带阻尼项的二维 Navier-Stokes 方程的 全局吸引子

考虑了带阻尼的二维Navier-Stokes方程在全空间中解的长时间行为,证明了相应的涡量方程存在(L~2(R~2),H~1(R~2))-吸引子.     

非线性应变波方程与薛定谔方程耦合组在R3上的极大吸引子

本文研究了非线性应变波方程与薛定谔方程耦合组在？3上的柯西问题与极大吸引子存在性.     

一类带奇异项的双调和方程的多解性

利用纤维方法及亏格理论对一类带奇异项的双调和方程进行了研究,证明了方程在两种不同情形下解的存在性及多解性.     

The Willmore functional and the containment problem in R4

Let ∑ be a convex hypersurface in the Euclidean space R4 with mean curvature H. We obtain a geometric lower bound for the Willmore functional f∑ H2dσ. This bound is an invariant involving the area of ∑, the volume and Minkowski quermassintegrals of the convex body that ∑bounds. We also obtain a sufficient condition for a convex body to contain another in the Euclidean space R4.     

A Multiplicity Result for a Singular and Nonhomogeneous Elliptic Problem in Rn

We establish sufficient conditions under which the quasilinear equation $$-div(|∇u|^{n-2}∇u)+V(x)|u|^{n-2}u=\frac{f(x,u)}{|x|^β}+εh(x) in \mathbb{R}^n,$$ has at least two nontrivial weak solutions in $W^{1,n} (\mathbb{R}^n)$ when ε > 0 is small enough, 0≤β < n, V is a continuous potential, f(x,u) behaves like $exp{γ|u|^{n/(n-1)}}$ as $|u|→∞$ for some γ > 0 and h≢ 0 belongs to the dual space of $W^{1,n} (\mathbb{R}^n)$.     

带有临界增长的Kirchhoff型问题的基态解

本文主要考虑如下Kirchhoff问题{-(a+b∫R_3|?u|~2dx)?u+u=f(x,u)+Q(x)|u|~4u,u∈H~1(R~3),其中a,b是正的常数.我们证明了基态解,即上述问题的极小能量解的存在性.同时,如果假定Q≡1,且h(x)满足一定的条件,可以证明下述问题{-(a+b∫R_3|?u|~2dx)?u+u=|u|~4u+h(x)u,u∈H~1(R~3)的基态解的存在性.     

Bochner-Martinelli Formula for Higher Spin Operators of Several R6 Variables*

The higher spin operator of several R⁶ variables is an analogue of the ■-operator in theory of several complex variables.The higher spin representation of so₆(C)is ⊙^kC⁴ and the higher spin operator D_k acts on ⊙^kC⁴-valued functions.In this paper,the authors establish the Bochner-Martinelli formula for higher spin operator D_k of several R⁶variables.The embedding of R⁶ⁿ into the space of co...     

Classification of complete 3-dimensional self-shrinkers in the Euclidean space R4

In this paper,we completely classify 3-dimensional complete self-shrinkers with the constant norm S of the second fundamental form and the constant f₃ in the Euclidean space R⁴,where h_ij’s are components of the second fundamental form,S=∑_i,j h_ij² and f₃=∑_i,j,k h_ijh_jkh_ki.     

R4(C)中平均曲率方向平行的Bonnet曲面

本文利用[5]的方法,对R⁴(C)中平均曲率方向平行的Bonnet曲面引入了半测地等温 参数,并给出了一个分类结果.     

Fractional Fourier transform on R2 and an application

We focus on the L^p(R²) theory of the fractional Fourier transform (FRFT) for 1 ≤ p ≤ 2. In L¹(R²), we mainly study the properties of the FRFT via introducing the two-parameter chirp operator. In order to get the point-wise convergence for the inverse FRFT, we introduce the fractional convolution and establish the corresponding approximate identities. Then the well-defined inverse FRFT is given via approximation by suitable means, such as fractional Gauss means and Able means. Furthermore, if the signal F_α,βf is received, we give the process of recovering the original signal f with MATLAB. In L²(R²), the general Plancherel theorem, direct sum decomposition, and the general Heisenberg inequality for the FRFT are obtained.     

含临界参数及临界指数的非线性双调和方程的多解问题(英文)

By using variational method, the multiplicity of solutions for nonlinear biharmonic equation involving critical parameter and critical exponent are established.     

Multiplicity Result for a Scalar Field Equation on Compact Surfaces

Abstract

We consider a scalar field equation on compact surfaces which has variational structure. When the surface is a torus and a physical parameter ρ belongs to (8π,4π²) we show under some extra assumptions that, besides a local minimum, the functional admits at least other two saddle points.     

Darboux transformation and solitons of Yang-Mills-Higgs equations in R2,1

The Darboux transformations for soliton equations are applied to the Yang-Mills-Higgs equations. New solutions can be obtained from a known one via universal and purely algebraic formulas. SU(N) soliton solutions are constructed with explicit formulas. The interaction of solitons is described by the splitting theorem: each p-soliton is splitting into p single solitons asymptotically as t → ±∞. The main contents of this work were reported at the Moshé Flato memorial conference (Dijon, September 2000) and W. L. Chow and K. T. Chen memorial conference (Tianjin, October 2000).     

丢番图方程aX~4-bY~2=1

应用Thue-Siegel方法,我们证明:对任意正整数a,b,不定方程aX~4-bY~2=1至多只有两组正整数解(X,Y),这证实了Walsh提的一个猜测.     

R2,1中类时Weingarten曲面的B?cklund变换

本文利用平行变换，得到３维Ｍｉｎｋｏｗｓｋｉ空间Ｒ^2,1中的Ｋ－２ｍＨ＋ｍ²－ｌ²＝０（Ｈ²＞ Ｋ，０≤ｍ＜ｌ）类时Ｗｅｉｎｇａｒｔｅｎ曲面的Ｂ?ｃｋｌｕｎｄ定理．     

LSC(X,R*)上Way-Below关系的一个注

本文指出文献［１］中关于下半连续函数格ＬＳＣ（Ｘ,Ｒ^*）上Ｗａｙ－ｂｅｌｏｗ关系的一个结论是错误的,并且给出一个修正的命题及其证明．     

R3中一类Kirchhoff型方程变号解的存在性及集中性

本文研究了一类Kirchhoff型方程。利用极大极小原理及惩罚函数方法,证明了上述方程变号解的存在性及集中性,我们的结果推广了文献[4]的结果。     

丢番图方程X~2-(a~2+1)Y~4=3-4a

设a≥2是正整数.本文证明了:当a=2时,方程X~2一(a~2+1)Y~4=3-4a仅有正整数解(X,Y)=(20,3);当a=3时,该方程仅有2组互素的正整数解(X,Y)=(1,1)和(79,5);当a≥4且4a+1非平方数时,该方程最多有4组互素的正整数解(X,Y);当a≥4且4a+1为平方数时,该方程最多有5组互素的正整数解(X,Y).