We also present a result of orbital instability of snoidal standing wave solutions to the Klein–Gordon equation
uttuxx+|u|2u=0.
The main tool to obtain these results is the classical Grillakis, Shatah and Strauss' theory in the periodic context.  相似文献   

14.
Weighted Hardy inequalities     
D.E. Edmunds  R. Hurri-Syrjnen 《Journal of Mathematical Analysis and Applications》2005,310(2):424-435
For bounded Lipschitz domains D in it is known that if 1<p<∞, then for all β[0,β0), where β0=p−1>0, there is a constant c<∞ with
for all . We show that if D is merely assumed to be a bounded domain in that satisfies a Whitney cube-counting condition with exponent λ and has plump complement, then the same inequality holds with β0 now taken to be . Further, we extend the known results (see [H. Brezis, M. Marcus, Hardy's inequalities revisited, Dedicated to Ennio De Giorgi, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 25 (1997–1998) 217–237; M. Hoffmann-Ostenhof, T. Hoffmann-Ostenhof, A. Laptev, A geometrical version of Hardy's inequality, J. Funct. Anal. 189 (2002) 537–548; J. Tidblom, A geometrical version of Hardy's inequality for W1,p(Ω), Proc. Amer. Math. Soc. 132 (2004) 2265–2271]) concerning the improved Hardy inequality
c=c(n,p), by showing that the class of domains for which the inequality holds is larger than that of all bounded convex domains.  相似文献   

15.
Hardy–Sobolev critical elliptic equations with boundary singularities     
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   

16.
Lower bounds for the merit factors of trigonometric polynomials from Littlewood classes     
Peter Borwein  Tams Erdlyi 《Journal of Approximation Theory》2003,125(2):190-197
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 satisfies
||p||L2(K)An1/2 and ||p′||L2(K)Bn3/2.
Then
M4(p)−M2(p)M2(p)
with
We also prove that
and
M2(p)−M1(p)10−31M2(p)
for every , where denotes the collection of all trigonometric polynomials of the form
  相似文献   

17.
Dissecting the Stanley partition function     
Alexander Berkovich  Frank G. Garvan 《Journal of Combinatorial Theory, Series A》2005,112(2):277-291
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   

18.
Positive solution to a special singular second-order boundary value problem   总被引:1,自引:0,他引:1  
Qingliu Yao   《Mathematical and Computer Modelling》2008,47(11-12):1284-1291
Let λ be a nonnegative parameter. The existence of a positive solution is studied for a semipositone second-order boundary value problem
where d>0,α≥0,β≥0,α+β>0, q(t)f(t,u,v)≥0 on a suitable subset of [0,1]×[0,+)×(−,+) and f(t,u,v) is allowed to be singular at t=0,t=1 and u=0. The proofs are based on the Leray–Schauder fixed point theorem and the localization method.  相似文献   

19.
BV solutions to a degenerate parabolic equation for image denoising     
孔令海郇中丹  郭柏灵 《数学物理学报(B辑英文版)》2007,27(1):169-179
In this article, the authors consider equation ut = div(ψ(Гu)A(|Du|2)Du) -(u- I), where ψ is strictly positive and Г is a known vector-valued mapping, A: R → R is decreasing and A(s) ~ 1/√a as s → ∞. This kind of equation arises naturally from image denoising. For an initial datum I ∈ BVloc ∩ L∞, the existence of BV solutions to the initial value problem of the equation is obtained.  相似文献   

20.
On positive solutions of some nonlinear fourth-order beam equations   总被引:3,自引:0,他引:3  
Zhanbing Bai  Haiyan Wang   《Journal of Mathematical Analysis and Applications》2002,270(2):357-368
The existence, uniqueness and multiplicity of positive solutions of the following boundary value problem is considered:
u(4)(t)−λf(t,u(t))=0, for 0<t<1,u(0)=u(1)=u″(0)=u″(1)=0,
where λ>0 is a constant, f :[0,1]×[0,+∞)→[0,+∞) is continuous.  相似文献   

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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   

2.
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
p(u))+f(t,u)=0, t(0,1),
  相似文献   

3.
Let be the radial subspace of the Besov space . We prove the independence of the asymptotic behavior of the entropy numbers
from the difference s0s1 as long as the embedding itself is compact. In fact, we shall show that
This is in a certain contrast to earlier results on entropy numbers in the context of Besov spaces Bp,qs(Ω) on bounded domains Ω.  相似文献   

4.
This paper deals with the existence of positive solutions for the one-dimensional p-Laplacian
subject to the boundary value conditions:
where p(s)=|s|p−2s,p>1. We show that it has at least one or two positive solutions under some assumptions by applying the fixed point theorem. The interesting points are that the nonlinear term f is involved with the first-order derivative explicitly and f may change sign.  相似文献   

5.
We consider a Cauchy problem for a semilinear heat equation
with p>pS where pS is the Sobolev exponent. If u(x,t)=(Tt)−1/(p−1)φ((Tt)−1/2x) for xRN and t[0,T), where φ is a regular positive solution of
(P)
then u is called a backward self-similar blowup solution. It is immediate that (P) has a trivial positive solution κ≡(p−1)−1/(p−1) for all p>1. Let pL be the Lepin exponent. Lepin obtained a radial regular positive solution of (P) except κ for pS<p<pL. We show that there exist no radial regular positive solutions of (P) which are spatially inhomogeneous for p>pL.  相似文献   

6.
A nonlocal p-Laplacian evolution equation with Neumann boundary conditions   总被引:1,自引:0,他引:1  
In this paper we study the nonlocal p-Laplacian type diffusion equation,
If p>1, this is the nonlocal analogous problem to the well-known local p-Laplacian evolution equation ut=div(|u|p−2u) with homogeneous Neumann boundary conditions. We prove existence and uniqueness of a strong solution, and if the kernel J is rescaled in an appropriate way, we show that the solutions to the corresponding nonlocal problems converge strongly in L(0,T;Lp(Ω)) to the solution of the p-Laplacian with homogeneous Neumann boundary conditions. The extreme case p=1, that is, the nonlocal analogous to the total variation flow, is also analyzed. Finally, we study the asymptotic behavior of the solutions as t goes to infinity, showing the convergence to the mean value of the initial condition.  相似文献   

7.
Let dλ(t) be a given nonnegative measure on the real line , with compact or infinite support, for which all moments exist and are finite, and μ0>0. Quadrature formulas of Chakalov–Popoviciu type with multiple nodes
where σ=σn=(s1,s2,…,sn) is a given sequence of nonnegative integers, are considered. A such quadrature formula has maximum degree of exactness dmax=2∑ν=1nsν+2n−1 if and only if
The proof of the uniqueness of the extremal nodes τ12,…,τn was given first by Ghizzetti and Ossicini (Rend. Mat. 6(8) (1975) 1–15). Here, an alternative simple proof of the existence and the uniqueness of such quadrature formulas is presented. In a study of the error term R(f), an influence function is introduced, its relevant properties are investigated, and in certain classes of functions the error estimate is given. A numerically stable iterative procedure, with quadratic convergence, for determining the nodes τν, ν=1,2,…,n, which are the zeros of the corresponding σ-orthogonal polynomial, is presented. Finally, in order to show a numerical efficiency of the proposed procedure, a few numerical examples are included.  相似文献   

8.
We study the large time asymptotic behavior, in Lp (1p∞), of higher derivatives Dγu(t) of solutions of the nonlinear equation
(1)
where the integers n and θ are bigger than or equal to 1, a is a constant vector in with . The function ψ is a nonlinearity such that and ψ(0)=0, and is a higher order elliptic operator with nonsmooth bounded measurable coefficients on . We also establish faster decay when .  相似文献   

9.
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(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   

10.
In this paper we prove the uniqueness of the positive solution for the boundary blow-up problem
where Ω is a C2 bounded domain in , under the hypotheses that f(t) is nondecreasing in t>0 and f(t)/tp is increasing for large t and some p>1. We also consider the uniqueness of a related problem when the equation includes a nonnegative weight a(x).  相似文献   

11.
We establish sufficient conditions for the persistence and the contractivity of solutions and the global asymptotic stability for the positive equilibrium N*=1/(a+∑i=0mbi) of the following differential equation with piecewise constant arguments:
where r(t) is a nonnegative continuous function on [0,+∞), r(t)0, ∑i=0mbi>0, bi0, i=0,1,2,…,m, and a+∑i=0mbi>0. These new conditions depend on a,b0 and ∑i=1mbi, and hence these are other type conditions than those given by So and Yu (Hokkaido Math. J. 24 (1995) 269–286) and others. In particular, in the case m=0 and r(t)≡r>0, we offer necessary and sufficient conditions for the persistence and contractivity of solutions. We also investigate the following differential equation with nonlinear delay terms:
where r(t) is a nonnegative continuous function on [0,+∞), r(t)0, 1−axg(x,x,…,x)=0 has a unique solution x*>0 and g(x0,x1,…,xm)C1[(0,+∞)×(0,+∞)××(0,+∞)].  相似文献   

12.
The purpose of this paper is to study the asymptotic behavior of the solutions of certain type of differential inclusions posed in Banach spaces. In particular, we obtain the abstract result on the asymptotic behavior of the solution of the boundary value problem
where Ω is a bounded open domain in with smooth boundary ∂Ω, f(t,x) is a given L1-function on ]0,∞[×Ω, γ1 and 1p<∞. Δp represents the p-Laplacian operator, is the associated Neumann boundary operator and β a maximal monotone graph in with 0β(0).  相似文献   

13.
In the present paper we show some results concerning the orbital stability of dnoidal standing wave solutions and orbital instability of cnoidal standing wave solutions to the following Klein–Gordon equation:
uttuxx+u−|u|2u=0.
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