Semilinear elliptic equations with uniform blow-up on the boundary

We prove the existence and the uniqueness of a solutionu of−Lu+h|u| ^α-1u=f in some open domain ℝ^d, whereL is a strongly elliptic operator,f a nonnegative function, and α>1, under the assumption that ∂G is aC ² compact hypersurface, lim _x→∂G (dist(x, ∂G))^2α/(α-1) f(x)=0, and lim _x→∂G u(x)=∞.     

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.

---

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.

     

On the growth of functions represented by Dirichlet series with complex coefficients on the real axis

We establish conditions under which, for a Dirichlet series F(z) = Σ _{n = 1} ^∞ d _n exp(λ _n z), the inequality ⋎F(x)⋎≤y(x),x≥x _o, implies the relation Σ _{n = 1} ^∞ |d _n exp(λ _n z)| ⪯ γ((1 + o(1))x) as x→+∞, where γ is a nondecreasing function on (−∞,+∞). Franko Drohobych State Pedagogic Institute, Drohobych. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 49, No. 12, pp. 1610–1616. December, 1997     

Fixed point theorems for non-Lipschitzian mappings of asymptotically nonexpansive type

LetX be a Banach space,K a nonempty, bounded, closed and convex subset ofX, and supposeT:K→K satisfies: for eachx∈K, lim sup _i→∞{sup _y∈K ‖t ⁱx−Tⁱy∼−‖x−y‖}≦0. IfT ^N is continuous for some positive integerN, and if either (a)X is uniformly convex, or (b)K is compact, thenT has a fixed point inK. The former generalizes a theorem of Goebel and Kirk for asymptotically nonexpansive mappings. These are mappingsT:K→K satisfying, fori sufficiently large, ‖Tⁱx−Tⁱy‖≦k _i‖x−y∼,x,y∈K, wherek _i→1 asi→∞. The precise assumption in (a) is somewhat weaker than uniform convexity, requiring only that Goebel’s characteristic of convexity, ɛ₀ (X), be less than one. Research supported by National Science Foundation Grant GP 18045.     

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.     

Two renewal theorems for general random walks tending to infinity

Summary. Necessary and sufficient conditions for the existence of moments of the first passage time of a random walk S _n into [x, ∞) for fixed x≧ 0, and the last exit time of the walk from (−∞, x], are given under the condition that S _n →∞ a.s. The methods, which are quite different from those applied in the previously studied case of a positive mean for the increments of S _n , are further developed to obtain the “order of magnitude” as x→∞ of the moments of the first passage and last exit times, when these are finite. A number of other conditions of interest in renewal theory are also discussed, and some results for the first time for which the random walk remains above the level x on K consecutive occasions, which has applications in option pricing, are given. Received: 18 September 1995/In revised form: 28 February 1996     

Penultimate limiting forms in extreme value theory

Summary Let {X _n}_n≧1 be a sequence of independent, identically distributed random variables. If the distribution function (d.f.) ofM _n=max (X ₁,…,X _n), suitably normalized with attraction coefficients {α_n}_n≧1(α_n>0) and {b _n}_n≧1, converges to a non-degenerate d.f.G(x), asn→∞, it is of interest to study the rate of convergence to that limit law and if the convergence is slow, to find other d.f.'s which better approximate the d.f. of(M _n−b_n)/a_n thanG(x), for moderaten. We thus consider differences of the formF ⁿ(a_nx+b_n)−G(x), whereG(x) is a type I d.f. of largest values, i.e.,G(x)≡Λ(x)=exp (-exp(−x)), and show that for a broad class of d.f.'sF in the domain of attraction of Λ, there is a penultimate form of approximation which is a type II [Ф _α(x)=exp (−x^−α), x>0] or a type III [Ψ _α(x)= exp (−(−x)^α), x<0] d.f. of largest values, much closer toF ⁿ(a_nx+b_n) than the ultimate itself.     

环域上一类拟线性椭圆型方程的正径向解的存在性

The existence of positive radial solutions of the equation -din( |Du|p-2Du)=f(u) is studied in annular domains in Rn,n≥2. It is proved that if f(0)≥0, f is somewherenegative in (0,∞), limu→0^ f‘ (u)=0 and limu→∞ (f(u)/u^p-1)=∞, then there is alarge positive radial solution on all annuli. If f(0)≤0 and satisfies certain conditions, then the equation has no radial solution if the annuli are too wide.     

Asymptotic behavior of solutions for a class of retarded difference equations

Consider the retarded difference equationx _n −x _n−1 =F(−f(x _n )+g(x _n−k )), wherek is a positive integer,F,f,g:R→R are continuous,F andf are increasing onR, anduF(u)>0 for allu≠0. We show that whenf(y)≥g(y) (resp. f(y)≤g(y)) fory∈R, every solution of (*) tends to either a constant or −∞ (resp. ∞) asn→∞. Furthermore, iff(y)≡g(y) fory∈R, then every solution of (*) tends to a constant asn→∞. Project supported by NNSF (19601016) of China and NSF (97-37-42) of Hunan     

CONVERGENCE RATE FOR ITERATES OF q-BERNSTEIN POLYNOMIALS

Recently, q-Bernstein polynomials have been intensively investigated by a number of authors. Their results show that for q ≠ 1, q-Bernstein polynomials possess of many interesting properties. In this paper, the convergence rate for iterates of both q-Bernstein polynomials and their Boolean sum are estimated. Moreover, the saturation of {Bn（., qn）} when n → ∞ and convergence rate of Bn（f,q;x） when f ∈ C^n-1 [0, 1], q → ∞ are also presented.     

HOMOCLINIC ORBITS FOR SECOND ORDER HAMILTONIAN SYSTEM WITH QUADRATIC GROWTH

ＨＯＭＯＣＬＩＮＩＣＯＲＢＩＴＳＦＯＲＳＥＣＯＮＤＯＲＤＥＲＨＡＭＩＬＴＯＮＩＡＮＳＹＳＴＥＭＷＩＴＨＱＵＡＤＲＡＴＩＣＧＲＯＷＴＨＷＵＳＨＡＯＰＩＮＧＡＮＤＬＩＵＪＩＡＱＵＡＮＡｂｓｔｒａｃｔ：Ｓｏｍｅｅｘｉｓｔｅｎｃｅａｎｄｍｕｌｔｉｐｌｉｃｉｔ...     

Singular integrals in the Cesàro sense

The existence of the singular integral ∫K(x, y)f(y)dy associated to a Calderón-Zygmund kernel where the integral is understood in the principal value sense TF(x)=lim_ε→0+∫_|x−y|>εK(x, y)f(y)dy has been well studied. In this paper we study the existence of the above integral in the Cesàro-α sense. More precisely, we study the existence of

Purely imaginary eigenvalues of operator semigroups

For aC ₀-contraction semigroup (S(t)) _t≥0 of bounded linear operators on a complex Banach spaceX, J. A. Goldstein and B. Nagy [6] have shown that, givenx∈X, S(t)x=e ^iλt x, t≥0, for some λ∈ℝ, provided lim _t→∞ |<S(t)x,x ^* >|=|<x,x ^* >| for allx ^*∈X^*. We present (a) an extension to the case of nonlinear nonexpansive mapsS(t), t≥0, and (b) various generalizations in the linear context.     

Analogues of the “zero-two” law for positive linear contractions inL p andC(X)

LetT be a positive linear contraction inL ^p (1≦p<∞), then we show that lim ‖T ^pf −T ⁿ⁺¹ f‖ _p ≦(1 − ε)2^1/p (f∈L _p ⁺ , ε>0 independent off) implies already lim_n n→∞ ‖T ⁿf −T ⁿ⁺¹ n+1f ‖_p p=0. Several other related results as well as uniform variants of these are also given. Finally some similar results inLsu/t8 andC(X) are shown.     

Scaling limits in the second Painlevé transcendent

By the isomonodromy deformation method, the asymptotics of the general solution for the second Painlevé equation y_xx=2y³+xy−α asReα→∞ are described for any x. The corresponding formulas are also presented. Bibliography: 23 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 209, 1994, pp. 60–101.     

Sharp estimation of the almost-sure Lyapunov exponent for the Anderson model in continuous space

In this article we study the exponential behavior of the continuous stochastic Anderson model, i.e. the solution of the stochastic partial differential equation u(t,x)=1+∫₀^tκΔ_xu (s,x) ds+∫₀^t W(ds,x) u (s,x), when the spatial parameter x is continuous, specifically x∈R, and W is a Gaussian field on R₊×R that is Brownian in time, but whose spatial distribution is widely unrestricted. We give a partial existence result of the Lyapunov exponent defined as lim_t→∞t⁻¹ log u(t,x). Furthermore, we find upper and lower bounds for lim sup_t→∞t⁻¹ log u(t,x) and lim inf_t→∞t⁻¹ log u(t,x) respectively, as functions of the diffusion constant κ which depend on the regularity of W in x. Our bounds are sharper, work for a wider range of regularity scales, and are significantly easier to prove than all previously known results. When the uniform modulus of continuity of the process W is in the logarithmic scale, our bounds are optimal. This author's research partially supported by NSF grant no. : 0204999     

The condensate 〈(A µ 2 )〉 in commutative and noncommutative theories

We show that the gauge invariance of the operator ∫ dx tr(A _μ ² −2/(gξ)x ^υθ_μυ A ^μ) in a noncommutative gauge theory does not lead to the gauge independence of its vacuum condensate. We obtain the generalized Ward identities for Green’s functions containing the operator lim_Ω→∞(1/Ω)∫_Ω dx tr (A _μ ² ) in commutative and noncommutative gauge theories. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 148, No. 3, pp. 350–356, September, 2006.     

p-Energia in spazi metrici generalizzati

Given a setS and a function σ:S x S→[0, +∞[ such that σ(x, x)=0, we define the generalizedp-energy of a curve γ: [a, b]→S, so that, ifS is a Hilbert space and σ(x, y)=‖x−y‖ we obtain . Sufficient conditions for the equality of the defined energies are also given. Moreover the case in whichS is a set of measurable parts of ℝⁿ and σ_r is a family of functions used in order to study the generalized minimizing motions, is discussed.

---

Conferenza tenuta il 30 ottobre 1995     

Weighted estimates for rough oscillatory singular integrals

Weighted L^p estimates (1<p<∞) are shown for oscillatory singular integral operators with polynomial phase and a rough kernel of the form e^iP(x,y)Ω(x−y)h(|x−y|)|x−y|⁻ⁿ. We assume that Ω∈L logL(Sⁿ⁻¹) is homogeneous of degree zero and ∫_Sn-1Ω=0. The radial factor h has bounded variation. The necessary condition on the weight is similar to the A_p condition but involves rectangles (instead of cubes) arising from a covering of a star-shaped set related to Ω.     

A problem on simultaneous approximation and a conjecture of Hasson

In 1980, M. Hasson raised a conjecture as follows: Let N≥1, then there exists a function f₀(x)∈C _[−1,1] ^2N , for N+1≤k≤2N, such that p _n ^(k) (f₀,1)→f ₀ ^(k) (1), n→∞, where p_n(f,x) is the algebraic polynomial of best approximation of degree ≤n to f(x). In this paper, a, positive answer to this conjecture is given.