共查询到20条相似文献,搜索用时 937 毫秒
1.
An Application of a Mountain Pass Theorem 总被引:3,自引:0,他引:3
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. 相似文献
2.
WU Hao & LI Weigu School of Mathematical Sciences Peking University Beijing China 《中国科学A辑(英文版)》2005,48(12):1670-1682
In this paper, we consider the following autonomous system of differential equations: x = Ax f(x,θ), θ = ω, where θ∈Rm, ω = (ω1,…,ωm) ∈ Rm, x ∈ Rn, A ∈ Rn×n is a constant matrix and is hyperbolic, f is a C∞ function in both variables and 2π-periodic in each component of the vector e which satisfies f = O(||x||2) as x → 0. We study the normal form of this system and prove that under some proper conditions this system can be transformed to an autonomous system: x = Ax g(x), θ = ω. Additionally, the proof of this paper naturally implies the extension of Chen's theory in the quasi-periodic case. 相似文献
3.
In this work we prove that the initial value problem of the Benney-Lin equation ut + uxxx + β(uxx + u xxxx) + ηuxxxxx + uux = 0 (x ∈ R, t ≥0 0), where β 〉 0 and η∈R, is locally well-posed in Sobolev spaces HS(R) for s ≥ -7/5. The method we use to prove this result is the bilinear estimate method initiated by Bourgain. 相似文献
4.
Consider the Cauchy problem ∂u(x, t)/∂t = ℋu(x, t) (x∈ℤd, t≥ 0) with initial condition u(x, 0) ≡ 1 and with ℋ the Anderson Hamiltonian ℋ = κΔ + ξ. Here Δ is the discrete Laplacian, κ∈ (0, ∞) is a diffusion constant,
and ξ = {ξ(x): x∈ℤ
d
} is an i.i.d.random field taking values in ℝ. G?rtner and Molchanov (1990) have shown that if the law of ξ(0) is nondegenerate,
then the solution u is asymptotically intermittent.
In the present paper we study the structure of the intermittent peaks for the special case where the law of ξ(0) is (in the
vicinity of) the double exponential Prob(ξ(0) > s) = exp[−e
s
/θ] (s∈ℝ). Here θ∈ (0, ∞) is a parameter that can be thought of as measuring the degree of disorder in the ξ-field. Our main result
is that, for fixed x, y∈ℤ
d
and t→∈, the correlation coefficient of u(x, t) and u(y, t) converges to ∥w
ρ∥−2
ℓ2Σz ∈ℤd
w
ρ(x+z)w
ρ(y+z). In this expression, ρ = θ/κ while w
ρ:ℤd→ℝ+ is given by w
ρ = (v
ρ)⊗
d
with v
ρ: ℤ→ℝ+ the unique centered ground state (i.e., the solution in ℓ2(ℤ) with minimal l
2-norm) of the 1-dimensional nonlinear equation Δv + 2ρv log v = 0. The uniqueness of the ground state is actually proved only for large ρ, but is conjectured to hold for any ρ∈ (0, ∞).
empty
It turns out that if the right tail of the law of ξ(0) is thicker (or thinner) than the double exponential, then the correlation
coefficient of u(x, t) and u(y, t) converges to δ
x, y
(resp.the constant function 1). Thus, the double exponential family is the critical class exhibiting a nondegenerate correlation
structure.
Received: 5 March 1997 / Revised version: 21 September 1998 相似文献
5.
Hydrodynamic large scale limit for the Ginzburg-Landau ∇φ interface model was established in [6]. As its next stage this
paper studies the corresponding large deviation problem. The dynamic rate functional is given by
for h=h(t,θ),t∈[0,T],θ∈?
d
, where σ=σ(u) is the surface tension for mean tilt u∈ℝ
d
. Our main tool is H
−1-method expoited by Landim and Yau [9]. The relationship to the rate functional obtained under the static situation by Deuschel
et al. [3] is also discussed.
Received: 22 February 2000 / Revised version: 19 October 2000 / Published online: 5 June 2001 相似文献
6.
Mohammad Ashraf Nadeem-ur-Rehman Shakir Ali Muzibur Rahman Mozumder 《Mathematica Slovaca》2012,62(3):451-460
The main purpose of this paper is to prove the following result: Let R be a 2-torsion free semiprime *-ring. Suppose that θ, φ are endomorphisms of R such that θ is onto. If there exists an additive mapping F: R → R associated with a (θ, φ)-derivation d of R such that F(xx*) = F(x)θ(x*) + φ(x)d(x*) holds for all x ∈ R, then F is a generalized (θ, φ)-derivation. Further, some more related results are obtained. 相似文献
7.
N. G. Khoma 《Ukrainian Mathematical Journal》1998,50(12):1917-1923
In three spaces, we find exact classical solutions of the boundary-value periodic problem utt - a2uxx = g(x, t) u(0, t) = u(π, t) = 0, u(x, t + T) = u(x, t), x ∈ ℝ, t ∈ ℝ. We study the periodic boundary-value problem for a quasilinear equation whose left-hand side is the d’Alembert operator
and whose right-hand side is a nonlinear operator.
Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 50, No. 12, pp. 1680–1685, December, 1998. 相似文献
8.
Summary. We consider the Cauchy problem for the mass density ρ of particles which diffuse in an incompressible fluid. The dynamical
behaviour of ρ is modeled by a linear, uniformly parabolic differential equation containing a stochastic vector field. This
vector field is interpreted as the velocity field of the fluid in a state of turbulence. Combining a contraction method with
techniques from white noise analysis we prove an existence and uniqueness result for the solution ρ∈C
1,2([0,T]×ℝ
d
,(S)*), which is a generalized random field. For a subclass of Cauchy problems we show that ρ actually is a classical random field,
i.e. ρ(t,x) is an L
2-random variable for all time and space parameters (t,x)∈[0,T]×ℝ
d
.
Received: 27 March 1995 / In revised form: 15 May 1997 相似文献
9.
We consider the parametric programming problem (Q
p
) of minimizing the quadratic function f(x,p):=x
T
Ax+b
T
x subject to the constraint Cx≤d, where x∈ℝ
n
, A∈ℝ
n×n
, b∈ℝ
n
, C∈ℝ
m×n
, d∈ℝ
m
, and p:=(A,b,C,d) is the parameter. Here, the matrix A is not assumed to be positive semidefinite. The set of the global minimizers and the set of the local minimizers to (Q
p
) are denoted by M(p) and M
loc
(p), respectively. It is proved that if the point-to-set mapping M
loc
(·) is lower semicontinuous at p then M
loc
(p) is a nonempty set which consists of at most ?
m,n
points, where ?
m,n
= is the maximal cardinality of the antichains of distinct subsets of {1,2,...,m} which have at most n elements. It is proved also that the lower semicontinuity of M(·) at p implies that M(p) is a singleton. Under some regularity assumption, these necessary conditions become the sufficient ones.
Received: November 5, 1997 / Accepted: September 12, 2000?Published online November 17, 2000 相似文献
10.
In this note, the authors resolve an evolutionary Wente's problem associated to heat equation, where the special integrability of det▽u for u ∈ H1(R2,R2) is used. 相似文献
11.
V. V. Kapustin 《Journal of Mathematical Sciences》2007,141(5):1538-1542
Let θ be an inner function, let K
θ
= H
2 ⊖ θH
2, and let Sθ : Kθ → Sθ be defined by the formula Sθf = Pθzf, where f ∈ Kθ is the orthogonal projection of H2 onto Kθ. Consider the set A of all trace class operators L : Kθ → Kθ, L = ∑(·,un)vn, ∑∥un∥∥vn∥ < ∞ (un, vn ∈ Kθ), such that ∑ūn vn ∈ H
0
1
. It is shown that trace class commutators of the form XSθ − SθX (where X is a bounded linear operator on Kθ) are dense in A in the trace class norm. Bibliography: 2 titles.
__________
Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 333, 2006, pp. 54–61. 相似文献
12.
A. Nemirovskii 《Mathematical Programming》1996,73(2):175-198
We establish polynomial time convergence of the method of analytic centers for the fractional programming problemt→min |x∈G, tB(x)−A(x)∈K, whereG ⊂ ℝ
n
is a closed and bounded convex domain,K ⊂ ℝ
m
is a closed convex cone andA(x):G → ℝ
n
,B(x):G→K are regular enough (say, affine) mappings.
This research was partly supported by grant #93-012-499 of the Fundamental Studies Foundation of Russian Academy of Sciences 相似文献
13.
Daniel W. Cunningham 《Archive for Mathematical Logic》2002,41(1):49-54
Jensen's celebrated Covering Lemma states that if 0# does not exist, then for any uncountable set of ordinals X, there is a Y∈L such that X⊆Y and |X| = |Y|. Working in ZF + AD alone, we establish the following analog: If ℝ# does not exist, then L(ℝ) and V have exactly the same sets of reals and for any set of ordinals X with |X| ≥Θ
L
(ℝ), there is a Y∈L(ℝ) such that X⊆Y and |X| = |Y|. Here ℝ is the set of reals and Θ is the supremum of the ordinals which are the surjective image of ℝ.
Received: 29 October 1999 / Published online: 12 December 2001 相似文献
14.
Mikhail A. Chebotar Wen-Fong Ke Pjek-Hwee Lee Ruibin Zhang 《Monatshefte für Mathematik》2006,162(1):91-101
Let R be a ring, A = M
n
(R) and θ: A → A a surjective additive map preserving zero Jordan products, i.e. if x,y ∈ A are such that xy + yx = 0, then θ(x)θ(y) + θ(y)θ(x) = 0. In this paper, we show that if R contains
\frac12\frac{1}{2}
and n ≥ 4, then θ = λϕ, where λ = θ(1) is a central element of A and ϕ: A → A is a Jordan homomorphism. 相似文献
15.
András Máthé 《Israel Journal of Mathematics》2008,164(1):285-302
We show that Hausdorff measures of different dimensions are not Borel isomorphic; that is, the measure spaces (ℝ, B, H
s
) and (ℝ, B, H
t
) are not isomorphic if s ≠ t, s, t ∈ [0, 1], where B is the σ-algebra of Borel subsets of ℝ and H
d
is the d-dimensional Hausdorff measure. This answers a question of B. Weiss and D. Preiss.
To prove our result, we apply a random construction and show that for every Borel function ƒ: ℝ → ℝ and for every d ∈ [0, 1] there exists a compact set C of Hausdorff dimension d such that ƒ(C) has Hausdorff dimension ≤ d.
We also prove this statement in a more general form: If A ⊂ ℝn is Borel and ƒ: A → ℝm is Borel measurable, then for every d ∈ [0, 1] there exists a Borel set B ⊂ A such that dim B = d·dim A and dim ƒ(B) ≤ d·dim ƒ (A).
Partially supported by the Hungarian Scientific Research Fund grant no. T 49786. 相似文献
16.
A subgroup D of GL (n, ℝ) is said to be admissible if the semidirect product of D and ℝ
n
, considered as a subgroup of the affine group on ℝ
n
, admits wavelets ψ ∈ L2(ℝ
n
) satisfying a generalization of the Calderón reproducing, formula. This article provides a nearly complete characterization
of the admissible subgroups D. More precisely, if D is admissible, then the stability subgroup Dx for the transpose action of D on ℝ
n
must be compact for a. e. x. ∈ ℝ
n
; moreover, if Δ is the modular function of D, there must exist an a ∈ D such that |det a| ≠ Δ(a). Conversely, if the last condition holds and for a. e. x ∈ ℝ
n
there exists an ε > 0 for which the ε-stabilizer D
x
ε
is compact, then D is admissible. Numerous examples are given of both admissible and non-admissible groups. 相似文献
17.
Yu. M. Semenov 《Differential Equations》2011,47(11):1668-1674
We describe the controllability sets of linear nonautonomous systems ẋ = A(t)x + B(t)u, x ∈ ℝ
n
, u ∈ U ⊆ ℝ
m
, with entire matrix functions A(t) and B(t) and with a linear set U of control constraints. We derive a criterion for the complete controllability of these linear systems in terms of derivatives
of the entire matrix functions A(t) and B(t) at zero. This complete controllability criterion is compared with the Kalman and Krasovskii criteria. 相似文献
18.
We consider the Cauchy problem of Navier-Stokes equations in weak Morrey spaces. We first define a class of weak Morrey type spaces Mp*,λ(Rn) on the basis of Lorentz space Lp,∞ = Lp*(Rn)(in particular, Mp*,0(Rn) = Lp,∞, if p > 1), and study some fundamental properties of them; Second,bounded linear operators on weak Morrey spaces, and establish the bilinear estimate in weak Morrey spaces. Finally, by means of Kato's method and the contraction mapping principle, we prove that the Cauchy problem of Navier-Stokes equations in weak Morrey spaces Mp*,λ(Rn) (1<p≤n) is time-global well-posed, provided that the initial data are sufficiently small. Moreover, we also obtain the existence and uniqueness of the self-similar solution for Navier-Stokes equations in these spaces, because the weak Morrey space Mp*,n-p(Rn) can admit the singular initial data with a self-similar structure. Hence this paper generalizes Kato's results. 相似文献
19.
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
20.
In this paper, sufficient conditions are obtained, so that the second order neutral delay differential equation
has a positive and bounded solution, where q, h, f ∈ C ([0, ∞), ℝ) such that q(t) ≥ 0, but ≢ 0, h(t) ≤ t, h(t) → ∞ as t → ∞, r ∈ C
(1) ([0, ∞), (0, ∞)), p ∈ C
(2) [0, ∞), ℝ), G ∈ C(ℝ, ℝ) and τ ∈ ℝ+. In our work r(t) ≡ 1 is admissible and neither we assume G is non-decreasing, xG(x) > 0 for x ≠ 0, nor we take G is Lipschitzian. Hence the results of this paper improve many recent results.
相似文献