Global existence and time decay of small solutions to the landau-ginzburg type equations

We study the Cauchy problem for the nonlinear dissipative equations (0.1) uo∂u-αδu + Β|u|^2/n u = 0,x ∃ Rⁿ,t } 0,u(0,x) = u0(x),x ∃ Rⁿ, where α,Β ∃ C, ℜα 0. We are interested in the dissipative case ℜα 0, and ℜδ(α,Β)≥ 0, θ = |∫ u0(x)dx| ⊋ 0, where δ(α, Β) = ##|α|^n-1n^n/2 / ((n + 1)|α|² + α^{2 n/2}. Furthermore, we assume that the initial data u₀ ∃ L^p are such that (1 + |x|)^αu0 ∃ L¹, with sufficiently small norm ∃ = (1 + |x|)^α u₀ 1 + u₀ p, wherep 1, α ∃ (0,1). Then there exists a unique solution of the Cauchy problem (0.1)u(t, x) ∃ C ((0, ∞); L^∞) ∩ C ([0, ∞); L¹ ∩ L^p) satisfying the time decay estimates for allt0 u(t)||_∞ Cɛt^-n/2(1 + η log 〈t〉)^-n/2, if hg = θ^2/n 2_π ℜδ(α, Β) 0; u(t)||_∞ Cɛt^-n/2(1 + Μ log 〈t〉)^-n/4, if η = 0 and Μ = θ^4/n 4π)² (ℑδ(α, Β))² ℜ((1 + 1/n) υ1-1 υ2) 0; and u(t)||_∞ Cɛt^-n/2(1 + κ log 〈t〉)^-n/6, if η = 0, Μ = 0, κ 0, where υl,l = 1,2 are defined in (1.2), κ is a positive constant defined in (2.31).     

On solutions of the Beltrami equation

In this paper we study the existence and uniqueness of solutions of the Beltrami equationf _-z (z) =Μ(z)f _z (z), whereΜ(z) is a measurable function defined almost everywhere in a plane domain ‡ with ‖ΜΜ∞ = 1-Here the partialsf _z andf _z of a complex valued functionf _z exist almost everywhere. In case ‖Μ‖∞ ≤9 < 1, it is well-known that homeomorphic solutions of the Beltrami equation are quasiconformal mappings. In case ‖Μ‖∞= 1, much less is known. We give sufficient conditions onΜ(z) which imply the existence of a homeomorphic solution of the Beltrami equation, which isACL and whose partial derivativesf _z andf _z are locally inL ^q for anyq < 2. We also give uniqueness results. The conditions we consider improve already known results.     

On a singular integral on product spaces

We show that if K(x,y)=Ω(x,y)/|x|ⁿ|y|^m is a Calder n-Zygmund kerned on Rⁿ×R^m, where Ω∈L²(Sⁿ⁻¹×S^m−1) and b(x,y) is any bounded function which is radial with x∈Rⁿ and y∈R^m respectively, then b(x,y)K(x,y) is the kernel of a convolution operator which is bounded on L^p(Rⁿ×R^m) for 1<p<∞ and n≧2, m≧2. Project supported by NSFC     

Mixing of Cartesian squares of positive operators

LetT be a power bounded positive operator inL ₁(X, Σ, m)of a probability space, given by a transition measureP (x, A). The Cartesian squareS is the operator onL ₁ (X × X, Σ × Σ, m × m) induced by the transition measure Q((x, y), A × B)=P(x, A)P(y, B).T iscompletely mixing if ∝u e dm=0 impliesT ⁿ u→0 weakly (where 0≦e ∈L _∞ withT ^* e=e).Theorem. IfT has no fixed points, thenT is completely mixing if and only ifS is completely mixing. Part of this research was done at The Hebrew University of Jerusalem.     

The best Lipschitz constants of Bernstein polynomials and Bezier nets over a given triangle

This paper proved the following three facts about the Lipschitz continuous property of Bernstein polynomials and Bezier nets defined on a triangle: suppose f(P) is a real valued function defined on a triangle T, (1) If f(P) satisfies Lipschitz continuous condition, i.e. f(P)∃Lip_Aα, then the corresponding Bernstein Bezier net f_n∃Lip _Asec ^αφα, here φ is the half of the largest angle of triangle T; (2) If Bernstein Bezier net f_n∃Lip _Bα, then its elevation Bezier net Ef_n∃Lip _Bα; and (3) If f(P)∃Lip _Aα, then the corresponding Bernstein polynomials B_n(f;P)∃Lip _Asec ^αφα, and the constant Asec^αφ is best in some sense. Supported by NSF and SF of National Educational Committee     

Extensions of the Menchoff-Rademacher theorem with applications to ergodic theory

We prove extensions of Menchoff's inequality and the Menchoff-Rademacher theorem for sequences {f _n } ∪L _p , based on the size of the norms of sums of sub-blocks of the firstn functions. The results are aplied to the study of a.e. convergence of series Σ _n a _n T ⁿ g/ n whenT is anL ₂ -contraction,g∃L ₂ , and {a _n } is an appropriate sequence. Given a sequence {f _n }∪L _p (Ω, μ), 1<p≤2, of independent centered random variables, we study conditions for the existence of a set ofx of μ-probability 1, such that for every contractionT on andg∈L ₂ (π), the random power series Σ _n f _n (x)T ⁿ g converges π-a.e. The conditions are used to show that for {f _n } centered i.i.d. withf ₁∃L log⁺ L, there exists a set ofx of full measure such that for every contractionT on andg∃L ₂ (π), the random series Σ _n f _n (x)T ⁿ g/n converges π-a.e. We use Menchoff's own spelling of his name in the papers he wrote in French. Dedicated to Hillel Furstenberg upon his retirement     

Quasilinear elliptic equations with boundary blow-up

Assume that Ω is a bounded domain in ℝ _N withN ≥2, which has aC ²-boundary. We show that forp ∃ (1, ∞) there exists a weak solutionu of the problem δ_p u(x) = f(u(x)), x ∃ Ω with boundary blow-up, wheref is a positive, increasing function which meets some natural conditions. The boundary blow-up ofu(x) is characterized in terms of the distance ofx from ∂Ω. For the Laplace operator, our results coincide with those of Bandle and Essén [1]. Finally, for a rather wide subclass of the class of the admissible functionsf, the solution is unique whenp ∃ (1, 2].     

On estimation of a density and its derivatives

Summary Letf _n ^(p) be a recursive kernel estimate off ^(p) thepth order derivative of the probability density functionf, based on a random sample of sizen. In this paper, we provide bounds for the moments of and show that the rate of almost sure convergence of to zero isO(n ^−α), α<(r−p)/(2r+1), iff ^(r),r>p≧0, is a continuousL ₂(−∞, ∞) function. Similar rate-factor is also obtained for the almost sure convergence of to zero under different conditions onf. This work was supported in part by the Research Foundation of SUNY.     

Non-oscillating Paley-Wiener functions

A non-oscillating Paley-Wiener function is a real entire functionf of exponential type belonging toL ₂(R) and such that each derivativef ⁽ⁿ⁾,n=0, 1, 2,…, has only a finite number of real zeros. It is established that the class of such functions is non-empty and contains functions of arbitrarily fast decay onR allowed by the convergence of the logarithmic integral. It is shown that the Fourier transform of a non-oscillating Paley-Wiener function must be infinitely differentiable outside the origin. We also give close to best possible asymptotic (asn→∞) estimates of the number of real zeros of then-th derivative of a functionf of the class and the size of the smallest interval containing these zeros.     

Convergence of mock Fourier series

For certain Cantor measures μ on ℝⁿ, it was shown by Jorgensen and Pedersen that there exists an orthonormal basis of exponentialse ^2πiγ·x for λεΛ. a discrete subset of ℝⁿ called aspectrum for μ. For anyL ¹ functionf, we define coefficientsc _γ(f)=∝f(y)e ^−2πiγiy dμ(y) and form the Mock Fourier series ∑_λ∈Λ^cλ(f)e ^2πiλ·x . There is a natural sequence of finite subsets Λ_n increasing to Λ asn→∞, and we define the partial sums of the Mock Fourier series by We prove, under mild technical assumptions on μ and Λ, thats _n(f) converges uniformly tof for any continuous functionf and obtain the rate of convergence in terms of the modulus of continuity off. We also show, under somewhat stronger hypotheses, almost everywhere convergence forfεL ¹. Research supported in part by the National Science Foundation, Grant DMS-0140194.     

Hardy spaces via distribution spaces

Let ℐ(ℝⁿ) be the Schwartz class on ℝⁿ and ℐ_∞(ℝⁿ) be the collection of functions ϕ ∊ ℐ(ℝⁿ) with additional property that

Multiparameter inverse problem of approximation by functions with given supports

Let L _p (S), 0 < p < +∞, be a Lebesgue space of measurable functions on S with ordinary quasinorm ∥·∥ _p . For a system of sets {B _t |t ∈ [0, +∞) ⁿ } and a given function ψ: [0, +∞) ⁿ ↦ [ 0, +∞), we establish necessary and sufficient conditions for the existence of a function f ∈ L _p (S) such that inf {∥f − g∥ _p ^p g ∈ L _p (S), g = 0 almost everywhere on S\B _t } = ψ (t), t ∈ [0, +∞) ⁿ . As a consequence, we obtain a generalization and improvement of the Dzhrbashyan theorem on the inverse problem of approximation by functions of the exponential type in L ₂. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 8, pp. 1116–1127, August, 2006.     

Some remarks on the identity between a variational integral and its relaxed functional

Summary Letf: (x, z)∈R ⁿ×Rⁿ→f(x, z)∈[0, +∞] be measurable inx and convex inz. It is proved, by an example, that even iff verifies a condition as|z| ^p≤f(x, z)≤Λ(a(x)+|z|^q) with 1<p<q,a∈L _loc ^s (R ⁿ),s>1, the functional that isL ¹(Ω)-lower semicontinuous onW ^1,1(Ω), does not agree onW ^1,1(Ω) with its relaxed functional in the topologyL ¹(Ω) given by inf

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Riassunto Siaf: (x, z)∈R ⁿ×Rⁿ→f(x, z)∈[0, +∞] misurabile inx e convessa inz. Si mostra con un esempio che anche sef verifica una condizione del tipo|z| ^p≤f(x, z)≤Λ(a(x)+|z|^q) con 1<p<q,a∈L _loc ^s (R ⁿ),s>1, il funzionale , che èL ¹(Ω)-semicontinuo inferiormente suW ^1,1(Ω), non coincide suW ^1,1(Ω) con il suo funzionale rilassato nella topologiaL ¹(Ω) definito da inf

     

Extensions of the measurable choice theorem by means of forcing

Using the method of forcing of set theory, we prove the following two theorems on the existence of measurable choice functions: LetT be the closed unit interval [0,1] and letm be the usual Lebesgue measure defined on the Borel subsets ofT. Theorem1. LetS⊂T×T be a Borel set such that for alltεT,S _t ^def={x|(t,x)εS} is countable and non-empty. Then there exists a countable series of Lebesgue-measurable functionsf _n: T→T such thatS _t={f_n(t)|nεω} for alltε[0,1],W _x={y|(x,y)εW} is uncountable. Then there exists a functionh:[0,1]×[0,1]→W with the following properties: (a) for each xε[0,1], the functionh(x,·) is one-one and ontoW _x and is Borel measurable; (b) for eachy, h(·, y) is Lebesgue measurable; (c) the functionh is Lebesgue measurable.     

Cyclic elements of the shift operator in weighted anisotropic spaces of holomorphic functions in the polydisk

Let ϕ(r) = (ϕ₁(r₁), …, ϕ_n(r_n)) be a vector-valued function on R ₊ ⁿ . A necessary and sufficient condition is obtained under which any function f ∈, H^∞ (D ⁿ ), f(z) ≠ 0, z ∈, D ⁿ , is cyclic in the corresponding weighted space L^p(ϕ), where D ⁿ is the unit polydisk in C ⁿ. Bibliography: 13 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 327, 2005, pp. 226–234.     

Construction of positive definite cubature formulae and approximation of functions via Voronoi tessellations

Let Ω ⊂ ℝ ^d be a compact convex set of positive measure. In a recent paper, we established a definiteness theory for cubature formulae of order two on Ω. Here we study extremal properties of those positive definite formulae that can be generated by a centroidal Voronoi tessellation of Ω. In this connection we come across a class of operators of the form L_n[f](x): = ?_i=1ⁿ f_i(x)(f(y_i) + á?f(y_i), x-y_i?)L_n[f](\boldsymbol{x}):= \sum_{i=1}^n \phi_i(\boldsymbol{x})(f(\boldsymbol{y}_i) + \langle\nabla f(\boldsymbol{y}_i), \boldsymbol{x}-\boldsymbol{y}_i\rangle), where y₁,..., y_n\boldsymbol{y}_1,\dots, \boldsymbol{y}_n are distinct points in Ω and {ϕ ₁, ..., ϕ _n } is a partition of unity on Ω. We present best possible pointwise error estimates and describe operators L _n with a smallest constant in an L ^p error estimate for 1 ≤ p < ∞ . For a generalization, we introduce a new type of Voronoi tessellation in terms of a twice continuously differentiable and strictly convex function f. It allows us to describe a best operator L _n for approximating f by L _n [f] with respect to the L ^p norm.     

A modulus of smoothness on the unit sphere

For functions onS ^d−1 (the unit sphere inR ^d) and, in particular, forf∈L _p(S ^d−1), we define new simple moduli of smoothness. We relate different orders of these moduli, and we also relate these moduli to best approximation by spherical harmonics of order smaller thann. Our new moduli lead to sharper results than those now available for the known moduli onL _p(S ^d−1). Supported by NSERC Grant A4816 of Canada.     

A note on the multilinear singular integral operators

Lp(Rn) boundedness is considered for the multilinear singular integral operator defined by TAf(x) = ∫Rn Ω(x - y)/|x - y|n 1 (A(x) - A(y) - (△)A(y)(x - y))f(y)dy,where Ω is homogeneous of degree zero, integrable on the unit sphere and has vanishing moment of order one. A has derivatives of order one in BMO(Rn). We give a smoothness condition which is fairly weaker than that Ω∈ Lipα(Sn-1) (0 ＜α≤ 1) and implies the Lp(Rn) (1 ＜ p ＜ oo) boundedness for the operator TA. Some endpoint estimates are also established.     

The influence of variables in product spaces

LetX be a probability space and letf: X ⁿ → {0, 1} be a measurable map. Define the influence of thek-th variable onf, denoted byI _f (k), as follows: Foru=(u ₁,u ₂,…,u _n−1) ∈X ⁿ⁻¹ consider the setl _k (u)={(u ₁,u ₂,...,u _k−1,t,u _k ,…,u _n−1):t ∈X}. More generally, forS a subset of [n]={1,...,n} let the influence ofS onf, denoted byI _f (S), be the probability that assigning values to the variables not inS at random, the value off is undetermined. Theorem 1:There is an absolute constant c ₁ so that for every function f: X ⁿ → {0, 1},with Pr(f ⁻¹(1))=p≤1/2,there is a variable k so that Theorem 2:For every f: X ⁿ → {0, 1},with Prob(f=1)=1/2, and every ε>0,there is S ⊂ [n], |S|=c ₂(ε)n/logn so that I _f (S)≥1−ε. These extend previous results by Kahn, Kalai and Linial for Boolean functions, i.e., the caseX={0, 1}. Work supported in part by grants from the Binational Israel-US Science Foundation and the Israeli Academy of Science.     

Interpolating sequences in the ball ofC n

LetB be the unit ball ofC ⁿ , I give necessary conditions on sequenceS of points inB to beH ^∞(B) interpolating in term of aC ⁿ valued holomorphic function zero onS (a substitute for the interpolating Blaschke product). These conditions are sufficient to prove that the sequenceS is interpolating for ∩ _p>1 (B) and is also interpolating forH ^p (B) for 1≤p<∞.