On one representation of analytic functions by harmonic functions

Let u(x) be a function analytic in some neighborhood D about the origin, $ \mathcal{D} Let u(x) be a function analytic in some neighborhood D about the origin, ⊂ ℝ ⁿ . We study the representation of this function in the form of a series u(x) = u ₀(x) + |x|² u ₁(x) + |x|⁴ u ₂(x) + …, where u _k (x) are functions harmonic in . This representation is a generalization of the well-known Almansi formula. Original Russian Text ? V. V. Karachik, 2007, published in Matematicheskie Trudy, 2007, Vol. 10, No. 2, pp. 142–162.     

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.     

The Dual Spaces of Sets of Difference Sequences of Order <Emphasis Type="Italic">m</Emphasis> and Matrix Transformations

Let p = （pk）k=0^∞ be a bounded sequence of positive reals, m C N and u be s sequence of nonzero terms. If x = （xk）k=0^∞ is any sequence of complex numbers we write Δ（m）x for the sequence of the m th order differences of x and Δu^（m）X = {x=（x）k=0^∞ uΔ（m）x ∈ X} for any set X of sequences. We determine the α-, β- and γ-duals of the sets Δμ^（m）X for X=co（p）,c（p）,l∞（p） and characterize some matrix transformations between these spaces Δ^（m）X.     

Existence of solutions with turning points for nonlinear singularly perturbed boundary-value problems

We consider the following singularly perturbed boundary-value problem:

Homogenization of two-phase metrics and applications

We consider two-phase metrics of the form ϕ(x, ξ) ≔ , where α,β are fixed positive constants and B _α, B _β are disjoint Borel sets whose union is ℝ^N, and prove that they are dense in the class of symmetric Finsler metrics ϕ satisfying

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.     

On a problem of D.H. Lehmer and pseudorandom binary sequences

Let p be an odd prime, and f(x), g(x) ∈ [x]. Define

Nonlinear maps of convex sets in Hilbert spaces with application to kinetic equations

Let be a separable Hilbert space, an open convex subset, and f: a smooth map. Let Ω be an open convex set in with , where denotes the closure of Ω in . We consider the following questions. First, in case f is Lipschitz, find sufficient conditions such that for ɛ > 0 sufficiently small, depending only on Lip(f), the image of Ω by I + ɛf, (I + ɛf)(Ω), is convex. Second, suppose df(u): is symmetrizable with σ(df(u)) ⊆ (0,∞), for all u ∈ , where σ(df(u)) denotes the spectrum of df(u). Find sufficient conditions so that the image f(Ω) is convex. We establish results addressing both questions illustrating our assumptions and results with simple examples. We also show how our first main result immediately apply to provide an invariance principle for finite difference schemes for nonlinear ordinary differential equations in Hilbert spaces. The main application of the theory developed in this paper concerns our second result and provides an invariance principle for certain convex sets in an L ²-space under the flow of a class of kinetic transport equations so called BGK model.      

On the fourth moment in the Rankin-Selberg problem

If

Lower bounds at infinity of solutions of partial differential equations in the exterior of a proper cone

An extension of a classical theorem of Rellich to the exterior of a closed proper convex cone is proved: Let Γ be a closed convex proper cone inR ⁿ and −Γ′ be the antipodes of the dual cone of Γ. Let be a partial differential operator with constant coefficients inR ⁿ, whereQ(ζ)≠0 onR ⁿ−iΓ′ andP _i is an irreducible polynomial with real coefficients. Assume that the closure of each connected component of the set {ζ∈R ⁿ−iΓ′;P _j(ζ)=0, gradP _j(ζ)≠0} contains some real point on which gradP _j≠0 and gradP _j∉Γ∪(−Γ). LetC be an open cone inR ⁿ−Γ containing both normal directions at some such point, and intersecting each normal plane of every manifold contained in {ξ∈R ⁿ;P(ξ)=0}. Ifu∈ℒ′∩L _loc ² (R ⁿ−Γ) and the support ofP(−i∂/∂x)u is contained in Γ, then the condition implies that the support ofu is contained in Γ.     

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.     

Remainder estimates for the approximation numbers of weighted Hardy operators acting onL 2

We consider the weighted Hardy integral operatorT:L ²(a, b) →L ²(a, b), −∞≤a<b≤∞, defined by . In [EEH1] and [EEH2], under certain conditions onu andv, upper and lower estimates and asymptotic results were obtained for the approximation numbersa _n(T) ofT. In this paper, we show that under suitable conditions onu andv, where ∥w∥_p=(∫ _a ^b |w(t)|^p dt)^1/p. Research supported by NSERC, grant A4021. Research supported by grant No. 201/98/P017 of the Grant Agency of the Czech Republic.     

Concentration and influences

Consider the discrete cube Ω={0,1} ^N , provided with the uniform probabilityP. We denote byd(x, A) the Hamming distance of a pointx of Ω and a subsetA of Ω. We define the influenceI(A) of theith coordinate onA as follows. Forx in Ω, consider the pointT _i (x) obtained by changing the value of theith coordinate. Then We prove that we always have Since it is easy to see that , this recovers the well known fact that ∫_Ω d(x, A)dP(x) is at most of order whenP(A)≥1/2. The new information is that ∫_Ω d(x, A)dP(x) can be of order only ifA reassembles the Hamming ball {x; ∑_1≤N x _i ≥N/2}.     

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     

A linear periodic boundary-value problem for a second-order hyperbolic equation

We study the boundary-value problemu _tt -u _xx =g(x, t),u(0,t) =u (π,t) = 0,u(x, t +T) =u(x, t), 0 ≤x ≤ π,t ∈ ℝ. We findexact classical solutions of this problem in three Vejvoda-Shtedry spaces, namely, in the classes of, and-periodic functions (q and s are natural numbers). We obtain the results only for sets of periods, and which characterize the classes of π-, 2π -, and 4π-periodic functions. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 51, No. 2, pp. 281–284, February, 1999.     

Maximal deviation theory of some estimators of prior distribution functions

Summary Let be a sequence of independent identically distributed random variables withθ ₁∼G and the conditional distribution ofx ₁ givenθ ₁=θ given by . HereG is unknown andF _θ(·) is known. This paper provides estimators ofG based onx ₁, …,x _n such that the random variable sup has an asymptotic distribution asn→∞ under certain on conditionsG and for certain choices ofF _θ. A simulation model has been discussed involving the uniform distribution on (0, θ) forF _θ and an exponential distribution forG. Research supported by the National Science Foundation under Grant #MCS77-26809.     

A general oscillation theorem for self-adjoint differential systems with applications to Sturm-Liouville eigenvalue problems and quadratic functionals

For given 2n×2n matricesS ₁₃,S ₂₄ with rank(S ₁₃,S ₂₄)=2n we consider the eigenvalue problem:u′=A(x)u+B(x)v,v′=C ₁(x;λ)u-A ^T(x)v with

Regularity of Weak Solutions of Nonlinear Equations with Discontinuous Coefficients

In this paper, we prove that the weak solutions u∈Wloc^1, p （Ω） （1 〈p〈∞） of the following equation with vanishing mean oscillation coefficients A（x）： -div[（A（x）△↓u·△↓u）p-2/2 A（x）△↓u＋│F（x）│^p-2 F（x）]=B（x, u, △↓u）, belong to Wloc^1, q （Ω）（A↓q∈（p, ∞）, provided F ∈ Lloc^q（Ω） and B（x, u, h） satisfies proper growth conditions where Ω ∪→R^N（N≥2） is a bounded open set, A（x）=（A^ij（x）） N×N is a symmetric matrix function.     

Oblique derivative problem for general Chaplygin-Rassias equations

The present paper deals with the oblique derivative problem for general second order equations of mixed (elliptic-hyperbolic) type with the nonsmooth parabolic degenerate line K_1(y)u_(xx) |K_2(x)|u_(yy) a(x,y)u_x b(x, y)u_y c(x,y)u=-d(x,y) in any plane domain D with the boundary D=Γ∪L_1∪L_2∪L_3∪L_4, whereΓ(■{y>0})∈C_μ~2 (0<μ<1) is a curve with the end points z=-1,1. L_1, L_2, L_3, L_4 are four characteristics with the slopes -H_2(x)/H_1(y), H_2(x)/H_1(y),-H_2(x)/H_1(y), H_2(x)/H_1(y)(H_1(y)=|k_1(y)|~(1/2), H_2(x)=|K_2(x)|~(1/2) in {y<0}) passing through the points z=x iy=-1,0,0,1 respectively. And the boundary condition possesses the form 1/2 u/v=1/H(x,y)Re[λuz]=r(z), z∈Γ∪L_1∪L_4, Im[λ(z)uz]|_(z=z_l)=b_l, l=1,2, u(-1)=b_0, u(1)=b_3, in which z_1, z_2 are the intersection points of L_1, L_2, L_3, L_4 respectively. The above equations can be called the general Chaplygin-Rassias equations, which include the Chaplygin-Rassias equations K_1(y)(M_2(x)u_x)_x M_1(x)(K_2(y)u_y)_y r(x,y)u=f(x,y), in D as their special case. The above boundary value problem includes the Tricomi problem of the Chaplygin equation: K(y)u_(xx) u_(yy)=0 with the boundary condition u(z)=φ(z) onΓ∪L_1∪L_4 as a special case. Firstly some estimates and the existence of solutions of the corresponding boundary value problems for the degenerate elliptic and hyperbolic equations of second order are discussed. Secondly, the solvability of the Tricomi problem, the oblique derivative problem and Frankl problem for the general Chaplygin- Rassias equations are proved. The used method in this paper is different from those in other papers, because the new notations W(z)=W(x iy)=u_z=[H_1(y)u_x-iH_2(x)u_y]/2 in the elliptic domain and W(z)=W(x jy)=u_z=[H_1(y)u_x-jH_2(x)u_y]/2 in the hyperbolic domain are introduced for the first time, such that the second order equations of mixed type can be reduced to the mixed complex equations of first order with singular coefficients. And thirdly, the advantage of complex analytic method is used, otherwise the complex analytic method cannot be applied.