奇异积分中的一些问题

设n≥2,Ω为R~n中单位球面S~(n-1)上的可积函数且Ω在S~(n-1)上的平均值为零,即∫_S~(n1)~Ω(x)dσ(x)=0.其中dσ为S~(n-1)上的体积元.定义奇异积分算子T_0,和相应的极大算子T~*,其中h∈L~∞(R~+).关于算子T和T~*已有许多研究([1]-[6]等).在1986年,Namazi利用Fourier变换的Hausdorff-Young不等式证明了     

有限元导数的一致超收敛估计

§1 引言 设ΩR~2是边界为Γ的有界区域,Ω=Ω∪Γ。Sobolev空间W_p~m(Ω)的范数、半范数分别用‖·‖_(m,p,Ω),|·|_(m,p,Ω)或‖·‖_(m,p),|·|_(m,p)表示。在W_p~1(Ω)×W_p~1(Ω) (1≤p≤∞,1/p+1/q=1)上定义双线性泛函: 我们假定系数a_(if)定义在Ω上且满足     

带有齐性核的分数次积分算子在Hardy型空间上的有界性

该文研究了带有齐性核的分数次积分算子T_(Ω,α)在一些Hardy空间上的映射性质,其中核Ω在球面S~(n-1)上满足一些L~S-Dini条件.作者将前人的一些结果改进到0αn情形,同时还得到了算子T_(Ω,α)在Herz型Hardy空间上的一个端点估计.     

抛物型方程的基本杂交元方法

一 引言 考虑以下问题: 其中I=(0,T_0],T_0>0是一常数,Ω是R~2中有界凸开域,边界Ω充分光滑,是Ω的单位外法向矢量。 假定     

B值鞅的Davis型不等式与Banach空间的几何性质

下面两个结果被证明: 1.Banach空间X同构于p光滑空间(10使得每个X值鞅f=(f_n)满足‖f~*‖≤C‖S~((p))(f)‖_1。 2.Banach空间X同构于q凸空间(2≤q<∞)当且仅当存在C>0使得每个X值鞅f=(f_n)满足‖S~((q))(f)‖_1≤C‖f~*‖_1。 此外,应用f~*与S~((p))(f)的弱(1,1)型不等式以及集合{f~*<∞}{S~((p))(f)<∞}的相互包含关系刻划了B空间的p光滑性和q凸性。作为推论,给出了超自反空间以及与Hilbert空间同构的Banach空间的特征。     

特征值问题有限元逼近的Lp估计与超收敛性

设Ω是R~2中的一个开域,边界Г满足李普希兹条件。W_(m,p)(Ω)表示通常意义下的obolev空间,其范数用‖·‖m,p表示,记     

有限元的L~p估计

为书写方便,仅考虑如下二次泛函数的极小问题[注1]:其中W_p~m(Ω)为通常的空间(范数记为‖·‖m,p).假定凸域ΩR~2的边界满足文[1]中的条件(详见以下的(6),(6)＇).并假定Ω=Ω∪Ω上给出了三角形(或四边形)网格族_h,(0相似文献   

L~(p(x))(Ω)中关于Luxemburg范数和共轭Orlicz范数间的一个最佳不等式

令 L~(p(x))(Ω)为变指数 Lebesgue空间,其中 p：Ω→[1,∞]．‖·‖_(p(x))和‖·‖_(p(x))~o 分别表示 L~(p(x))(Ω)中的 Luxemburg 范数和共轭 Orlicz 范数．本文证明成立最佳不等式‖·‖_(p(x))≤‖·‖_(p(x))~o ≤ d_(p-,p )‖·‖_(p(x)),其中 d_(p-,p )是一个依赖于 p-=essinf_Ωp(x)和 p =esssup_Ωp(x)的常数．当1＜p-＜p ＜∞时, (?) 当 p-=1或 p =∞时,d(p-,p )是相应的极限形式．     

deg≥2的圆周自映射

<正> 记 R=(-∞,+∞),I=[0,1]和 S~1{e~(2πxi)|x∈I}.S~1是复平面上中心在原点的单位圆周.S~1上全体连续自映射的集合记为 C~0(S~1,S~1).设 f∈C~0(S~1,S~1),f 的周期集合,不动点集,周期点集,非游荡集和拓扑熵分别记为 p(f),F(f),P(f),Ω(f)和ent(f).此外,用 deg(f)记 f 的拓扑度或层数(一种定义见§2).关于圆周自映射所产生的动力系统性质已有很多人进行了讨论.据作者所知,所有这种讨论还仅限于在某种条件下寻求拓扑熵下限的最好估计以及 Sharkovskii 和 Li Yorke     

反应扩散方程在H~2(Ω)和L~(2p-2)(Ω)中的指数吸引子

该文讨论一类具有任意多项式增长非线性项和非齐次项的反应扩散方程指数吸引子的存在性.首先,对R~3中的有界开子集Ω,分别选取解半群S(t)在L~2(Ω)和H~2(Ω)中的有界正不变吸收集来构造H~2(Ω)中的指数吸引子.然后证明对某个足够大的时间T_1,S(T_1)在这两个吸收集之间是Lipschitz连续的.最后由一种新的逼近技巧证明了对任意的g∈L~2(Ω),S(t)在L~(2p-2)(Ω)中存在指数吸引子.该结论推广了已有文献中的结果.     

Jacobian-determinant equation in the critical Besov spaces

Let Ω be a bounded domain in R~n with smooth boundary. Here we consider the following Jacobian-determinant equation det u(x)=f(x),x∈Ω;u(x)=x,x∈?Ω where f is a function on Ω with min_Ω f = δ 0 and Ωf(x)dx = |Ω|. We prove that if f ∈B_(p1)~(np)(Ω) for some p∈(n,∞), then there exists a solution u ∈ B_(p1)~(np+1)(Ω)C~1(Ω) to this equation. On the other hand, we give a simple example such that u ∈ C_0~1(R~2, R~2) while detu does not lie in B_(p1)~(2p)(R~2) for any p∞.     

The Degree of Proper Holomorphic MappingsBetween Special Domains in C^n

Let Gi be closed Lie groups of U (n), Ω i be bounded Gi-invariant domains in C^n which contains 0, and O(C^n)^Gi = C, for i = 1, 2. It is known that if f : Ω 1 → Ω 2 is a proper holomorphic mapping, and f^-1{0} = {0}, then f is a polynomial mapping. In this paper, we provide an upper bound for the degree of such a polynomial mapping using the multiplicity of f .     

Multiple Positive Solutions for a Nonlinear Elliptic Equation Involving Hardy–Sobolev–Maz'ya Term

In this paper, we study the existence and nonexistence of multiple positive solutions for the following problem involving Hardy–Sobolev–Maz'ya term:-Δu- λu/|y|2=|u|pt-1u/|y|t+ μf(x), x ∈Ω,where Ω is a bounded domain in RN(N ≥ 2), 0 ∈Ω, x =(y, z) ∈ Rk× RN-kand pt =N +2-2t N-2(0 ≤ t ≤2). For f(x) ∈ C1(Ω)\{0}, we show that there exists a constant μ* 0 such that the problem possessesat least two positive solutions if μ∈(0, μ*) and at least one positive solution if μ = μ*. Furthermore,there are no positive solutions if μ∈(μ*, +∞).     

双线性Fourier乘子交换子的有界性与紧性

假定T_σ是关于乘子σ的双线性Fourier乘子算子,其中σ满足如下Sobolev正则条件:对某个s∈(n,2n],有sup_(κ∈Z)‖σ_k‖W~s(R~(2m))∞.对于p_1,p_2,p∈(1,∞)且满足1/p=1/p_1+1/p_2和ω=(ω_1,ω_2)∈A_(p/t)(R~(2n)),建立了T_σ及其与函数b=(b_1,b_2)∈(BMO(R~n))~2生成的交换子T_(σ,b)由L~(p_1,λ)(ω_1)×L~(p_2,λ)(ω_2)到L~(p,λ)(v_w)的有界性;同时,在b_1,b_2∈CMO(R~n)(C_c~∞(R~n)在BMO拓扑下的闭包)的条件下,证明交换子T_(σ,b)是L~(p_1,λ)(ω_1)×L~(p_2,λ)(ω_2)到L~(p,λ)(v_w)的紧算子.为了得到主要结果,我们先后建立了几个双(次)线性极大函数在加多权Morrey空间上的有界性以及该空间中准紧集的判定.     

Projective Dirichlet boundary condition with applications to a geometric problem

Given a domain Ω ? R~n, let λ 0 be an eigenvalue of the elliptic operator L :=Σ!(i,j)~n =1?/?xi(a~(ij0 ?/?xj) on Ω for Dirichlet condition. For a function f ∈ L~2(Ω), it is known that the linear resonance equation Lu + λu = f in Ω with Dirichlet boundary condition is not always solvable.We give a new boundary condition P_λ(u|? Ω) = g, called to be pro jective Dirichlet condition, such that the linear resonance equation always admits a unique solution u being orthogonal to all of the eigenfunctions corresponding to λ which satisfies ||u||2,2 ≤ C(||f ||_2 +|| g||_(2,2)) under suitable regularity assumptions on ?Ω and L, where C is a constant depends only on n, Ω, and L. More a priori estimates,such as W~(2,p)-estimates and the C~(2,α)-estimates etc., are given also. This boundary condition can be viewed as a generalization of the Dirichlet condition to resonance equations and shows its advantage when applying to nonlinear resonance equations. In particular, this enables us to find the new indicatrices with vanishing mean(Cartan) torsion in Minkowski geometry. It is known that the geometry of indicatries is the foundation of Finsler geometry.     

三维Boussinesq方程关于水平速度的一个爆破准则

本文主要讨论三维Boussinesq方程当扩散系数κ=0时光滑解的爆破准则.利用Littlewood-Paley分解和能量方法证明了如果方程关于水平速度场ū=（u₁,u₂,0）的水平导数满足▽_hū=（₁ū,₁ū,0）∈L¹（0,T;B0∞,∞（R³））.则解（u,θ）可以连续到T₁>T.     

关于电磁场方程组解的W~(1,p)正则性研究

本文研究了R~3中有界区域Ω上的电磁场方程组弱解的W~(1,p)估计.该方程组来自于磁场所满足的稳态麦克斯韦方程组.在假定系数矩阵的逆属于VMO空间的条件下,利用R~3中向量场的旋度和散度的性质,将该方程组转化为标量椭圆型方程组,从而根据椭圆型方程组的正则性理论,得到解的W~(1,p)估计,其中1

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Accretivity of the General Second Order Linear Differential Operator

For the general second order linear differential operator ■ with complex-valued distributional coefficients a_(jk), b_j, and c in an open set Ω ? R~n(n ≥ 1), we present conditions which ensure that-L~0 is accretive, i.e., Re-L_0φ, φ ≥ 0 for all φ∈ C_0~∞(Ω).     

On n-widths of a Sobolev function class in Orlicz spaces

This paper considers the problem of n-widths of a Sobolev function class Ωr∞ determined by Pr(D) = Dσ∏lj =1(D2- t2jI) in Orlicz spaces. The relationship between the extreme value problem and width theory is revealed by using the methods of functional analysis. Particularly, as σ = 0 or σ = 1, the exact values of Kolmogorov's widths, Gelfand's widths, and linear widths are obtained respectively, and the related extremal subspaces and optimal linear operators are given.     

四维空间中一类带奇异位势的非局部临界指数问题的正解

本文研究了如下非局部临界指数问题{(-a+b∫_Ω|?_u|~2dx)△u=μu~3+λ|x|β~/u~(q-1),x∈Ω,x∈δΩ,其中Ω?R~4是一个有界光滑区域且0∈Ω,a≥0,b,λ,μ0,1q2,0β2.利用变分方法,我们获得了一些存在性与多重性结果.