共查询到20条相似文献,搜索用时 46 毫秒
1.
I. V. Sadovnichaya 《Differential Equations》2008,44(5):675-684
We study the Sturm-Liouville operator L = ?d 2/dx 2 + q(x) in the space L 2[0, π] with the Dirichlet boundary conditions. We assume that the potential has the form q(x) = u′(x), u ∈ W 2 θ [0, π], 0 < θ < 1/2. We consider the problem on the uniform (on the entire interval [0, π]) equiconvergence of the expansion of a function f(x) in a series in the system of root functions of the operator L with its Fourier expansion in the system of sines. We show that if the antiderivative u(x) of the potential belongs to any of the spaces W 2 θ [0, π], 0 < θ < 1/2, then the equiconvergence rate can be estimated uniformly over the ball u(x) ∈ B R = {v(x) ∈ W 2 θ [0, π] | ∥v∥W 2 θ ≤ R} for any function f(x) ∈ L 2[0, π]. 相似文献
2.
V. P. Kostov 《Functional Analysis and Its Applications》2016,50(2):153-156
We consider the partial theta function θ(q, x) := ∑j=0∞qj(j+1)/2xj, where x ∈ ? is a variable and q ∈ ?, 0 < |q| < 1, is a parameter. We show that, for any fixed q, if ζ is a multiple zero of the function θ(q, · ), then |ζ| ≤ 811. 相似文献
3.
We investigate the equiconvergence on TN = [?π, π)N of expansions in multiple trigonometric Fourier series and in the Fourier integrals of functions f ∈ Lp(TN) and g ∈ Lp(RN), p > 1, N ≥ 3, g(x) = f(x) on TN, in the case where the “partial sums” of these expansions, i.e., Sn(x; f) and Jα(x; g), respectively, have “numbers” n ∈ ZN and α ∈ RN (nj = [αj], j = 1,..., N, [t] is the integral part of t ∈ R1) containing N ? 1 components which are elements of “lacunary sequences.” 相似文献
4.
L. V. Kritskov 《Differential Equations》2018,54(8):1032-1048
For the system of root functions of an operator defined by the differential operation ?u″ + p(x)u′ + q(x)u, x ∈ G = (0, 1), with complex-valued singular coefficients, sufficient conditions for the Bessel property in the space L2(G) are obtained and a theorem on the unconditional basis property is proved. It is assumed that the functions p(x) and q(x) locally belong to the spaces L2 and W2?1, respectively, and may have singularities at the endpoints of G such that q(x) = qR(x) +q′S(x) and the functions qS(x), p(x), q 2 S (x)w(x), p2(x)w(x), and qR(x)w(x) are integrable on the whole interval G, where w(x) = x(1 ? x). 相似文献
5.
We prove that the mixed problem for the Klein–Gordon–Fock equation u tt (x, t) ? u xx (x, t) + au(x, t) = 0, where a ≥ 0, in the rectangle Q T = [0 ≤ x ≤ l] × [0 ≤ t ≤ T] with zero initial conditions and with the boundary conditions u(0, t) = μ(t) ∈ L p [0, T ], u(l, t) = 0, has a unique generalized solution u(x, t) in the class L p (Q T ) for p ≥ 1. We construct the solution in explicit analytic form. 相似文献
6.
In this paper, we study the following stochastic Hamiltonian system in ?2d (a second order stochastic differential equation): where b(x; v) : ?2d → ?d and σ(x; v): ?2d → ?d ? ?d are two Borel measurable functions. We show that if σ is bounded and uniformly non-degenerate, and b ∈ H p 2/3,0 and ?σ ∈ Lp for some p > 2(2d+1), where H p α, β is the Bessel potential space with differentiability indices α in x and β in v, then the above stochastic equation admits a unique strong solution so that (x, v) ? Zt(x, v) := (Xt, ?t)(x, v) forms a stochastic homeomorphism flow, and (x, v) ? Zt(x, v) is weakly differentiable with ess.supx, v E(supt∈[0, T] |?Zt(x, v)|q) < ∞ for all q ? 1 and T ? 0. Moreover, we also show the uniqueness of probability measure-valued solutions for kinetic Fokker-Planck equations with rough coefficients by showing the well-posedness of the associated martingale problem and using the superposition principle established by Figalli (2008) and Trevisan (2016).
相似文献
$$d{\dot X_t} = b({X_t},{\dot X_t})dt + \sigma ({X_t},{\dot X_t})d{W_t},({X_0},{\dot X_0}) = (x,v) \in \mathbb{R}^{2d},$$
7.
We prove the existence of a completely integrable Pfaffian system ?x/?t i = A i (t)x, x ∈ R n , t = (t 1, t 2, t 3) ∈ R + 3 , i = 1, 2, 3, with infinitely differentiable bounded coefficients and with lower characteristic set of positive three-dimensional Lebesgue measure. 相似文献
8.
I. V. Sadovnichaya 《Proceedings of the Steklov Institute of Mathematics》2008,261(1):243-252
We consider the Sturm-Liouville operator L = ?d 2/dx 2 + q(x) with the Dirichlet boundary conditions in the space L 2[0, π] under the assumption that the potential q(x) belongs to W 2 ?1 [0, π]. We study the problem of uniform equiconvergence on the interval [0, π] of the expansion of a function f(x) in the system of eigenfunctions and associated functions of the operator L and its Fourier sine series expansion. We obtain sufficient conditions on the potential under which this equiconvergence holds for any function f(x) of class L 1. We also consider the case of potentials belonging to the scale of Sobolev spaces W 2 ?θ [0, π] with ½ < θ ≤ 1. We show that if the antiderivative u(x) of the potential belongs to some space W 2 θ [0, π] with 0 < θ < 1/2, then, for any function in the space L 2[0, π], the rate of equiconvergence can be estimated uniformly in a ball lying in the corresponding space and containing u(x). We also give an explicit estimate for the rate of equiconvergence. 相似文献
9.
A. M. Savchuk 《Proceedings of the Steklov Institute of Mathematics》2008,261(1):237-242
In the space L 2[0, π], the Sturm-Liouville operator L D(y) = ?y″ + q(x)y with the Dirichlet boundary conditions y(0) = y(π) = 0 is analyzed. The potential q is assumed to be singular; namely, q = σ′, where σ ∈ L 2[0, π], i.e., q ∈ W 2 ?1 [0, π]. The inverse problem of reconstructing the function σ from the spectrum of the operator L D is solved in the subspace of odd real functions σ(π/2 ? x) = ?σ(π/2 + x). The existence and uniqueness of a solution to this inverse problem is proved. A method is proposed that allows one to solve this problem numerically. 相似文献
10.
Malkhaz Ashordia 《Czechoslovak Mathematical Journal》2017,67(3):579-608
A general theorem (principle of a priori boundedness) on solvability of the boundary value problem dx = dA(t) · f(t, x), h(x) = 0 is established, where f: [a, b]×R n → R n is a vector-function belonging to the Carathéodory class corresponding to the matrix-function A: [a, b] → R n×n with bounded total variation components, and h: BVs([a, b],R n ) → R n is a continuous operator. Basing on the mentioned principle of a priori boundedness, effective criteria are obtained for the solvability of the system under the condition x(t1(x)) = B(x) · x(t 2(x))+c 0, where t i: BVs([a, b],R n ) → [a, b] (i = 1, 2) and B: BVs([a, b], R n ) → R n are continuous operators, and c 0 ∈ R n . 相似文献
11.
Let ?? m be the m-dimensional unit torus, m ∈ ?. The torsional rigidity of an open set Ω ? ?? m is the integral with respect to Lebesgue measure over all starting points x ∈ Ω of the expected lifetime in Ω of a Brownian motion starting at x. In this paper we consider Ω = ?? m \β[0, t], the complement of the path ß[0, t] of an independent Brownian motion up to time t. We compute the leading order asymptotic behaviour of the expectation of the torsional rigidity in the limit as t → ∞. For m = 2 the main contribution comes from the components in ??2\β0, t] whose inradius is comparable to the largest inradius, while for m = 3 most of ??3\β[0, t] contributes. A similar result holds for m ≥ 4 after the Brownian path is replaced by a shrinking Wiener sausage W r(t)[0, t] of radius r(t) = o(t -1/(m-2)), provided the shrinking is slow enough to ensure that the torsional rigidity tends to zero. Asymptotic properties of the capacity of ß[0, t] in ?3 and W 1[0, t] in ? m , m ≥ 4, play a central role throughout the paper. Our results contribute to a better understanding of the geometry of the complement of Brownian motion on ?? m , which has received a lot of attention in the literature in past years. 相似文献
12.
We study the blow-up and/or global existence of the following p-Laplacian evolution equation with variable source power where Ω is either a bounded domain or the whole space ? N , q(x) is a positive and continuous function defined in Ω with 0 < q ? = inf q(x) ? q(x) ? sup q(x) = q+ < ∞. It is demonstrated that the equation with variable source power has much richer dynamics with interesting phenomena which depends on the interplay of q(x) and the structure of spatial domain Ω, compared with the case of constant source power. For the case that Ω is a bounded domain, the exponent p ? 1 plays a crucial role. If q+ > p ? 1, there exist blow-up solutions, while if q + < p ? 1, all the solutions are global. If q ? > p ? 1, there exist global solutions, while for given q ? < p ? 1 < q +, there exist some function q(x) and Ω such that all nontrivial solutions will blow up, which is called the Fujita phenomenon. For the case Ω = ? N , the Fujita phenomenon occurs if 1 < q ? ? q + ? p ? 1 + p/N, while if q ? > p ? 1 + p/N, there exist global solutions.
相似文献
$${s_j} = {\beta _j} + \overline {{\beta _{n - j}}}p$$
13.
This paper is concerned with the oscillatory behavior of the damped half-linear oscillator (a(t)?p(x′))′ + b(t)?p(x′) + c(t)?p(x) = 0, where ?p(x) = |x|p?1 sgn x for x ∈ ? and p > 1. A sufficient condition is established for oscillation of all nontrivial solutions of the damped half-linear oscillator under the integral averaging conditions. The main result can be given by using a generalized Young’s inequality and the Riccati type technique. Some examples are included to illustrate the result. Especially, an example which asserts that all nontrivial solutions are oscillatory if and only if p ≠ 2 is presented. 相似文献
14.
Basudeb Dhara 《Czechoslovak Mathematical Journal》2018,68(1):95-119
Let R be a noncommutative prime ring of characteristic different from 2 with Utumi quotient ring U and extended centroid C, let F, G and H be three generalized derivations of R, I an ideal of R and f(x1,..., x n ) a multilinear polynomial over C which is not central valued on R. If for all r = (r1,..., r n ) ∈ I n , then one of the following conditions holds:
相似文献
$$F(f(r))G(f(r)) = H(f(r)^2 )$$
- (1)there exist a ∈ C and b ∈ U such that F(x) = ax, G(x) = xb and H(x) = xab for all x ∈ R
- (2)there exist a, b ∈ U such that F(x) = xa, G(x) = bx and H(x) = abx for all x ∈ R, with ab ∈ C
- (3)there exist b ∈ C and a ∈ U such that F(x) = ax, G(x) = bx and H(x) = abx for all x ∈ R
- (4)f(x1,..., x n )2 is central valued on R and one of the following conditions holds
- (a)there exist a, b, p, p’ ∈ U such that F(x) = ax, G(x) = xb and H(x) = px + xp’ for all x ∈ R, with ab = p + p’
- (b)there exist a, b, p, p’ ∈ U such that F(x) = xa, G(x) = bx and H(x) = px + xp’ for all x ∈ R, with p + p’ = ab ∈ C.
- (a)
15.
V. M. Badkov 《Proceedings of the Steklov Institute of Mathematics》2009,264(1):39-43
Let {p n (t)} n=0 t8 be a system of algebraic polynomials orthonormal on the segment [?1, 1] with a weight p(t); let {x n,ν (p) } ν=1 n be zeros of a polynomial p n (t) (x x,ν (p) = cosθ n,ν (p) ; 0 < θ n,1 (p) < θ n,2 (p) < ... < θ n,n (p) < π). It is known that, for a wide class of weights p(t) containing the Jacobi weight, the quantities θ n,1 (p) and 1 ? x n,1 (p) coincide in order with n ?1 and n ?2, respectively. In the present paper, we prove that, if the weight p(t) has the form p(t) = 4(1 ? t 2)?1{ln2[(1 + t)/(1 ? t)] + π 2}?1, then the following asymptotic formulas are valid as n → ∞:
相似文献
$$\theta _{n,1}^{(p)} = \frac{{\sqrt 2 }}{{n\sqrt {\ln (n + 1)} }}\left[ {1 + {\rm O}\left( {\frac{1}{{\ln (n + 1)}}} \right)} \right],x_{n,1}^{(p)} = 1 - \left( {\frac{1}{{n^2 \ln (n + 1)}}} \right) + O\left( {\frac{1}{{n^2 \ln ^2 (n + 1)}}} \right).$$
16.
We consider integrals of the form , where h is a small positive parameter and S(x, θ) and f(τ, x, θ) are smooth functions of variables τ ∈ ?, x ∈ ? n , and θ ∈ ? k ; moreover, S(x, θ) is real-valued and f(τ, x, θ) rapidly decays as |τ| →∞. We suggest an approach to the computation of the asymptotics of such integrals as h → 0 with the use of the abstract stationary phase method.
相似文献
$$I\left( {x,h} \right) = \frac{1}{{{{\left( {2\pi h} \right)}^{k/2}}}}\int_{{\mathbb{R}^k}} {f\left( {\frac{{S\left( {x,\theta } \right)}}{h},x,\theta } \right)} d\theta $$
17.
Let R be a prime ring of char R ≠ 2, let d be a nonzero derivation of R, and let ρ be a nonzero right ideal of R such that [[d(x)x n , d(y)] m , [y, x] s ] t = 0 for all x, y ? ρ, where n ≥ 1, m ≥ 0, s ≥ 0, and t ≥ 1 are fixed integers. If [ρ, ρ]ρ ≠ 0 then d(ρ)ρ = 0. 相似文献
18.
We study connecting orbits of a natural Lagrangian system defined on a complete Riemannian manifold subjected to the action of a nonstationary force field with potential U(q, t) = f(t)V(q). It is assumed that the factor f(t) tends to ∞ as t→±∞ and vanishes at a unique point t 0 ∈ ?. Let X +, X ? denote the sets of isolated critical points of V (x) at which U(x, t) as a function of x distinguishes its maximum for any fixed t > t 0 and t < t 0, respectively. Under nondegeneracy conditions on points of X ± we prove the existence of infinitely many doubly asymptotic trajectories connecting X ? and X +. 相似文献
19.
Luisa Carini Vincenzo De Filippis Giovanni Scudo 《Mediterranean Journal of Mathematics》2016,13(1):53-64
Let R be a non-commutative prime ring of characteristic different from 2 with extended centroid C, F ≠ 0 a generalized skew derivation of R, and n ≥ 1 such that [F(x), x] n = 0, for all x ∈ R. Then there exists an element λ ∈ C such that F(x) = λx, for all x ∈ R. 相似文献
20.
The aim of this paper is to investigate the relations between Seifert manifolds and (1, 1)-knots. In particular, we prove that each orientable Seifert manifold with invariants has the fundamental group cyclically presented by G n ((x 1 q ...x n q l x n ?p ) and, moreover, it is the n-fold strongly-cyclic covering of the lens space L(|nlq ? p|, q) which is branched over the (1, 1)-knot K(q, q(nl ? 2), p ? 2q, p ? q) if p ≥ 2q and over the (1, 1)-knot K(p? q, 2q ? p, q(nl ? 2), p ? q) if p< 2q.
相似文献
$\{ Oo,0| - 1;\underbrace {(p,q),...,(p,q)}_{n times},(l,l - 1)\} $