Convergence of weighted sums of independent random variables

Let {X_k} be a sequence of i.i.d. random variables with d.f. F(x). In the first part of the paper the weak convergence of the d.f.'s F_n(x) of sums is studied, where 0<α≤2, a_nk>0, 1≤k≤m_n, and, as n→∞, bothmax _1≤k≤mn_{a nk}→0 and . It is shown that such convergence, with suitably chosen A_n's and necessarily stable limit laws, holds for all such arrays {α_nk} provided it holds for the special case α_nk=1/n, 1≤k≤n. Necessary and sufficient conditions for such convergence are classical. Conditions are given for the convergence of the moments of the sequence {F_n(x)}, as well as for its convergence in mean. The second part of the paper deals with the almost sure convergence of sums , where a_n≠0, b_n>0, andmax _1≤k≤n a_k/b_n→0. The strong law is said to hold if there are constants A_n for which S_n→0 almost surely. Let N(0)=0 and N(x) equal the number of n≥1 for which b_n/|a_n|<x if x>0. The main result is as follows. If the strong law holds,EN (|X₁|)<∞. If for some 0<p≤2, then the strong law holds with if 1≤p≤2 and A_n=0 if 0<p<1. This extends the results of Heyde and of Jamison, Orey, and Pruitt. The strong law is shown to hold under various conditions imposed on F(x), the coefficients a_n and b_n, and the function N(x). Proceedings of the Seminar on Stability Problems for Stochastic Models, Moscow, 1993.     

Asymptotic behavior of the solution to the two-dimensional stationary problem of flow past a body far from it

In the exterior domain Ω⊂ℝ² we consider the two-dimensional Navier-stokes system Δu-▽p=(u,▽)u, div u=0 whose solution possesses a finite Dirichlet integral and satisfies the condition lim_|x|→∞ u(x)=(1, 0). For this solution, we establish the estimate |u(x)−(1, 0)|≤c|x| ^−α, where α>1/4. This estimate implies an asymptotic expression for the solution indicating the presence of a track behind the body. Translated fromMatematicheskie Zametki, Vol. 65, No. 2, pp. 246–253, February, 1999.     

Continuous spectrum for a class of nonhomogeneous differential operators

We study the boundary value problem in Ω, u = 0 on ∂Ω, where Ω is a bounded domain in with smooth boundary, λ is a positive real number, and the continuous functions p ₁, p ₂, and q satisfy 1 < p ₂(x) < q(x) < p ₁(x) < N and for any . The main result of this paper establishes the existence of two positive constants λ₀ and λ₁ with λ₀ ≤ λ₁ such that any is an eigenvalue, while any is not an eigenvalue of the above problem.     

Potential space estimates in local Hardy spaces for Green potentials in convex domains

Let Ω be a bounded co.nvex domain in Rn(n≥3) and G(x,y) be the Green function of the Laplace operator -△ on Ω. Let hrp(Ω) = {f ∈ D'(Ω) :(E)F∈hp(Rn), s.t. F|Ω = f}, by the atom characterization of Local Hardy spaces in a bounded Lipschitz domain, the bound of f→(△)2(Gf) for every f ∈ hrp(Ω) is obtained, where n/(n 1)＜p≤1.     

Controllability in a filled domain for the multidimensional wave equation with a singular boundary control

In accordance with the demands of the so-called local approach to inverse problems, the set of “waves” u^f (·, T) is studied, where u^f (x,t) is the solution of the initial boundary-value problem u_tt−Δu=0 in Ω×(0,T), u|_t<0=0, u|_∂Ω×(0,T)=f, and the (singular) control f runs over the class L²((0,T); H^−m (∂Ω)) (m>0). The following result is established. Let Ω^T={x ∈ Ω : dist(x, ∂Ω)<T)} be a subdomain of Ω ⊂ ℝⁿ (diam Ω<∞) filled with waves by a final instant of time t=T, let T_*=inf{T : Ω^T=Ω} be the time of filling the whole domain Ω. We introduce the notation D_m=Dom((−Δ)^m/2), where (−Δ) is the Laplace operator, Dom(−Δ)=H²(Ω)∩H ₀ ¹ (Ω);D_−m=(D_m)′;D_−m(Ω^T)={y∈D_−m:supp y ⋐ Ω^T. If T<T., then the reachable set R _m ^T ={u^t(·, T): f ∈ L₂((0,T), H^−m (∂Ω))} (∀m>0), which is dense in D_−m(Ω^T), does not contain the class C ₀ ^∞ (Ω^T). Examples of a ∈ C ₀ ^∞ , a ∈ R _m ^T , are presented. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 210, 1994, pp. 7–21. Translated by T. N. Surkova.     

A capacitary method for the asymptotic analysis of dirichlet problems for monotone operators

Given a non-linear elliptic equation of monotone type in a bounded open set Ω ⊂ Rⁿ, we prove that the asymptotic behaviour, asj → ∞, of the solutions of the Dirichlet problems corresponding to a sequence (Ω_j) of open sets contained in Ω is uniquely determined by the asymptotic behaviour, asj → ∞, of suitable non-linear capacities of the setsKΩ _j, whereK runs in the family of all compact subsets of Ω.     

Generalized bisymmetry on a restricted domain

Let X ⊂ ℝ be an interval of positive length and define the set Δ = {(x, y) ∈ X × X | x ≧ y}. We give the solution of the equation

On the boundedness of solutions of nonlinear differential equations in Hilbert spaces

LetH be a complex Hilbert space and letB be the space of all bounded linear operators fromH intoH with the strong operator topology. We will give a boundedness result for the solutions of the differential equationx′=A(t)x+f(t,x) whereA: I=[t ₀, ∞)→B is continuous,f: I×H→H is also continuous and for every bounded setS⊂I×H there exists a constantM(S)>0 such that |f(t,x)−f(t,y)|≤M(S)|x−y|,(t,x), (t,y)∈S.

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Sunto SiaH uno spazio di Hilbert complesso e siaB lo spazio degli operatori lineari limitati daH inH, con la topologia forte. In questo lavoro si prova un risultato di limitatezza per le soluzioni dell'equazione differenzialex′=A(t)x+f(t,x), doveA: I=[t ₀, ∞)→B è continua,f: I×H→H è continua e per ogni insieme limitatoS⊂I×H esiste una costanteM(S)>0 tale che |f(t,x)−f(t,y)|≤M(S)|x−y| per ogni(t,x), (t,y)∈S.

     

Sharp stability results for almost conformal maps in even dimensions

Let Ω, ⊂R ⁿ and n ≥ 4 be even. We show that if a sequence {u^j} in W^1,n/2(Ω;R ⁿ) is almost conformal in the sense that dist (∇u^j,R ⁺SO(n)) converges strongly to 0 in L^n/2 and if u^j converges weakly to u in W^1,n/2, then u is conformal and ∇u^j → ∇u strongly in L _loc ^q for all 1 < -q < n/2. It is known that this conclusion fails if n/2 is replaced by any smaller exponent p. We also prove the existence of a quasiconvex function f(A) that satisfies 0 ≤ f(A) ≤ C (1 + |A|^n/2) and vanishes exactly onR ⁺ SO(n). The proof of these results involves the Iwaniec-Martin characterization of conformal maps, the weak continuity and biting convergence of Jacobians, and the weak-L¹ estimates for Hodge decompositions.     

Superstability of the generalized orthogonality equation on restricted domains

Chmielinski has proved in the paper [4] the superstability of the generalized orthogonality equation |〈f(x), f(y)〉| = |〈x,y〉|. In this paper, we will extend the result of Chmielinski by proving a theorem: LetD _n be a suitable subset of ℝⁿ. If a function f:D _n → ℝⁿ satisfies the inequality ∥〈f(x), f(y)〉| |〈x,y〉∥ ≤ φ(x,y) for an appropriate control function φ(x, y) and for allx, y ∈ D _n, thenf satisfies the generalized orthogonality equation for anyx, y ∈ D _n.     

A note on regularity for a class of quasilinear elliptic equations

The following regularity of weak solutions of a class of elliptic equations of the form are investigated.     

Unique solvability of the Cauchy problem for certain quasilinear pseudoparabolic equations

We study the Cauchy problem in the layer Π _T =ℝ ⁿ ×[0,T] for the equationu _t =cGΔu _t +Δϕ(u), wherec is a positive constant and the functionϕ(p) belongs toC ¹(ℝ₊) and has a nonnegative monotone non-decreasing derivative. The unique solvability of this Cauchy problem is established for the class of nonnegative functionsu(x,t) ∈C _x,t ^2,1 (Π _T ) with the properties: , . Translated fromMatematicheskie Zametki, Vol. 60, No. 3, pp. 356–362, September, 1996. This research was partially supported by the International Science Foundation under grant No. MX6000.     

Higher integrability for minimizers of variational integrals with nonstandard growth

We consider a (possibly) vector-valued function u: Ω→R ^N, Ω⊂R ⁿ, minimizing the integral , whereD _iu=∂u/∂x _i, or some more general functional retaining the same behaviour; we prove higher integrability forDu:D ₁u,…,D_n−1u∈L^q, under suitable assumptions ona _i(x).

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Sunto Consideriamo una funzione u: Ω→R ^N, Ω⊂R ⁿ che minimizzi l'integrale , doveD _iu=∂u/∂x_i, o un funzionale con un comportamento simile; sotto opportune ipotesi sua _i(x), dimostriamo la seguente maggiore integrabilità perDu:D ₁u,…,D_n−1uεL^q.

     

Equipartition of mass distributions by hyperplanes

We consider the problem of determining the smallest dimensiond=Δ(j, k) such that, for anyj mass distributions inR ^d , there arek hyperplanes so that each orthant contains a fraction 1/2 ^k of each of the masses. The case Δ(1,2)=2 is very well known. The casek=1 is answered by the ham-sandwich theorem with Δ(j, 1)=j. By using mass distributions on the moment curve the lower bound Δ(j, k)≥j(2 ^k −1)/k is obtained. We believe this is a tight bound. However, the only general upper bound that we know is Δ(j, k)≤j2 ^k−1. We are able to prove that Δ(j, k)=⌈j(^2k−1/k⌉ for a few pairs (j, k) ((j, 2) forj=3 andj=2 ⁿ withn≥0, and (2, 3)), and obtain some nontrivial bounds in other cases. As an intermediate result of independent interest we prove a Borsuk-Ulam-type theorem on a product of balls. The motivation for this work was to determine Δ(1, 4) (the only case forj=1 in which it is not known whether Δ(1,k)=k); unfortunately the approach fails to give an answer in this case (but we can show Δ(1, 4)≤5). This research was supported by the National Science Foundation under Grant CCR-9118874.     

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

     

Existence of singular solutions of a degenerate equation in R 2

Let a₁,a₂, . . . ,a_m ∈ ℝ², 2≤f ∈ C([0,∞)), g_i ∈ C([0,∞)) be such that 0≤g_i(t)≤2 on [0,∞) ∀i=1, . . . ,m. For any p>1, we prove the existence and uniqueness of solutions of the equation u_t=Δ(logu), u>0, in satisfying and logu(x,t)/log|x|→−f(t) as |x|→∞, logu(x,t)/log|x−a_i|→−g_i(t) as |x−a_i|→0, uniformly on every compact subset of (0,T) for any i=1, . . . ,m under a mild assumption on u₀ where We also obtain similar existence and uniqueness of solutions of the above equation in bounded smooth convex domains of ℝ² with prescribed singularities at a finite number of points in the domain.     

Convergence of double Fourier series

Summary Let a2π-periodic function f(x, y) be continuous in some neighbourhood of the point (x, y) except possibly along finitely many lines l ₁ , l ₂ , ..., l_k terminating at (x, y). The problem of convergence of the Fourier series of f(x, y) at the point (x, y) is examined in some detail. It is established that under certain restrictions on the variation of f(x, y), and also on the lines l ₁ , l ₂ , ..., l_k, the fourier series converges to a value bounded above by the limit superior, and below by the limit inferior of f(x+u, y+v), u, v →0, this value depending on the manner in which the series is summed. The preparation of this paper was financed, in part, by a Canadian Mathematical Congress Summer Research Grant (1968). Entrata in Redazione il 3 febbraio 1969.     

On the existence and nonexistence of global solutions for the porous medium equation with strongly nonlinear sources in a cone

In this paper, we study the initial-boundary value problem of the porous medium equation u _t  = Δu ^m  + V(x)u ^p in a cone D = (0, ∞) × Ω, where V(x) ~ (1 + |x|) ^σ . Let ω ₁ denote the smallest Dirichlet eigenvalue for the Laplace–Beltrami operator on Ω and let l denote the positive root of l ² + (n − 2)l = ω ₁. We prove that if m ≤ p ≤ m + (2 + σ)/(n + l), then the problem has no global nonnegative solutions for any nonnegative u ₀ unless u ₀ = 0; if p > m + (2 + σ)/n, then the problem has global solutions for some u ₀ ≥ 0.     

A question of Gol’dberg concerning entire functions with prescribed zeros

Let (z_j) be a sequence of complex numbers satisfying |z_j|→ ∞ asj → ∞ and denote by n(r) the number of z_j satisfying |z_j|≤ r. Suppose that lim inf_{r → ⇈} log n(r)/ logr > 0. Let ϕ be a positive, non-decreasing function satisfying ∫^∞ (ϕ(t)t logt)⁻¹ dt < ∞. It is proved that there exists an entire functionf whose zeros are the z_j such that log log M(r,f) = o((log n(r))²ϕ(log n(r))) asr → ∞ outside some exceptional set of finite logarithmic measure, and that the integral condition on ϕ is best possible here. These results answer a question by A. A. Gol’dberg.     

Isometric approximation

Suppose thatA ⊂R ⁿ is a bounded set of diameter 1 and that:f:A →l ₂ is a map satisfying the nearisometry condition |x−y|−ɛ≤|fx−fy|≤|x−y|+ɛ withɛ≤1. Then there is an isometryS:A →l ₂ such that |Sx−fx|≤c _n √ɛ for allx inA. IfA satisfies a thickness condition and iff:A →R ⁿ , then there is an isometryS:R ⁿ →R ⁿ with |Sx−fx|≤c _nɛ/q, whereq is a thickness parameter.