HARMONIC MAPS AND FUNDAMENTAL GROUPS OF NONPOSITIVELY CURVEDRIEMANNIAN MANIFOLDS

Ｔｈｅｔｈｅｏｒｙｏｆｈａｒｍｏｎｉｃｍａｐｓｉｎｔｏａｃｏｍｐａｃｔｎｏｎｐｏｓｉｔｉｖｅｌｙｃｕｒｖｅｄｔａｒｇｅｔｉｓｗｅｌｌ-ｄｅｖｅｌｏｐｅｄ.ＴｈｅｅｘｉｓｔｅｎｃｅｏｆｓｕｃｈａｈａｒｍｏｎｉｃｍａｐｐｉｎｇｉｎａｎｙｇｉｖｅｎｈｏｍｏｔｏｐｙｃｌａｓｓｉｓｅｓｔａｂｌｉｓｈｅｄｂｙＥｅｌｌｓａｎｄＳａｍｐｓｏｎ[1]ｉｎ1964,ａｎｄｌａｔｅｒ,ｉｎ1967,Ｈａｒｔｍａｎ[2]ｇａｖｅａｐｒｏｏｆｏｆｔｈｅｕｎｉｑｕｅｎｅｓｓ.Ｔｈｉｓｐａｐｅｒｉｓｄｅｖｏｔｅｄｔｏａｎａｐｐｌｉｃａｔｉｏｎｏ…     

Toeplitz Algebras on Dirichlet Spaces

§ 1.Ｉｎｔｒｏｄｕｃｔｉｏｎ　ＬｅｔＢｎｂｅｔｈｅｕｎｉｔｂａｌｌｉｎＣｎ,ａｎｄｄｖｔｈｅｎｏｒｍａｌｉｚｅｄＬｅｂｅｓｇｕｅｍｅａｓｕｒｅｏｎＢｎ.ＳｏｂｏｌｅｖｓｐａｃｅＬ2 ,1ｉｓｔｈｅｓｐａｃｅｏｆｆｕｎｃｔｉｏｎｓｕ :Ｂｎ→Ｃ ,ｆｏｒｗｈｉｃｈｔｈｅｎｏｒｍ‖ｕ‖1/ 2 =∑ｎｉ=1 ∫Ｂｎ ｕ ｚｉ(ｚ) 2 + ｕ ｚｉ(ｚ) 2 ｄν1/ 2 <∞ .Ｌ2 ,1/ 2 ｉｓａＨｉｌｂｅｒｔｓｐａｃｅｗｉｔｈｔｈｅｉｎｎｅｒｐｒｏｄｕｃｔ〈ｕ ,ｖ〉1/ 2 =∑ｎｉ=1〈 ｕ ｚｉ, ｕ ｚｉ〉Ｌ2 (ｄν) +∑ｎｉ=1〈 ｕ ｚｉ, ｕ …     

The Minimal Solutions of Boolean Matrix-equation A~k=J

ＬｅｔＢ ={0 ,1 }ｂｅｔｈｅｕｓｕａｌｂｉｎａｒｙＢｏｏｌｅａｎａｌｇｅｂｒａ .ＴｈｅｍａｔｒｉｃｅｓｏｖｅｒＢａｒｅｃａｌｌｅｄＢｏｏｌｅａｎｍａｔｒｉｃｅｓ.Ａｎｎ×ｎＢｏｏｌｅａｎｍａｔｒｉｘＡｉｓｃａｌｌｅｄａｐｒｉｍｉｔｉｖｅｍａｔｒｉｘｉｆｔｈｅｒｅｅｘｉｓｔｓａｐｏｓｉｔｉｖｅｉｎｔｅｇｅｒｋｓｕｃｈｔｈａｔＡｋ=Ｊ (ｗｈｅｒｅＪｉｓｔｈｅｕｎｉｖｅｒｓａｌｍａｔｒｉｘ) .ＴｈｅｌｅａｓｔｓｕｃｈｋｉｓｃａｌｌｅｄｔｈｅｅｘｐｏｎｅｎｔｏｆＡ ,ｄｅｎｏｔｅｄｂｙγ(Ａ) .　Ｉｎ ｐｒｏｊ…     

EXISTENCE TIME FOR THE SEMILINEAR SCHR?DINGER EQUATION

§ 1　ＩｎｔｒｏｄｕｃｔｉｏｎＩｎｔｈｉｓｐａｐｅｒ ,ｗｅｗｉｌｌｃｏｎｓｉｄｅｒｔｈｅｓｅｍｉｌｉｎｅａｒＳｃｈｒ ｄｉｎｇｅｒｅｑｕａｔｉｏｎｉｎｏｎｅｓｐａｃｅｄｉｍｅｎｓｉｏｎｏｆｔｈｅｔｙｐｅｕｔ-ｉｕｘｘ =Ｆ(ｕ) .　 (ｘ ,ｔ)∈Ｒ×Ｒ+,( 1 )ｕ(ｘ ,0 ) =ｕ0 ,　ｘ ∈Ｒ ,( 2 )ｗｈｅｒｅｕ =ｕ(ｘ ,ｔ)ｉｓｃｏｍｐｌｅｘ ｖａｌｕｅｄｆｕｎｃｔｉｏｎ ,ａｎｄＦｉｓａｓｍｏｏｔｈｆｕｎｃｔｉｏｎｏｆｕｓｕｃｈｔｈａｔ｜Ｆ(ｕ) ｜ =Ｏ( ｜ｕ｜α+1)ｆｏｒ |ｕ|ｓｕｆｆｉｃｉｅｎｔｌｙｓｍａｌｌａｎｄ…     

ERROR ESTIMATES OF GALERKIN METHOD FOR HIGH DIMENSIONAL STEADY STATE KURAMOTO-SIVASHINSKY EQUATION

1　ＩｎｔｒｏｄｕｃｔｉｏｎＩｎｔｈｅｐａｐｅｒ,ｗｅｃｏｎｓｉｄｅｒｔｈｅｆｏｌｌｏｗｉｎｇＫ -Ｓｅｑｕａｔｉｏｎ (ｆｏｒｅｘａｍｐｌｅ,ｓｅｅ [1 ] -[3]ａｎｄｒｅｆｅｒｅｎｃｅｓｔｈｅｒｅｉｎ)ｉｎｓｐａｃｅｄｉｍｅｎｓｉｏｎ≥ 3:Δ2 ｕ+Δｕ+ 12 ｜ ｕ｜2 -12 Ｎｉ=1ｌｉ∫ＩＮ｜ ｕ｜2 ｄｘ=0 ,ｘ∈ＩＮ,(1 )ｗｉｔｈＬ ｐｅｒｉｏｄｉｃｂｏｕｎｄａｒｙｃｏｎｄｉｔｉｏｎｓ,Ｌ =(ｌ1,ｌ2 ,… ,ｌＮ)ｂｅｉｎｇｔｈｅｓｉｚｅｏｆａｔｙｐｉｃａｌｐａｔｔｅｒｎｃｅｌｌ (ａｌｓｏｂｉｆｕｒｃａｔｉｏｎｐａｒ…     

Monotone Iterative Method for Set valued Quasilinear Elliptic Boundary Value Problems

§ 1.ＰｒｏｂｌｅｍａｎｄＡｓｓｕｍｐｔｉｏｎｓ　Ｔｈｉｓｐａｐｅｒｄｅａｌｓｗｉｔｈｔｈｅｓｏｌｕｔｉｏｎｓｏｆｔｈｅｆｏｌｌｏｗｉｎｇｄｉｆｆｅｒｅｎｔｉａｌｉｎｃｌｕｓｉｏｎｐｒｏｂｌｅｍ :Ａｕ∈ｆ(ｘ,ｕ) ,ｘ∈Ω ;ｕ=0 ,ｘ∈ Ω ,(1 )ｗｈｅｒｅＡｕ(ｘ) =-∑Ｎｉ=1Ｄｉ[ａｉ(ｘ ,Ｄｕ(ｘ) ) ] ,Ω ＲＮｉｓａｂｏｕｎｄｅｄｄｏｍａｉｎｗｉｔｈｐｉｅｃｅｗｉｓｅＬｉｐｓｃｈｉｔｚｂｏｕｎｄａｒｙ Ω ,Ｄｕ =(Ｄ1ｕ ,Ｄ2 ｕ ,… ,ＤＮｕ) ,Ｄｉｕ = ｕ ｘｉ,ｉ=1 ,2 ,… ,Ｎ ,ａｎｄｆ:Ω×Ｒ→ 2 Ｒｉｓａ…     

Strong and Weak Property of Travelling Wave Front Solutions for a Class of Degenerate Parabolic Equations

§ 1.Ｉｎｔｒｏｄｕｃｔｉｏｎ　Ｉｎｔｈｉｓｐａｐｅｒｗｅｓｔｕｄｙｔｈｅｓｔｒｏｎｇａｎｄｗｅａｋｐｒｏｐｅｒｔｙｏｆｔｒａｖｅｌｌｉｎｇｗａｖｅｆｒｏｎｔｓｏｌｕｔｉｏｎｓ (ＴＷＦＳ)ｆｏｒｔｈｅｆｏｌｌｏｗｉｎｇｄｅｇｅｎｅｒａｔｅｐａｒａｂｏｌｉｃｅｑｕａｔｉｏｎ : ψ(ｕ) ｔ = 2 ｕ ｘ2 +ｆ(ｕ) . ( 1 .1 )　Ｗｅｌｏｏｋｆｏｒｓｏｌｕｔｉｏｎｓｏｆ( 1 .1 )ｏｆｔｈｅｆｏｒｍｕ(ｘ,ｔ) =ｑ(ｘ -ｃｔ) (ｃｉｓｔｈｅｗａｖｅｓｐｅｅｄ) ,ｗｈｅｒｅｑ( ξ) ,ξ=ｘ -ｃｔ,ａｓａｆｕｎｃｔｉｏｎｏｆａｓ…     

Characterization of V-modules by Relative Quasi-continuity

§ 1.Ｉｎｔｒｏｄｕｃｔｉｏｎ　ＬｅｔＲｂｅａｒｉｎｇｗｉｔｈｉｄｅｎｔｉｔｙａｎｄＭａｌｅｆｔＲ ｍｏｄｕｌｅ .ＡｌｅｆｔＲ ｍｏｄｕｌｅＵｉｓｃａｌｌｅｄＭ ｉｎｊｅｃｔｉｖｅｉｆ,ｆｏｒｅｖｅｒｙｓｕｂｍｏｄｕｌｅＮｏｆＭａｎｄｅｖｅｒｙｈｏｍｏｍｏｒｐｈｉｓｍ :Ｎ→Ｕ ,ｃａｎｂｅｅｘｔｅｎｄｅｄｔｏａｈｏｍｏｍｏｒｐｈｉｓｍ ψ :Ｍ→Ｕ .ＡｌｅｆｔＲ ｍｏｄｕｌｅＭｉｓｃａｌｌｅｄａＶ ｍｏｄｕｌｅｂｙＲａｍａｍｕｒｔｈｉ[1]ａｎｄＴｏｍｉｎａｇａ[2 ](ａｎｄｉｓｃａｌｌｅｄａｃｏ ｓｅｍｉｓ…     

SOME EXTENSIONS OF PALEY-WIENNER THEOREM

§１．ＩｎｔｒｏｄｕｃｔｉｏｎａｎｄＭｏｔｉｖａｔｉｏｎＴｈｅｃｌａｓｉｃａｌＳｈａｎｎｏｎ’ｓｓａｍｐｌｉｎｇｔｈｅｏｒｅｍｈｏｌｄｓｄｕｅｔｏｔｈｅｆｏｌｏｗｉｎｇｔｗｏｒｅａｓｏｎｓ：（ｉ）ｑ（ｔ，ｕ）＝ｓｉｎπ（ｔ－ｕ）π（ｔ－ｕ）ｉｓｔｈｅ...     

Heisenberg群上四阶非线性次椭圆方程的可解性

§ 1　ＩｎｔｒｏｄｕｃｔｉｏｎＷｅｃｏｎｓｉｄｅｒｔｈｅｆｏｕｒｔｈｏｒｄｅｒｓｅｍｉｌｉｎｅａｒｓｕｂｅｌｌｉｐｔｉｃｂｏｕｎｄａｒｙｖａｌｕｅｐｒｏｂｌｅｍΔ2 Ｈｕ +ｃΔＨｕ =ｆ( (ｚ ,ｔ) ,ｕ)　ｉｎＤ ,ｕ｜Ｄ =ΔＨｕ｜Ｄ =0 ,( 1 .1 )ｗｈｅｒｅＤｉｓａｂｏｕｎｄｅｄｏｐｅｎｓｕｂｓｅｔｏｆｔｈｅＨｅｉｓｅｎｂｅｒｇｇｒｏｕｐＨｎａｎｄΔＨｉｓｔｈｅｓｕｂｅｌｌｉｐｔｉｃＬａｐｌａ ｃｉａｎｏｎＨｎ.ＷｅｒｅｃａｌｌｔｈａｔＨｎｉｓｔｈｅＬｉｅｇｒｏｕｐｗｈｏｓｅｕｎｄｅｒｌｙｉｎｇｍａｎｉ…     

Existence and nonexistence of global solutions for nonlinear evolution equation of fourth order

Abstract. The existence and uniqueness of classical global solutions and the nonexistence of global solutions to the first boundary value problem and the second boundary value problem for the equation ua-a1uxx-a2uxxl-a3uxxu=ψ(ux)x are proved     

Solutions for the prescribing mean curvature equation

By variational methods, for a kind of Yamabe problem whose scalar curvature vanishes in the unit ball BN and on the boundary S^N-1 the mean curvature is prescribed, we construct multi-peak solutions whose maxima are located on the boundary as the parameter tends to 0^＋ under certain assumptions. We also obtain the asymptotic behaviors of the solutions.     

On Eigenvalue Intervals and Eigenfunctions of Nonresonance Singular Dirichlet Boundary Value Problems

In this paper we shall consider the nonresonance Dirichlet boundary value problemwhere λ>0 is a parameter, p>0 is a constant. Intervals of A are determined to ensure the existence of a nonnegative solution of the boundary value problem. For λ=1, we shall also offer criteria for the existence of eigenfunctions. The main results include and improve on those of [2,4,6,8].     

Asymptotic Smoothing Effect of Solutions to Davey-Stewartson Systems on the Whole Plane

This paper discusses the Cauchy problem of elliptic-elliptic-type Davey-Stewartson systems with zero-order dissipation on R^2. Making use of the Fourier spectral projector, together with a long time comparison between the solutions to the Davey-Stewartson systems and to an auxiliary problem, we prove that the global attractor in H^1 （R^2） for the addressed Davey-Stewartson systems is a compact subset of H^2（R^2）, and thus reveal the asymptotic smoothing effect of the solutions for the systems.     

控制调幅波耗散方程的整体吸引子的有限维

§ 1　IntroductionIn this paperwe shall study successively the equationsgiven in [1 ,2 ] .Underthe sameconditions(i.e.the initial boundary value and other conditions) as in[1 ,2 ] ,the upperbound of the dimension of their global attractor will be obtained respectively.Theexistence of the global attractors have been proved in[1 ,2 ] .This paper is organized as follows:In§ 2 ,we derive the Frechet differential of the solution operator s(t) for theperturbed system.In§ 3 ,we show the finite di…     

Properties of positive solutions to a nonlinear parabolic problem

This paper deals with the properties of positive solutions to a quasilinear parabolic equation with the nonlinear absorption and the boundary flux. The necessary and sufficient conditions on the global existence of solutions are described in terms of different parameters appearing in this problem. Moreover, by a result of Chasseign and Vazquez and the comparison principle, we deduce that the blow-up occurs only on the boundary (?)Ω. In addition, for a bounded Lipschitz domainΩ, we establish the blow-up rate estimates for the positive solution to this problem with a= 0.     

Weak and Strong Solutions for the Stokes Approximation of Non-homogeneous Incompressible Navier-Stokes Equations

In this paper,the Dirichlet problem of Stokes approximate of non-homogeneous incompressibleNavier-Stokes equations is studied.It is shown that there exist global weak solutions as well as global andunique strong solution for this problem,under the assumption that initial density ρ_0(x)is bounded away from0 and other appropriate assumptions(see Theorem 1 and Theorem 2).The semi-Galerkin method is applied toconstruct the approximate solutions and a prior estimates are made to elaborate upon the compactness of theapproximate solutions.     

Blow-up of the Solution for a <Emphasis Type="Italic">p</Emphasis>-Laplacian Equation with Positive Initial Energy

In this paper we consider a p-Laplacian equation with the homogeneous Dirichlet boundary condition. We establish a blow-up result for certain solution with positive initial energy. This work was supported by the National Natural Science Foundation of China 10771032, and the Natural Science Foundation of Jiangsu province BK2006088.     

Boundary layer to a system of viscous hyperbolic conservation laws

In this paper, we investigate the large-time behavior of solutions to the initial-boundary value problem for n × n hyperbolic system of conservation laws with artificial viscosity in the half line （0, ∞）. We first show that a boundary layer exists if the corresponding hyperbolic part contains at least one characteristic field with negative propagation speed. We further show that such boundary layer is nonlinearly stable under small initial perturbation. The proofs are given by an elementary energy method.     

Existence of Single and Multiple Solutions for First Order Periodic Boundary Value Problems

This paper is devoted to the study ofthe existence of single and multiple positive solutions forthe first order boundary value problem x′= f(t,x),x(0)=x(T),where f ∈ C([0,T]×R).In addition,weapply our existence theorems to a class of nonlinear periodic boundary value problems with a singularity at theorigin.Our proofs are based on a fixed point theorem in cones.Our results improve some recent results in theliteratures.