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.     

The Number of Lattice Points Above the Catenary

For fixed c > 1 and for arbitrary and independent a,b ≧ 1 let Z ²|b( cosh(x/a)−c) ≦ y < 0}. We investigate the asymptotic behaviour of R(a,b) for a,b → ∞. In the special case b = o(a ^5/6) the lattice rest has true order of magnitude . This revised version was published online in June 2006 with corrections to the Cover Date.     

Asymptotic behavior of solutions to viscous conservation laws with slowly varying external forces

This paper considers the existence and large time behavior of solutions to the convection-diffusion equation u _t −Δu+b(x)·∇(u|u| ^{q −1})=f(x, t) in ℝ ⁿ ×[0,∞), where f(x, t) is slowly decaying and q≥1+1/n (or in some particular cases q≥1). The initial condition u ₀ is supposed to be in an appropriate L ^p space. Uniform and nonuniform decay of the solutions will be established depending on the data and the forcing term.This work is partially supported by an AMO Grant     

On the behavior of solutions of a semilinear parabolic or elliptic equation satisfying a nonlinear boundary condition in a cylindrical domain

One considers a semilinear parabolic equation u _t = Lu − a(x)f(u) or an elliptic equation u _tt + Lu − a(x)f(u) = 0 in a semi-infinite cylinder Ω × ℝ₊ with the nonlinear boundary condition , where L is a uniformly elliptic divergent operator in a bounded domain Ω ∈ ℝⁿ; a(x) and b(x) are nonnegative measurable functions in Ω. One studies the asymptotic behavior of solutions of such boundary-value problems for t → ∞. __________ Translated from Trudy Seminara imeni I. G. Petrovskogo, No. 26, pp. 368–389, 2007.     

On mappings preserving pentagons

Summary We introduce a new characterization of linear isometries. More precisely, we prove that if a one-to-one mapping f:<span style='font-size:10.0pt;font-family:"Lucida Sans Unicode"'>ℝⁿ→ℝⁿ(2<span style='font-size:10.0pt;font-family:"Lucida Sans Unicode"'>≦n<<span style='font-size:10.0pt;font-family:"Lucida Sans Unicode"'>∞) maps every regular pentagon of side length a> 0 onto a pentagon with side length b> 0, then there exists a linear isometry I :<span style='font-size:10.0pt;font-family:"Lucida Sans Unicode"'>ℝⁿ→ℝⁿup to translation such that f(x) = (b/a) I(x).     

Reciprocal polynomials with all zeros on the unit circle

Let f(x)=a _d x ^d +a _d−1 x ^d−1+⋅⋅⋅+a ₀∈ℝ[x] be a reciprocal polynomial of degree d. We prove that if the coefficient vector (a _d ,a _d−1,…,a ₀) or (a _d−1,a _d−2,…,a ₁) is close enough, in the l ¹-distance, to the constant vector (b,b,…,b)∈ℝ ^d+1 or ℝ ^d−1, then all of its zeros have moduli 1.     

带非局部源的退化半线性抛物方程的解的爆破性质

This paper deals with the blow-up properties of the positive solutions to the nonlocal degenerate semilinear parabolic equation u _t − (x ^a u _x ) _x =∫ ₀ ^a f(u)dx in (0,a) × (0,T) under homogeneous Dirichlet conditions. The local existence and uniqueness of classical solution are established. Under appropriate hypotheses, the global existence and blow-up in finite time of positve solutions are obtained. It is also proved that the blow-up set is almost the whole domain. This differs from the local case. Furthermore, the blow-up rate is precisely determined for the special case: f(u)=u ^p , p>1.     

Higher order Riesz transforms associated with Bessel operators

In this paper we investigate Riesz transforms R _μ ^(k) of order k≥1 related to the Bessel operator Δ_μ f(x)=-f”(x)-((2μ+1)/x)f’(x) and extend the results of Muckenhoupt and Stein for the conjugate Hankel transform (a Riesz transform of order one). We obtain that for every k≥1, R _μ ^(k) is a principal value operator of strong type (p,p), p∈(1,∞), and weak type (1,1) with respect to the measure dλ(x)=x ^2μ+1  dx in (0,∞). We also characterize the class of weights ω on (0,∞) for which R _μ ^(k) maps L ^p (ω) into itself and L ¹(ω) into L ^1,∞(ω) boundedly. This class of weights is wider than the Muckenhoupt class of weights for the doubling measure dλ. These weighted results extend the ones obtained by Andersen and Kerman.     

On Small Deviations of Series of Weighted Random Variables

Let ξ,ξ ₁,ξ ₂,… be positive i.i.d. random variables, S=∑ _j=1^∞ a(j)ξ _j , where the coefficients a(j)≥0 are such that P(S<∞)=1. We obtain an explicit form of the asymptotics of −ln P(S<x) as x→0 for the following three cases:

Equations fonctionnelles et estimations de normes de matrices

We solve independently the equations 1/θ(x)θ(y)=ψ(x)−ψ(y)+φ(x−y)/θ(x−y) and 1/θ(x)θ(y)=σ(x)−σ(y)/θ(x−y)+τ(x)τ(y), τ(0)=0. In both cases we find θ²=aθ⁴+bθ²+c. We deduce estimates for the spectral radius of a matrix of type(1/θ(x _r −x _s )) (the accent meaning that the coefficients of the main diagonal are zero) and we study the case where thex _r are equidistant.

Dédié to à Monsieur le Professeur Otto Haupt à l'occasion de son cententiare avec les meilleurs voeux     

Polynomials of best approximation inC[−1,1]

Equivalences between the condition |P _n ^(k) (x)|≦K(n ⁻¹√1−x ²+1/n ²) ^k n ^-a, whereP _n(x) is the bestn-th degree polynomial approximation tof(x), and the Peetre interpolation space betweenC[−1,1] and the space (1−x ²) ^k f ^(2k)(x)∈C[−1,1] is established. A similar result is shown forE _n(f)= ‖f−P _n‖ _C[−1,1]. Rates other thann ^-a are also discussed. Supported by NSERC grant A4816 of Canada.     

Improving on the minimum risk equivariant estimator of a location parameter which is constrained to an interval or a half-interval

For location families with densitiesf ₀(x−θ), we study the problem of estimating θ for location invariant lossL(θ,d)=ρ(d−θ), and under a lower-bound constraint of the form θ≥a. We show, that for quite general (f ₀, ρ), the Bayes estimator δ _U with respect to a uniform prior on (a, ∞) is a minimax estimator which dominates the benchmark minimum risk equivariant (MRE) estimator. In extending some previous dominance results due to Katz and Farrell, we make use of Kubokawa'sIERD (Integral Expression of Risk Difference) method, and actually obtain classes of dominating estimators which include, and are characterized in terms of δ _U . Implications are also given and, finally, the above dominance phenomenon is studied and extended to an interval constraint of the form θ∈[a, b]. Research supported by NSERC of Canada.     

Uniqueness results and asymptotic behavior of solutions with boundary blow-up for logistic-type porous media equations

Under the proper structure conditions on the nonlinear term f(u) and weight function b(x), the paper shows the uniqueness and asymptotic behavior near the boundary of boundary blow-up solutions to the porous media equations of logistic type −Δu = a(x)u ^1/m − b(x)f(u) with m > 1.     

On the Greatest Common Divisor of u ? 1 and v ? 1 with u and v Near S{\cal S}-units

In [2], it was shown that if a and b are multiplicatively independent integers and ɛ > 0, then the inequality gcd (aⁿ − 1,bⁿ − 1) < exp(ɛn) holds for all but finitely many positive integers n. Here, we generalize the above result. In particular, we show that if f(x),f₁(x),g(x),g₁(x) are non-zero polynomials with integer coefficients, then for every ɛ > 0, the inequality gcd (f(n)aⁿ+g(n), f₁(n)bⁿ+g₁(n)) < exp(ne){\rm gcd}\, (f(n)a^n+g(n), f_1(n)b^n+g_1(n)) < \exp(n\varepsilon) holds for all but finitely many positive integers n.     

Norm inequalities for a certain class of ∞ functions

In 1936 the author showed that the function sin(π(x+1)/4) is the entire function of least exponential type (=π/4) among all entire functionsf(z) with the property thatf ⁽ⁿ⁾(z) vanishes somewhere in the real interval [−1, 1] (n=0, 1,2,…). Now more precise results of this kind are obtained by working within the class ∞[−1, 1]. For Paul Montel on his 95th birthday     

An Application of a Mountain Pass Theorem

We are concerned with the following Dirichlet problem: −Δu(x) = f(x, u), x∈Ω, u∈H ¹ ₀(Ω), (P) where f(x, t) ∈C (×ℝ), f(x, t)/t is nondecreasing in t∈ℝ and tends to an L ^∞-function q(x) uniformly in x∈Ω as t→ + ∞ (i.e., f(x, t) is asymptotically linear in t at infinity). In this case, an Ambrosetti-Rabinowitz-type condition, that is, for some θ > 2, M > 0, 0 > θF(x, s) ≤f(x, s)s, for all |s|≥M and x∈Ω, (AR) is no longer true, where F(x, s) = ∫ ^s ₀ f(x, t)dt. As is well known, (AR) is an important technical condition in applying Mountain Pass Theorem. In this paper, without assuming (AR) we prove, by using a variant version of Mountain Pass Theorem, that problem (P) has a positive solution under suitable conditions on f(x, t) and q(x). Our methods also work for the case where f(x, t) is superlinear in t at infinity, i.e., q(x) ≡ +∞. Received June 24, 1998, Accepted January 14, 2000.     

Almost automorphic solutions to integral equations on the line

Given a∈L ¹(ℝ) and A the generator of an L ¹-integrable family of bounded and linear operators defined on a Banach space X, we prove the existence of almost automorphic solution to the semilinear integral equation u(t)=∫ _−∞ ^t a(t−s)[Au(s)+f(s,u(s))]ds for each f:ℝ×X→X almost automorphic in t, uniformly in x∈X, and satisfying diverse Lipschitz type conditions. In the scalar case, we prove that a∈L ¹(ℝ) positive, nonincreasing and log-convex is already sufficient.     

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.     

Triangle-free distance-regular graphs

Let Γ denote a distance-regular graph with diameter d≥3. By a parallelogram of length 3, we mean a 4-tuple xyzw consisting of vertices of Γ such that ∂(x,y)=∂(z,w)=1, ∂(x,z)=3, and ∂(x,w)=∂(y,w)=∂(y,z)=2, where ∂ denotes the path-length distance function. Assume that Γ has intersection numbers a ₁=0 and a ₂≠0. We prove that the following (i) and (ii) are equivalent. (i) Γ is Q-polynomial and contains no parallelograms of length 3; (ii) Γ has classical parameters (d,b,α,β) with b<−1. Furthermore, suppose that (i) and (ii) hold. We show that each of b(b+1)²(b+2)/c ₂, (b−2)(b−1)b(b+1)/(2+2b−c ₂) is an integer and that c ₂≤b(b+1). This upper bound for c ₂ is optimal, since the Hermitian forms graph Her₂(d) is a triangle-free distance-regular graph that satisfies c ₂=b(b+1). Work partially supported by the National Science Council of Taiwan, R.O.C.     

Explosive solutions of elliptic equations with absorption and nonlinear gradient term

Letf be a non-decreasing C¹-function such that andF(t)/f ² a(t)→ 0 ast → ∞, whereF(t)=∫ ₀ ^t f(s) ds anda ∈ (0, 2]. We prove the existence of positive large solutions to the equationΔu +q(x)|Δu| ^a =p(x)f(u) in a smooth bounded domain Ω ⊂R^N, provided thatp, q are non-negative continuous functions so that any zero ofp is surrounded by a surface strictly included in Ω on whichp is positive. Under additional hypotheses onp we deduce the existence of solutions if Ω is unbounded.