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1.
Biharmonic equations with asymptotically linear nonlinearities 总被引:1,自引:1,他引:0
This article considers the equation △2u = f(x, u)with boundary conditions either u|(a)Ω = (a)u/(a)n|(a)Ω = 0 or u|(a)Ω = △u|(a)Ω = 0, where f(x,t) is asymptotically linear with respect to t at infinity, and Ω is a smooth bounded domain in RN, N > 4. By a variant version of Mountain Pass Theorem, it is proved that the above problems have a nontrivial solution under suitable assumptions of f(x, t). 相似文献
2.
Let p(n) denote the number of unrestricted partitions of n. For i=0, 2, let pi(n) denote the number of partitions π of n such that . Here denotes the number of odd parts of the partition π and π′ is the conjugate of π. Stanley [Amer. Math. Monthly 109 (2002) 760; Adv. Appl. Math., to appear] derived an infinite product representation for the generating function of p0(n)-p2(n). Recently, Swisher [The Andrews–Stanley partition function and p(n), preprint, submitted for publication] employed the circle method to show that(i) and that for sufficiently large n (ii) In this paper we study the even/odd dissection of the Stanley product, and show how to use it to prove (i) and (ii) with no restriction on n. Moreover, we establish the following new result:Two proofs of this surprising inequality are given. The first one uses the Göllnitz–Gordon partition theorem. The second one is an immediate corollary of a new partition inequality, which we prove in a combinatorial manner. Our methods are elementary. We use only Jacobi's triple product identity and some naive upper bound estimates. 相似文献
3.
With the notation
,we prove the following result.Theorem 1. Assume that p is a trigonometric polynomial of degree at most n with real coefficients that satisfiesThenwithWe also prove thatandfor every
, where
denotes the collection of all trigonometric polynomials of the form 相似文献
||p||L2(K)An1/2 and ||p′||L2(K)Bn3/2.
M4(p)−M2(p)M2(p)
M2(p)−M1(p)10−31M2(p)
4.
Let B denote the unit ball of . For 0<p<∞, the holomorphic function spaces Qp and Qp,0 on the unit ball of are defined as and In this paper, we give some derivative-free, mixture and oscillation characterizations for Qp and Qp,0 spaces in the unit ball of . 相似文献
5.
N. Ghoussoub X. S. Kang 《Annales de l'Institut Henri Poincaré (C) Analyse Non Linéaire》2004,21(6):3934-793
Unlike the non-singular case s=0, or the case when 0 belongs to the interior of a domain Ω in
(n3), we show that the value and the attainability of the best Hardy–Sobolev constant on a smooth domain Ω, when 0<s<2,
, and when 0 is on the boundary ∂Ω are closely related to the properties of the curvature of ∂Ω at 0. These conditions on the curvature are also relevant to the study of elliptic partial differential equations with singular potentials of the form: where f is a lower order perturbative term at infinity and f(x,0)=0. We show that the positivity of the sectional curvature at 0 is relevant when dealing with Dirichlet boundary conditions, while the Neumann problems seem to require the positivity of the mean curvature at 0. 相似文献
6.
Vladimir Koltchinskii 《Annales de l'Institut Henri Poincaré (B) Probabilités et Statistiques》2003,39(6):1143-978
Let
be a probability space and let Pn be the empirical measure based on i.i.d. sample (X1,…,Xn) from P. Let
be a class of measurable real valued functions on
For
define Ff(t):=P{ft} and Fn,f(t):=Pn{ft}. Given γ(0,1], define n,γ(δ):=1/(n1−γ/2δγ). We show that if the L2(Pn)-entropy of the class
grows as −α for some α(0,2), then, for all
and all δ(0,Δn), Δn=O(n1/2), and where
and c(σ)↓1 as σ↓0 (the above inequalities hold for any fixed σ(0,1] with a high probability). Also, define Then for all
uniformly in
and with probability 1 (for
the above ratio is bounded away from 0 and from ∞). The results are motivated by recent developments in machine learning, where they are used to bound the generalization error of learning algorithms. We also prove some more general results of similar nature, show the sharpness of the conditions and discuss the applications in learning theory. 相似文献
7.
In this paper, we afford some sufficient conditions to guarantee the existence of multiple positive solutions for the nonlinear m-point boundary-value problem for the one-dimensional p-Laplacian 相似文献
(φp(u′))′+f(t,u)=0, t(0,1),
8.
Blow-up analysis for a system of heat equations with nonlinear flux which obey different laws 总被引:1,自引:0,他引:1
We consider a system of heat equations ut=Δu and vt=Δv in Ω×(0,T) completely coupled by nonlinear boundary conditions We prove that the solutions always blow up in finite time for non-zero and non-negative initial values. Also, the blow-up only occurs on ∂Ω with for p,q>0, 0≤α<1 and 0≤β<p. 相似文献
9.
Zili Wu 《Journal of Approximation Theory》2002,119(2):181-192
We prove that in a Banach space X with rotund dual X* a Chebyshev set C is convex iff the distance function dC is regular on X\C iff dC admits the strict and Gâteaux derivatives on X\C which are determined by the subdifferential ∂x−
x" height="11" width="10"> for each xX\C and
x" height="11" width="10">PC(x)\{cC:x−c=dC(x)}. If X is a reflexive Banach space with smooth and Kadec norm then C is convex iff it is weakly closed iff PC is continuous. If the norms of X and X* are Fréchet differentiable then C is convex iff dC is Fréchet differentiable on X\C. If also X has a uniformly Gâteaux differentiable norm then C is convex iff the Gâteaux (Fréchet) subdifferential ∂−dC(x) (∂FdC(x)) is nonempty on X\C. 相似文献
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10.
Fordyce A. Davidson Bryan P. Rynne 《Journal of Mathematical Analysis and Applications》2004,300(2):491-504
Let TR be a time-scale, with a=infT, b=supT. We consider the nonlinear boundary value problem (2) (4)
u(a)=u(b)=0,