Three-dimensional sets with small sumset

We describe the structure of three dimensional sets of lattice points, having a small doubling property. Let be a finite subset of ℤ³ such that dim = 3. If and , then lies on three parallel lines. Moreover, for every three dimensional finite set that lies on three parallel lines, if , then is contained in three arithmetic progressions with the same common difference, having together no more than terms. These best possible results confirm a recent conjecture of Freiman and cannot be sharpened by reducing the quantity υ or by increasing the upper bounds for .     

On nonexistence of Baras-Goldstein type without positivity assumptions for singular linear and nonlinear parabolic equations

The celebrated result by Baras and Goldstein (1984) established that the heat equation with the inverse square potential in the unit ball B ₁ ⊂ ℝ ^N , N ≥ 3, u _t = Δ _u + in B ₁ × (0,T), u|∂B ₁ = 0, in the supercritical range c > c _Hardy = does not have a solution for any nontrivial L ¹ initial data u ₀(x) ≥ 0 in B ₁ (or for a positive measure u ₀). More precisely, it was proved that a regular approximation of a possible solution by a sequence {u _n (x,t)} of classical solutions corresponding to truncated bounded potentials given by V(x) = ↦ V _n (x) = min{, n} (n ≥ 1) diverges; i.e., as n → ∞, u _n (x,t) → + ∞ in B ₁ × (0, T). Similar features of “nonexistence via approximation” for semilinear heat PDEs were inherent in related results by Brezis-Friedman (1983) and Baras-Cohen (1987). The main goal of this paper is to justify that this nonexistence result has wider nature and remains true without the positivity assumption on data u ₀(x) that are assumed to be regular and positive at x = 0. Moreover, nonexistence as the impossibility of regular approximations of solutions is true for a wide class of singular nonlinear parabolic problems as well as for higher order PDEs including, e.g., u _t = , and , N > 4. Dedicated to Professor S.I. Pohozaev on the occasion of his 70th birthday     

On the weighted elliptic problems involving multi-singular potentials and multi-critical exponents

Suppose Ω belong to R^N（N≥3） is a smooth bounded domain,ξi∈Ω,0〈ai〈√μ,μ：=（（N-1）/2）^2,0≤μi〈（√μ-ai）^2,ai〈bi〈ai＋1 and pi：=2N/N-2（1＋ai-bi）are the weighted critical Hardy-Sobolev exponents, i = 1, 2,..., k, k ≥ 2. We deal with the conditions that ensure the existence of positive solutions to the multi-singular and multi-critical elliptic problem ∑i=1^k（-div（｜x-ξi｜^-2ai△↓u）-μiu/｜x-ξi｜^2（1＋ai）-u^pi-1/｜x-ξi｜^bipi）=0with Dirichlet boundary condition, which involves the weighted Hardy inequality and the weighted Hardy-Sobolev inequality. The results depend crucially on the parameters ai, bi and #i, i -- 1, 2,..., k.     

Local and Global Existence of Solutions to Initial Value Problems of Nonlinear Kaup-Kupershmidt Equations

This paper is devoted to studying the initial value problems of the nonlinear Kaup Kupershmidt equations δu/δt ＋ α1 uδ^2u/δx^2 ＋ βδ^3u/δx^3 ＋ γδ^5u/δx^5 = 0, （x,t）∈ E R^2, and δu/δt ＋ α2 δu/δx δ^2u/δx^2 ＋ βδ^3u/δx^3 ＋ γδ^5u/δx^5 = 0, （x, t） ∈R^2. Several important Strichartz type estimates for the fundamental solution of the corresponding linear problem are established. Then we apply such estimates to prove the local and global existence of solutions for the initial value problems of the nonlinear Kaup- Kupershmidt equations. The results show that a local solution exists if the initial function u0（x） ∈ H^s（R）, and s ≥ 5/4 for the first equation and s≥301/108 for the second equation.     

Binormal operator and *-Aluthge transformation

Let T = U｜T｜ be the polar decomposition of a bounded linear operator T on a Hilbert space. The transformation T = ｜T｜^1/2 U｜T｜^1/2 is called the Aluthge transformation and Tn means the n-th Aluthge transformation. Similarly, the transformation T（＊）=｜T＊｜^1/2 U｜T＊｜＆1/2 is called the ＊-Aluthge transformation and Tn^（＊） means the n-th ＊-Aluthge transformation. In this paper, firstly, we show that T（＊） = UV｜T^（＊）｜ is the polar decomposition of T（＊）, where ｜T｜^1/2 ｜T^＊｜^1/2 = V｜｜T｜^1/2 ｜T^＊｜^1/2｜ is the polar decomposition. Secondly, we show that T（＊） = U｜T^（＊）｜ if and only if T is binormal, i.e., [｜T｜, ｜T^＊｜]=0, where [A, B] = AB - BA for any operator A and B. Lastly, we show that Tn^（＊） is binormal for all non-negative integer n if and only if T is centered, and so on.     

The Asymptotics of the Two-Dimensional Wave Equation for a General Multi-Connected Vibrating Membrane with Piecewise Smooth Robin Boundary Conditions

The asymptotic expansion for small |t| of the trace of the wave kernel ∧↑μ(t) =∑v=1^∞exp(-it μv^1/2), where i= √-1 and {μv}v=1^∞ are the eigenvalues of the negative Laplacian -△=-∑β=1^2(δ/δx^β)^2 in the (x^1, x^2)-plane, is studied for a multi-connected vibrating membrane Ω in R^2 surrounded by simply connected bounded domains Ωj with smooth boundaries δΩj(j=1,...,n), where a finite number of piecewise smooth Robin boundary conditions on the piecewise smooth components Гi(i=1 κj-1,...,κj) of the boundaries δΩj are considered, such that δΩj=∪i=1 κj-1^κj Гi and κ0=0. The basic problem is to extract information on the geometry of Ω using the wave equation approach. Some geometric quantities of Ω (e.g. the area of Ω, the total lengths of its boundary, the curvature of its boundary, the number of the holes of Ω, etc.) are determined from the asymptotic expansion of the trace of the wave kernel ∧↑μ(t) for small |t|.     

Propagation of oscillations to 2D incompressible Euler equations

The asymptotic expansions are studied for the vorticity to 2D incompressible Euler equations with-initial vorticity , where ϕ₀(x) satisfies |d ϕ₀(x)|≠0 on the support of and is sufficiently smooth and with compact support in ℝ² (resp. ℝ²×T) The limit,v(t,x), of the corresponding velocity fields {v ^ɛ(t,x)} is obtained, which is the unique solution of (E) with initial vorticity ω₀(x). Moreover, (ℤ²)) for all 1≽p∞, where and ϕ(t,x) satisfy some modulation equation and eikonal equation, respectively.     

Solutions to Some Open Problems in Fluid Dynamics

Let u=u（x,t,uo）represent the global solution of the initial value problem for the one-dimensional fluid dynamics equation ut-εuxxt＋δux＋γHuxx＋βuxxx＋f（u）x=αuxx,u（x,0）=uo（x）, whereα〉0,β〉0,γ〉0,δ〉0 andε〉0 are constants.This equation may be viewed as a one-dimensional reduction of n-dimensional incompressible Navier-Stokes equations. The nonlinear function satisfies the conditions f（0）=0,｜f（u）｜→∞as ｜u｜→∞,and f∈C^1（R）,and there exist the following limits Lo=lim sup/u→o f（u）/u^3 and L∞=lim sup/u→∞ f（u）/u^5 Suppose that the initial function u0∈L^I（R）∩H^2（R）.By using energy estimates,Fourier transform,Plancherel＇s identity,upper limit estimate,lower limit estimate and the results of the linear problem vt-εv（xxt）＋δvx＋γHv（xx）＋βv（xxx）=αv（xx）,v（x,0）=vo（x）, the author justifies the following limits（with sharp rates of decay） lim t→∞[（1＋t）^（m＋1/2）∫｜uxm（x,t）｜^2dx]=1/2π（π/2α）^（1/2）m!!/（4α）^m[∫R uo（x）dx]^2, if∫R uo（x）dx≠0, where 0!!=1,1!!=1 and m!!=1·3…（2m-3）…（2m-1）.Moreover lim t→∞[（1＋t）^（m＋3/2）∫R｜uxm（x,t）｜^2dx]=1/2π（x/2α）^（1/2）（m＋1）!!/（4α）^（m＋1）[∫Rρo（x）dx]^2, if the initial function uo（x）=ρo′（x）,for some functionρo∈C^1（R）∩L^1（R）and∫Rρo（x）dx≠0.     

On a class of nonlinear problems involving a p(x)-Laplace type operator

We study the boundary value problem in Ω, u = 0 on ∂Ω, where Ω is a smooth bounded domain in ℝ ^N . Our attention is focused on two cases when , where m(x) = max{p ₁(x), p ₂(x)} for any x ∈ or m(x) < q(x) < N · m(x)/(N − m(x)) for any x ∈ . In the former case we show the existence of infinitely many weak solutions for any λ > 0. In the latter we prove that if λ is large enough then there exists a nontrivial weak solution. Our approach relies on the variable exponent theory of generalized Lebesgue-Sobolev spaces, combined with a ℤ₂-symmetric version for even functionals of the Mountain Pass Theorem and some adequate variational methods.     

Many solutions for elliptic equations with critical exponents

Let Ω be an open bounded domain in ℝ^N(N ≥ 3) and . We are concerned with two kinds of critical elliptic problems. The first one is

Eigenvalue estimate on a compact Riemann manifold

LetM be a compact Riemann manifold with the Ricci curvature ≽ - R(R = const. > 0) . Denote by d the diameter ofM. Then the first eigenvalue λ₁ ofM satisfies . Moreover if , then      

Forbidden paths and cycles in ordered graphs and matrices

Assume % MathType!End!2!1! and let Ω⊂R ^N(N≥4) be a smooth bounded domain, 0∈Ω. We study the semilinear elliptic problem: % MathType!End!2!1!. By investigating the effect of the coefficientQ, we establish the existence of nontrivial solutions for any λ>0 and multiple positive solutions with λ,μ>0 small.     

A note on the crossed product $$ \mathcal{R}(\mathcal{A},\alpha ) $$ associated with PSL2(ℝ)

Given the hyperbolic measure dxdy/y ² on the upper half plane ℍ, the rational actions of PSL₂(ℝ) on ℍ induces a continuous unitary representation α of this group on the Hilbert space L ²(ℍ, dxdy/y ²). Supposing that = {M _f : f ∈ L ^∞ (ℍ, dxdy/y ²)}, we show that the crossed product is of type I. In fact, the crossed product is *-isomorphic to the von Neumann algebra , where is the abelian group von Neumann algebra generated by the left regular representation of K. This work was supported by the Youth Foundation of Sichuan Education Department of China (Grant No. 2003B017)     

Oscillation theorems for second-order nonlinear neutral delay dynamic equations on time scales

By employing the generalized Riccati transformation technique, we will establish some new oscillation criteria and study the asymptotic behavior of the nonoscillatory solutions of the second-order nonlinear neutral delay dynamic equation

Well-posedness in the energy space for non-linear system of wave equations with critical growth

The authors consider the well-posedness in energy space of the critical non-linear system of wave equations with Hamiltonian structure{utt-△u=-F1（｜u｜^2,｜v｜^2）u,utt-△u=-F2（｜u｜^2,｜v｜^2）u where there exists a function F（λ,μ） such that δF（λ,μ）/δλ=F1（λ,μ）.δF（λ,μ）/δμ=F2（λ,μ） By showing that the energy and dilation identities hold for weak solution under some assumptions on the non-linearities, we prove the global well-posedness in energy space by a similar argument to that for global regularity as shown in ＂Shatah and Struwe＇s paper, Ann. of Math. 138, 503-518 （1993）＂.     

Boundary functions on a bounded balanced domain

We solve the following Dirichlet problem on the bounded balanced domain with some additional properties: For p > 0 and a positive lower semi-continuous function u on ∂Ω with u(z) = u(λ z) for |λ| = 1, z ∈ ∂Ω we construct a holomorphic function f ∈ (Ω) such that for z ∈ ∂Ω, where = {λ ∈ ℂ: |λ| < 1}.      

The hardy and bellman operators in spaces connected with H( \mathbb{T} ) and BMO( \mathbb{T} )

Assume that 1 ≤ p < ∞ and a function f ∈ L ^p [0, π] has the Fourier series $ \sum\limits_{n = 1}^\infty {a_n } Assume that 1 ≤ p < ∞ and a function f ∈ L ^p [0, π] has the Fourier series cos nx. According to one result of G.H. Hardy, the series cos nx is the Fourier series for a certain function (f) ∈ L ^p [0, π]. But if 1 < p ≤ ∞ and f ∈ L ^p [0, π], then the series cos nx is the Fourier series for a certain function (f) ∈ L ^p [0, π]. Similar assertions are true for sine series. This allows one to define the Hardy operator on L ^p (), 1 ≤ p < ∞, and to define the Bellman operator on L ^p (), 1 < p ≤ ∞. In this paper we prove that the Bellman operator boundedly acts in VMO(), and the Hardy operator also maps a certain subspace C() onto VMO(). We also prove the invariance of certain classes of functions with given majorants of modules of continuity or best approximations in the spaces H(), L(), VMO() with respect to the Hardy and Bellman operators. Original Russian Text ? S.S. Volosivets and B.I. Golubov, 2008, published in Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, 2008, No. 5, pp. 4–13.     

Riesz multiwavelet bases generated by vector refinement equation

In this paper, we investigate compactly supported Riesz multiwavelet sequences and Riesz multiwavelet bases for L ₂(ℝ ^s ). Suppose ψ = (ψ¹,..., ψ ^r ) ^T and are two compactly supported vectors of functions in the Sobolev space (H ^μ(ℝ ^s )) ^r for some μ > 0. We provide a characterization for the sequences {ψ _jk ^l : l = 1,...,r, j ε ℤ, k ε ℤ ^s } and to form two Riesz sequences for L ₂(ℝ ^s ), where ψ _jk ^l = m ^j/2ψ ^l (M ^j ·−k) and , M is an s × s integer matrix such that lim _n→∞ M ⁻ⁿ = 0 and m = |detM|. Furthermore, let ϕ = (ϕ¹,...,ϕ ^r ) ^T and be a pair of compactly supported biorthogonal refinable vectors of functions associated with the refinement masks a, and M, where a and are finitely supported sequences of r × r matrices. We obtain a general principle for characterizing vectors of functions ψ^ν = (ψ^ν1,...,ψ^νr ) ^T and , ν = 1,..., m − 1 such that two sequences {ψ _jk ^νl : ν = 1,..., m − 1, l = 1,...,r, j ε ℤ, k ε ℤ ^s } and { : ν=1,...,m−1,ℓ=1,...,r, j ∈ ℤ, k ∈ ℤ ^s } form two Riesz multiwavelet bases for ^L ₂(ℝ ^s ). The bracket product [f, g] of two vectors of functions f, g in (L ₂(ℝ ^s )) ^r is an indispensable tool for our characterization. This work was supported by National Natural Science Foundation of China (Grant Nos. 10771190, 10471123)     

<Emphasis Type="Italic">K</Emphasis><Subscript>0</Subscript>-group and approximate similarity invariants of strongly irreducible operators

Let H olenote a complex separable Hilbert space and L(H) denote the collection of bounded linear operators on H. An operator T ∈ L(H) is said to be strongly irreducible if T does not commute with any nontrivial idempotent. Herrero and Jiang showed that the norm-closure of the class of all strongly irreducible operators is the class of all operators with connected spectrum. This result can be considered as an approximate inverse of the Riesz decomposition theorem. In the paper, we give a more precise charact...