带有线性记忆的波方程在R~n上的时间依赖吸引子

本文基于Conti M,Di Plinio F等人提出的关于时间依赖全局吸引子的概念,研究了无界域上带有线性记忆的波方程解的长时间行为.利用尾部估计和压缩函数的方法证明了过程的渐近紧性,进而获得了H~1(R~n)×L~2(R~n)×L_u~2(R~+;H~1(R~n))上时间依赖吸引子的存在性.     

向量值哈代-洛伦茨空间

对于1r∞与巴拿哈空间B=L~r(Ω,F,μ),我们研究了欧几里得空间R~n上B-值缓分布构成的哈代-洛伦茨空间H~(p,q)(R~n,B)及哈代-洛伦茨空间之间的内插,其中0p∞和0q≤∞,获得了H~(p,q)(R~n,B)的一系列等价的刻画及其原子分解.若Ω={1},则H~(p,q)(R~n,B)=H~(p,q)(R~n)是经典的情形;若Ω=Z是整数集且μ是Z上的计数测度并且r=2,0p∞及q=∞,则H~(p,q)(R~n,B)=H~(p,∞)(R~n,e~2)转化为Grafakos和He在文[Weak Hardy spaces,Preprint,2014]中讨论的情形.     

Anisotropic Hardy-Lorentz spaces and their applications

Let p ∈(0, 1], q ∈(0, ∞] and A be a general expansive matrix on Rn. We introduce the anisotropic Hardy-Lorentz space H~(p,q)_A(R~n) associated with A via the non-tangential grand maximal function and then establish its various real-variable characterizations in terms of the atomic and the molecular decompositions, the radial and the non-tangential maximal functions, and the finite atomic decompositions. All these characterizations except the ∞-atomic characterization are new even for the classical isotropic Hardy-Lorentz spaces on Rn.As applications, we first prove that Hp,q A(Rn) is an intermediate space between H~(p1,q1)_A(Rn) and H~(p2,q2)_A(R~n) with 0 p1 p p2 ∞ and q1, q, q2 ∈(0, ∞], and also between H~(p,q1)_A(Rn) and H~(p,q2)_A(R~n) with p ∈(0, ∞)and 0 q1 q q2 ∞ in the real method of interpolation. We then establish a criterion on the boundedness of sublinear operators from H~(p,q)_A(R~n) into a quasi-Banach space; moreover, we obtain the boundedness of δ-type Calder′on-Zygmund operators from H~(p,∞)_A(R~n) to the weak Lebesgue space L~(p,∞)(R~n)(or to H~p_A(R~n)) in the ln λcritical case, from H~(p,q)_A(R~n) to L~(p,q)(R~n)(or to H~(p,q)_A(R~n)) with δ∈(0,(lnλ)/(ln b)], p ∈(1/(1+,δ),1] and q ∈(0, ∞], as well as the boundedness of some Calderon-Zygmund operators from H~(p,q)_A(R~n) to L~(p,∞)(R~n), where b := | det A|,λ_:= min{|λ| : λ∈σ(A)} and σ(A) denotes the set of all eigenvalues of A.     

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.     

数学学报英文版Vol．23(2007)，No．9论文摘要

<正>Parametrized Area Integrals on Hardy Spaces and Weak Hardy Spaces Yong DING Shan Zhen LU Qing Ying XUE In this paper,the authors prove that ifΩsatisfies a class of thc integral Dini condition,then the parametrized area integralμ_(Ω,S)~ρis a bounded operator from the Hardy space H~1(R~n)to L~1(R~n)and from the weak Hardy space H~(1,∞)(R~n)to L~(1,∞)(R~n),respectively.As corollaries of the above results,it is shown thatμ_(Ω,S)~ρis also an operator of weak type(1,1)and of type(p,p)for 1相似文献   

L~2(R~n)空间上的不确定原理

该文主要研究L~2(R~n)空间上的不确定原理,将时频分析中信号的时间和Fourier的频率定义推广到了L~2(R~n)空间上.得到了L~2(R~n)空间上一个更强形式的不确定原理.并得到了不确定原理等式成立的条件.     

Hardy空间H~1(R)的紧小波框架系数刻画

利用标准正交小波基下函数的展开系数来刻画Hardy空间H~1(R)已经得到了很好的证明.该文利用紧小波框架与标准正交小波基的关系及其性质,给出了Hardy空间H~1(R)在紧小波框架下函数展开系数的一个刻画.     

无界区域非自治随机sine-Gordon方程的D-周期吸引子

研究非自治随机sine-Gordon方程所生成的随机动力系统φ的D-周期吸引子的存在唯一性.运用一致估计得到了D的D-吸收集的存在性,并用分解技巧,证明了φ的渐近紧性,建立了动力系统φ在H~1(R~n)×L~2(R~n)中的D-周期吸引子的存在唯一性.当非自治外力项具有周期性时,该D-吸引子也呈现相同的周期性.     

奇异积分算子在 H~1(R_+~2×R_+~2)上的有界性

本文用 H~1(R_+~2×R_+~2)函数原子分解的方法,证明由 R.Fefferman 及 E.M.Stein 在[1]、[2]中引进的乘积空间上的奇异积分算子,当核函数 K(x_1,x_2)满足的条件中的 η>1/2时,在 H~2(R_+~2×R_+~2)上是有界的.即存在与 f 无关的常数 C,使‖K*f‖_H~1≤C‖f‖_(H~1).     

R~n上p(x)-Laplace型椭圆问题的无穷多解

讨论了涉及一般散度型椭圆算子(p(x)-Laplace算子为其特例)非线性偏微分方程的弱解存在性和多解性问题,假定非线性项f_1,f_2其中之一是超线性的,且满足Ambrosetti-Rabinowitz条件,另一项是次线性的.所采用的方法依赖于变指数Sobolev空间W~(1,p(x))(R~n)理论.主要结果的证明基于喷泉定理和对偶喷泉定理.     

LONG-TIME DYNAMICS OF THE STRONGLY DAMPED SEMILINEAR PLATE EQUATION IN R~N

We investigate the initial-value problem for the semilinear plate equation containing localized strong damping, localized weak damping and nonlocal nonlinearity. We prove that if nonnegative damping coefficients are strictly positive almost everywhere in the exterior of some ball and the sum of these coefficients is positive a.e. in R~n, then the semigroup generated by the considered problem possesses a global attractor in H~2(R~n) × L~2(R~n).We also establish the boundedness of this attractor in H~3(R~n) × H~2(R~n).     

HILBERT TRANSFORHS FOR CONVEX CURVES IN R~n

Let г:[-1,1]→R~n be a curve in R~n,n≧2,with г(0)=0.To г we associate the Hilbert transform operator H defined bythe principal value integralHf(x)=P.V.f(x-г(t))(dt)/t(x∈R~n),(1)where f is an arbitrary function in an appropriate class.A pro-     

Real-variable characterizations of anisotropic product Musielak-Orlicz Hardy spaces

Let A :=(A_1, A_2) be a pair of expansive dilations and φ : R~n×R~m×[0, ∞) → [0, ∞) an anisotropic product Musielak-Orlicz function. In this article, we introduce the anisotropic product Musielak-Orlicz Hardy space H~φ_A(R~n× R~m) via the anisotropic Lusin-area function and establish its atomic characterization, the g-function characterization, the g_λ~*-function characterization and the discrete wavelet characterization via first giving out an anisotropic product Peetre inequality of Musielak-Orlicz type. Moreover, we prove that finite atomic decomposition norm on a dense subspace of H~φ_A(R~n× R~m) is equivalent to the standard infinite atomic decomposition norm. As an application, we show that, for a given admissible triplet(φ, q, s), if T is a sublinear operator and maps all(φ, q, s)-atoms into uniformly bounded elements of some quasi-Banach spaces B, then T uniquely extends to a bounded sublinear operator from H~φ_A(R~n× R~m) to B. Another application is that we obtain the boundedness of anisotropic product singular integral operators from H~φ_A(R~n× R~m) to L~φ(R~n× R~m)and from H~φ_A(R~n×R~m) to itself, whose kernels are adapted to the action of A. The results of this article essentially extend the existing results for weighted product Hardy spaces on R~n× R~m and are new even for classical product Orlicz-Hardy spaces.     

Sobolev方程一个新的H~1-Galerkin混合有限元分析

研究了Sobolev方程的H~1-Galerkin混合有限元方法.利用不完全双二次元Q_2~-和一阶BDFM元,建立了一个新的混合元模式,通过Bramble-Hilbert引理,证明了单元对应的插值算子具有的高精度结果.进一步,对于半离散和向后欧拉全离散格式,分别导出了原始变量u在H~1-模和中间变量p在H(div)-模意义下的超逼近性质.     

R~n上的调和函数是 C~n上的解析函数

令Δ_n=sum from j=1 to (?)((?)~2)/((?)x_j~2)为 R~n 上的 Laplace 算子,设Δ_n~ku(x_1,…,x_n)=0,(x_1…,x_n)∈R~n,k≥1,即 u(x_1,…,x_n)是 k 级调和函数。早已知道,u 是实解析函数,因而可延拓成 R~n 在 C~n 的一个邻域的解析函数 u(z_1,…,z_n)(可参看[1])。在这篇短文中,我们将证明 u 是整函数,即可延拓成 C~n 上的解析函数(定理1)。设 u(x_1,…,x_n)是 R~n 上的调和函数,则因 u(z_1,…,z_n)是 C~n 上的解析函数,故sum from j=1 to n ((?)~u)/((?)z_j~2)是 C~n 上的解析函数,因它在 R~n 上为零,故在 C~n 上为零。因此,我们的结果表明R~n 上的调和函数空间与 C~n 上满足:sum from j=1 to n ((?)~2u)/((?)z_j~2)=0的解析函数 u(我们不妨称之为复调和函数)的空间是一致的。同理 R~(n 1)上对最后一个变量为偶的调和函数空间与 C~(n 1)上对最     

对角形椭圆组H~1∩L~(n(r-1)/(2-r))弱解的处处正规性

考虑对角性拟线性椭园组 -D_β(A~(αβ)(x,u)D_αu~i)=B_i(x,u,Du),x∈Ω,i=1,…,N。 (1)这里Ω是R~n中的有界域。重复上、下标α、β表示从1到n求和,D_u={D_αu~i|α=1,…     

电报方程的H~1-Galerkin非协调混合有限元逼近

在几乎均匀矩形剖分下取双线性Q_(11)元和类Wilson元为逼近空间,研究了一类电报方程的H~1-Galerkin非协调混合有限元方法.利用单元的特殊性质,积分恒等式和平均值技巧,在不需要验证LBB相容性条件及抛弃传统的Ritz投影的情形下,得到了半离散和全离散格式下原始变量及流量分别在H~1模和H(div,Ω)模意义下的超逼近性质.进一步地,借助插值后处理技术,导出了相应的整体超收敛结果.     

NON-RADIAL SOLUTIONS OF A SEMILINEAR ELLIPTIC EQUATION

In this paper we consider the eigenvalue problem of semilinear elliptic equation in R~n(n≥~3)-△u+α(x)u=λf(x, u), u∈H~1(R~n).     

半线性伪双曲方程最低阶的H~1-Galerkin混合元方法

研究在半离散和全离散格式下,半线性伪双曲方程最低阶的协调H~1-Galerkin混合有限元逼近.具体地,用双线性元逼近原始变量u,用零阶Raviart-Thomas(R-T)元逼近流量p.首先通过泰勒展式和积分恒等式技巧得到了p的一个新的误差估计式.然后,导出了u在H~1模和p在H(div;Ω)模意义下的超逼近性质,改进了已有文献的结果.     

On the Harmonic Maps from R~2 into H~2

In [1], Eells and Lemaire arise a problem of the existence of harmonic mapsfrom R~2 into H~2. Using the fact that H~2 = {x = (x_1,x_2,x_3)∈R~(2,1):x_1~2+x_2~2- x_3~2= - 1,x_3≥1}, we can rewrite maps into H~2 as maps from R~2 into R~(2,1) valued in H~2.Then, we can write the equation of harmonic map as follows:φ:R~2→H~2 R~(2,1),