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1.
M. N. Yakovlev 《Journal of Mathematical Sciences》2007,141(6):1710-1722
It is proved that the boundary-value problem
, has a solution, provided that the following conditions are fulfilled:
, and, for ϕ(u) ≡ 0, the Galerkin method converges in the norm of the space H1(a, b; a). Several theorems of a similar kind are presented. Bibliography: 4 titles.
__________
Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 334, 2006, pp. 246–266. 相似文献
2.
Ruyun Ma 《Czechoslovak Mathematical Journal》2006,56(4):1243-1263
We study the existence of nodal solutions of the m-point boundary value problem
where η
i
∈ ℚ (i = 1, 2, ..., m − 2) with 0 < η
1 < η
2 < ... < η
m−2 < 1, and α
i
∈ ℝ (i = 1, 2, ..., m − 2) with α
i
> 0 and
< 1. We give conditions on the ratio f(s)/s at infinity and zero that guarantee the existence of nodal solutions. The proofs of the main results are based on bifurcation
techniques. 相似文献
3.
Mihai Mihăilescu 《Czechoslovak Mathematical Journal》2008,58(1):155-172
We study the boundary value problem in Ω, u = 0 on ∂Ω, where Ω is a smooth bounded domain in ℝ
N
. Our attention is focused on two cases when , where m(x) = max{p
1(x), p
2(x)} for any x ∈ or m(x) < q(x) < N · m(x)/(N − m(x)) for any x ∈ . In the former case we show the existence of infinitely many weak solutions for any λ > 0. In the latter we prove that if λ is large enough then there exists a nontrivial weak solution. Our approach relies on the variable exponent theory of generalized
Lebesgue-Sobolev spaces, combined with a ℤ2-symmetric version for even functionals of the Mountain Pass Theorem and some adequate variational methods. 相似文献
4.
M. N. Yakovlev 《Journal of Mathematical Sciences》2008,150(2):1998-2004
It is proved that under certain assumptions on the functions q(t) and f(t), there is one and only one function u0(t) ∈
at which the functional
attains its minimum. An error bound for the finite element method for computing the function u0(t) in terms of q(t), f(t), and the meshsize h is presented. Bibliography: 3 titles.
__________
Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 346, 2007, pp. 149–159. 相似文献
5.
Yong-ping Sun 《应用数学学报(英文版)》2011,27(2):233-242
Using the Leggett-Williams fixed point theorem,we will obtain at least three symmetric positive solutions to the second-order nonlocal boundary value problem of the form u(t)+g(t)f(t,u(t))=0,0相似文献
6.
We study the vector p-Laplacian
We prove that there exists a sequence (u
n
) of solutions of (*) such that u
n
is a critical point of ϕ and another sequence (u
n
*
) of solutions of (*) such that u
n
*
is a local minimum point of ϕ, where ϕ is a functional defined below.
The research is supported by NNSF of China (10301033). 相似文献
7.
In this paper, we consider the following nonlinear fractional three-point boundary-value problem:
*20c D0 + a u(t) + f( t,u(t) ) = 0, 0 < t < 1, u(0) = u¢(0) = 0, u¢(1) = ò0h u(s)\textds, \begin{array}{*{20}{c}} {D_{0 + }^\alpha u(t) + f\left( {t,u(t)} \right) = 0,\,\,\,\,0 < t < 1,} \\ {u(0) = u'(0) = 0,\,\,\,\,u'(1) = \int\limits_0^\eta {u(s){\text{d}}s,} } \\ \end{array} 相似文献
8.
Local and Global Existence of Solutions to Initial Value Problems of Nonlinear Kaup-Kupershmidt Equations 总被引:6,自引:0,他引:6
Shuang Ping TAO Shang Bin CUI 《数学学报(英文版)》2005,21(4):881-892
This paper is devoted to studying the initial value problems of the nonlinear Kaup Kupershmidt equations δu/δt + α1 uδ^2u/δx^2 + βδ^3u/δx^3 + γδ^5u/δx^5 = 0, (x,t)∈ E R^2, and δu/δt + α2 δu/δx δ^2u/δx^2 + βδ^3u/δx^3 + γδ^5u/δx^5 = 0, (x, t) ∈R^2. Several important Strichartz type estimates for the fundamental solution of the corresponding linear problem are established. Then we apply such estimates to prove the local and global existence of solutions for the initial value problems of the nonlinear Kaup- Kupershmidt equations. The results show that a local solution exists if the initial function u0(x) ∈ H^s(R), and s ≥ 5/4 for the first equation and s≥301/108 for the second equation. 相似文献
9.
Abdelmajid Siai 《Potential Analysis》2006,24(1):15-45
Let Ω be an open bounded set in ℝN, N≥3, with connected Lipschitz boundary ∂Ω and let a(x,ξ) be an operator of Leray–Lions type (a(⋅,∇u) is of the same type as the operator |∇u|p−2∇u, 1<p<N). If τ is the trace operator on ∂Ω, [φ] the jump across ∂Ω of a function φ defined on both sides of ∂Ω, the normal derivative
∂/∂νa related to the operator a is defined in some sense as 〈a(⋅,∇u),ν〉, the inner product in ℝN, of the trace of a(⋅,∇u) on ∂Ω with the outward normal vector field ν on ∂Ω. If β and γ are two nondecreasing continuous real functions everywhere
defined in ℝ, with β(0)=γ(0)=0, f∈L1(ℝN), g∈L1(∂Ω), we prove the existence and the uniqueness of an entropy solution u for the following problem,
10.
E. G. Goluzina 《Journal of Mathematical Sciences》1998,89(1):958-966
Let TR be the class of functions
that are regular and typically real in the disk E={z:⋱z⋱<1}. For this class, the region of values of the system {f(z0), f(r)} for z0 ∈ ℝ, r∈(-1,1) is studied. The sets Dr={f(z0):f∈TR, f(r)=a} for −1≤r≤1 and Δr={(c2, c3): f ∈ TR, −f(−r)=a} for 0<r≤1 are found, where aε(r(1+r)−2, r(1−r)−2) is an arbitrary fixed number. Bibliography: 11 titles.
Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 226, 1996, pp. 69–79. 相似文献
11.
Let (M, g) be a smooth compact Riemannian manifold of dimension n≥5, and
2
2
(M) be the Sobolev space consisting of functions in L2(M) whose derivatives up to the order two are also in L2(M). Thanks to the Sobolev embedding theorem, there exist positive constants A and B such that for any U ∈ H
2
2
(M),
12.
Here, we solve non-convex, variational problems given in the form
13.
We consider the weighted Hardy integral operatorT:L
2(a, b) →L
2(a, b), −∞≤a<b≤∞, defined by
. In [EEH1] and [EEH2], under certain conditions onu andv, upper and lower estimates and asymptotic results were obtained for the approximation numbersa
n(T) ofT. In this paper, we show that under suitable conditions onu andv,
where ∥w∥p=(∫
a
b
|w(t)|p
dt)1/p.
Research supported by NSERC, grant A4021.
Research supported by grant No. 201/98/P017 of the Grant Agency of the Czech Republic. 相似文献
14.
Yu. A. Grigor'ev 《Ukrainian Mathematical Journal》1997,49(8):1276-1278
For a function u(x, y) harmonic in the upper half-plane y>0 and represented by the Poisson integral of a function v(t) ∈ L
2
(−∞,∞), we prove that the inequality is true. A similar inequality is obtained for a function harmonic in a disk.
Odessa Marine University, Odessa. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 49, No. 8, pp. 1135–1136, August,
1997. 相似文献
15.
Alessandra Pagano 《Annali dell'Universita di Ferrara》1993,39(1):1-17
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,…,Dn−1u∈Lq, under suitable assumptions ona
i(x).
Sunto Consideriamo una funzione u: Ω→R N, Ω⊂R n che minimizzi l'integrale , doveD iu=∂u/∂xi, o un funzionale con un comportamento simile; sotto opportune ipotesi sua i(x), dimostriamo la seguente maggiore integrabilità perDu:D 1u,…,Dn−1uεLq.相似文献 16.
Linghai ZHANG 《数学年刊B辑(英文版)》2008,29(2):179-198
Let u=u(x,t,uo)represent the global solution of the initial value problem for the one-dimensional fluid dynamics equation ut-εuxxt+δux+γHuxx+βuxxx+f(u)x=αuxx,u(x,0)=uo(x), whereα〉0,β〉0,γ〉0,δ〉0 andε〉0 are constants.This equation may be viewed as a one-dimensional reduction of n-dimensional incompressible Navier-Stokes equations. The nonlinear function satisfies the conditions f(0)=0,|f(u)|→∞as |u|→∞,and f∈C^1(R),and there exist the following limits Lo=lim sup/u→o f(u)/u^3 and L∞=lim sup/u→∞ f(u)/u^5 Suppose that the initial function u0∈L^I(R)∩H^2(R).By using energy estimates,Fourier transform,Plancherel's identity,upper limit estimate,lower limit estimate and the results of the linear problem vt-εv(xxt)+δvx+γHv(xx)+βv(xxx)=αv(xx),v(x,0)=vo(x), the author justifies the following limits(with sharp rates of decay) lim t→∞[(1+t)^(m+1/2)∫|uxm(x,t)|^2dx]=1/2π(π/2α)^(1/2)m!!/(4α)^m[∫R uo(x)dx]^2, if∫R uo(x)dx≠0, where 0!!=1,1!!=1 and m!!=1·3…(2m-3)…(2m-1).Moreover lim t→∞[(1+t)^(m+3/2)∫R|uxm(x,t)|^2dx]=1/2π(x/2α)^(1/2)(m+1)!!/(4α)^(m+1)[∫Rρo(x)dx]^2, if the initial function uo(x)=ρo′(x),for some functionρo∈C^1(R)∩L^1(R)and∫Rρo(x)dx≠0. 相似文献
17.
Cao Jiading 《分析论及其应用》1989,5(2):99-109
Let an≥0 and F(u)∈C [0,1], Sikkema constructed polynomials:
, ifα
n
≡0, then Bn (0, F, x) are Bernstein polynomials.
Let
, we constructe new polynomials in this paper:
Q
n
(k)
(α
n
,f(t))=d
k
/dx
k
B
n+k
(α
n
,F
k
(u),x), which are called Sikkema-Kantorovic polynomials of order k. Ifα
n
≡0, k=1, then Qn
(1) (0, f(t), x) are Kantorovič polynomials Pn(f). Ifα
n
=0, k=2, then Qn
(2), (0, f(t), x) are Kantorovič polynomials of second order (see Nagel). The main result is:
Theorem 2. Let 1≤p≤∞, in order that for every f∈LP [0, 1],
, it is sufficient and necessary that
,
§ 1. Let f(t) de a continuous function on [a, b], i. e., f∈C [a, b], we define[1–2],[8–10]:
.
As usual, for the space Lp [a,b](1≤p<∞), we have
and L[a, b]=l1[a, b].
Letα
n
⩾0and F(u)∈C[0,1],Sikkema-Bernstein polynomials
[3] [4].
The author expresses his thanks to Professor M. W. Müller of Dortmund University at West Germany for his supports. 相似文献
18.
Let f∈C
[−1,1]
″
(r≥1) and Rn(f,α,β,x) be the generalized Pál interpolation polynomials satisfying the conditions Rn(f,α,β,xk)=f(xk),Rn
′(f,α,β,xk)=f′(xk)(k=1,2,…,n), where {xk} are the roots of n-th Jacobi polynomial Pn(α,β,x),α,β>−1 and {x
k
″
} are the roots of (1−x2)Pn″(α,β,x). In this paper, we prove that
holds uniformly on [0,1].
In Memory of Professor M. T. Cheng
Supported by the Science Foundation of CSBTB and the Natural Science Foundatioin of Zhejiang. 相似文献
19.
Let Ω be a bounded domain with a smooth C
2 boundary in ℝn (n ≥ 3), 0 ∈
, and υ denote the unit outward normal to ∂Ω. In this paper, we are concerned with the following class of boundary value problems:
20.
Emin Özçag 《Proceedings Mathematical Sciences》1999,109(1):87-94
The distributionF(x
+, −r) Inx+ andF(x
−, −s) corresponding to the functionsx
+
−r lnx+ andx
−
−s respectively are defined by the equations
|