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The Gleason's problem on mixed norm spaces in convex domains

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Science China Mathematics 2003年第6期46卷 827-837页

作者：胡璋剑 Department of Mathematics Huzhou Teachers College Huzhou 313000 China Let Ωbe a bounded convex domain with C2 boundary in C2 and for given 0 < p q ≤∞and normal weight function (r) let Hpq be the mixed norm space on Ω. In this paper we prove that the Gleason's problem (Ω a Hpq) is solvable for any fixed point a ∈Ω. While solving the Gleason's problem we obtain the boundedness of certain integral operator on Hpq.

Let Ω be a bounded convex domain with C2 boundary in Cn and for given 0 ＜ p, q ≤∞ and normal weight function ψ(r) let Hp,q,ψ be the mixed norm space on Ω. In this paper we prove that the Gleason's problem (Ω, a, Hp,q,ψ) is solvable for any fixed point a ∈Ω. While solving the Gleason's problem we obtain the boundedness of certain integral operator on H-p,q,ψ.

关键词： Gleason's problem, mixed norm space, normal weight.

A mixed finite element method for the unilateral contact problem in elasticity

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Science China Mathematics 2006年第4期49卷 513-524页

作者： HUA Dongying & WANG Lieheng The First Fundamental Department, Beijing Information Technology Institute, Beijing 100101, China Institute of Computational Mathematics and Scientific/Engineering Computing, Academy of Mathematics and System Sciences, Chinese Academy of Sciences, Beijing 100080, China The First Fundamental Department Beijing Information Technology Institute Beijing China Institute of Computational Mathematics and Scientific/Engineering Computing Academy of Mathematics and System Sciences Chinese Academy of Sciences Beijing China

In this paper, we provide a new mixed finite element approximation of the variational inequality resulting from the unilateral contact problem in elasticity. We use the continuous piecewise P2-P1 finite element to approximate the displacement field and the normal stress component on the contact region. Optimal convergence rates are obtained under the reasonable regularity hypotheses. Numerical example verifies our results.

关键词： unilateral contact problem, mixed finite element method.

Successive projection method for solving the unbalanced Procrustes problem

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Science China Mathematics 2006年第7期49卷 971-986页

作者： ZHANG Zhenyue DU Keqin Department of Mathematics Zhejiang University Hangzhou 310027 China Department of Mathematics Zhejiang University Hangzhou 310027 China

We present a successive projection method for solving the unbalanced Procrustes problem: given matrix A ∈ Rn×n and B ∈ Rn×κ, n ＞κ, minimize the residual ‖AQ - B‖F with the orthonormal constraint QTQ = Iκ on the variant Q ∈ Rn×κ. The presented algorithm consists of solving k least squares problems with quadratic constraints and an expanded balance problem at each sweep. We give a detailed convergence analysis. Numerical experiments reported in this paper show that our new algorithm is superior to other existing methods.

关键词： Procrustes problem, least squares problems, projection method, orthogonal constraint.

The generalized solution of ill-posed boundary problem

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Science China Mathematics 2006年第7期49卷 902-911页

作者： CAO Weiping & MA Jipu Department of Mathematics and Physics, Huaihai Institute of Technology, Lianyungang 222005, China Y. Y. Tseng Functional Analysis Research Centre, Harbin Normal University, Harbin 150080, China Department of Mathematics and Physics Huaihai Institute of Technology Lianyungang China Y. Y. Tseng Functional Analysis Research Centre Harbin Normal University Harbin China

In this paper, we define a kind of new Sobolev spaces, the relative Sobolev spaces Wk0,p(Ω, Σ). Then an elliptic partial differential equation of the second order with an ill-posed boundary is discussed. By utilizing the ideal of the generalized inverse of an operator, we introduce the generalized solution of the ill-posed boundary problem. Eventually, the connection between the generalized inverse and the generalized solution is studied. In this way, the non-instability of the minimal normal least square solution of the ill-posed boundary problem is avoided.

关键词： generalized inverse of operator, generalized solution, ill-posed boundary problem, elliptic partial differential equations of second order.

The Gleason's problem for some polyharmonic and hyperbolic harmonic function spaces

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Science China Mathematics 2006年第8期49卷 1128-1145页

作者： HU Zhangjian TANG Xiaomin Department of Mathematics Huzhou Teachers College Huzhou 313000 China Department of Mathematics Huzhou Teachers College Huzhou 313000 China

Let Ω(∈) Rn be a bounded convex domain with C2 boundary. For 0 ＜ p,q ≤∞ and a normal weight ψ, the mixed norm space Hp,q,ψk,(Ω) consists of all polyharmonic functions f of order k for which the mixed norm ||·||p,q,ψ＜∞.In this paper, we prove that the Gleason's problem (Ω,a,Hp,q,ψk) is always solvable for any reference point a ∈Ω. Also, the Gleason's problem for the polyharmonic ψ-Bloch (little ψ-Bloch) space is solvable. The parallel results for the hyperbolic harmonic mixed norm space are obtained.

关键词： Gleason's problem, mixed norm space, Bloch-type space.

Search-extension method for finding multiple solutions of nonlinear problem

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Science China Mathematics 2005年第6期48卷 721-734页

作者： CHEN Chuanmiao XIE Ziqing Institute of Computation Hunan Normal University Changsha 410081 China Institute of Computation Hunan Normal University Changsha 410081 China

Based on the eigensystem {λj, φj} of -△, the multiple solutions for nonlinear problem △u + f(u)= 0 in Ω,u = 0 on Ω are approximated. A new search-extension method (SEM) is proposed, which consists of three algorithms in three level subspaces.Numerical experiments for f(u)= u3 in a square and L-shape domain are presented. The results show that there exist at least 3κ - 1 distinct nonzero solutions corresponding to each κ-ple eigenvalue of -△ (Conjecture 1).

关键词： nonlinear problem, multiple solutions, search-extension method, conjecture.

On exponential sums over primes and application in Waring-Goldbach problem

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Science China Mathematics 2005年第6期48卷 785-797页

作者： REN Xiumin Department of Mathematics Shandong University Jinan 250100 China

In this paper, we prove the following estimateon exponential sums over primes: Let κ≥ 1, βκ = 1/2 + log κ/log 2, x ≥ 2 and α = a/q + λ subject to (a, q) = 1, 1 ≤ a ≤ q,and λ R. ThenΣx＜m≤2x∧(m)e(αmk)＜＜(d(q))βk(logx)c(x1/2√q(1+|λ|xk)+x4/5+x/√q(1+|λ|xk)).As an application, we prove that with at most O(N7/8+ε) exceptions, all positive integers up to N satisfying some necessary congruence conditions are the sum of three squares of primes. This result is as strong as what has previously been established under the generalized Riemann hypothesis.

关键词： exponential sums over primes, Waring-Goldbach problem, circle method.

The GPS-gravimetry boundary value problem

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Science China Earth Sciences 2005年第3期48卷 398-405页

作者： YU Jinhai1,2 & ZHANG Chuanding2 1. Institute of Geodesy and Geophysics of Chinese Academy of Sciences, Wuhan 430077, China 2. Zhengzhou Information Engineering University, Zhengzhou 450052, China Institute of Geodesy and Geophysics of Chinese Academy of Sciences Wuhan China Zhengzhou Information Engineering University Zhengzhou China

How to determine the earth’s external gravity field with the accuracy of O(T2) by making use of GPS data and gravity values measured on the earth’s surface is dealt with in this paper. There are two main steps: to extend these measured values on the earth’s surface onto the reference ellipsoid at first and then to seek for the integral solution of the external Neumann problem outside the ellipsoid. In addition, the corresponding judging criteria of accuracy to solve the GPS-gravity boundary value problem are established. The integral solution given in the paper not only contains all frequency-spectral information of the gravity field with the accuracy of O(T 2), but is also easily computed. In fact, the solution has great significance for both theory and prac- tice.

关键词： Neumann boundary value problem, integral solution, gravity disturbance.

Cauchy problem of semi-linear parabolic equations with weak data in homogeneous space and application to the Navier-Stokes equations

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Science China Mathematics 2003年第5期46卷 641-661页

作者：苗长兴 Institute of Applied Physics and Computational Mathematics Beijing 100088 China

In this paper we study the Cauchy problem for a class of semi-linear parabolic type equations withweak data in the homogeneous spaces. We give a method which can be used to construct local mild solutionsof the abstract Cauchy problem in Cσ,s,p and Lq([O, T);Hs,p) by introducing the concept of both admissiblequintuplet and compatible space and establishing time-space estimates for solutions to the linear parabolic typeequations. For the small data, we prove that these results can be extended globally in time. We also study theregularity of the solution to the abstract Cauchy problem for nonlinear parabolic type equations in Cσ,s,p. Asan application, we obtain the same result for Navier-Stokes equations with weak initial data in homogeneousSobolev spaces.

关键词： Cauchy problem, parabolic equation, Naiver-Stokes equation, admissible quintuplet,compatible space, time-space estimates.