f
1 · f
2) = ah
I
(f
1) + ah
I
(f
2), where I = {0}. We show that, for ideals I of general form, this equality does not generally hold, but ah
I
(f
n
) = n · ah
I
(f) holds for any I. We present many nontrivial ideals for which the general form of the Titchmarsh theorem is true. 相似文献
20.
E is a Banach lattice that is weakly sequentially complete and has a weak unitu. TLf
n=ϕ means that the infimum of |f
n−ϕ| andu converges strongly to zero.T is a positive contraction operator onE andA
n=(1/n)(I+T+...+T
n−1). Without an additional assumption onE, the “truncated limit” TLA
nf need not exist forf inE. This limit exists for eachf ifE satisfies the following additional assumption (C): For everyf inE
+ and for every numberα>0, there is a numberβ=β(f, α) such that ifg is inE
+, ‖g‖≦1, 0≦f′≦f and ‖f′‖>α then ‖f′+g‖≧‖g‖+β.
Research of this author is partially supported by NSERC Grant A3974.
Research of this author is partially supported by NSF Grant 8301619. 相似文献