共查询到20条相似文献,搜索用时 46 毫秒
1.
《Journal of Computational and Applied Mathematics》1998,88(2):377-399
A form (linear functional) u is called regular if there exists a sequence of polynomials {Pn}n⩾0, deg Pn=n which is orthogonal with respect to u. Such a form is said to be semi-classical, if there exist polynomials Φ and Ψ such that D(Φu) + Ψu = 0, where D designs the derivative operator.On certain regularity conditions, the product of a semi-classical form by a polynomial, gives a semi-classical form. In this paper, we consider the inverse problem: given a semi-classical form v, find all regular forms u which satisfy the relation x2u = −λv, . We give the structure relation (differential-recurrence relation) of the orthogonal polynomial sequence relatively to u. An example is treated with a nonsymmetric form v. 相似文献
2.
G. A. Dzyubenko 《Ukrainian Mathematical Journal》1996,48(3):367-376
We prove that if a functionf ∈C (1) (I),I: = [?1, 1], changes its signs times (s ∈ ?) within the intervalI, then, for everyn > C, whereC is a constant which depends only on the set of points at which the function changes its sign, andk ∈ ?, there exists an algebraic polynomialP n =P n (x) of degree ≤n which locally inherits the sign off(x) and satisfies the inequality $$\left| {f\left( x \right) - P_n \left( x \right)} \right| \leqslant c\left( {s,k} \right)\left( {\frac{1}{{n^2 }} + \frac{{\sqrt {1 - x^2 } }}{n}} \right)\omega _k \left( {f'; \frac{1}{{n^2 }} + \frac{{\sqrt {1 - x^2 } }}{n}} \right), x \in I$$ , where ω k (f′;t) is thekth modulus of continuity of the functionf’. It is also shown that iff ∈C (I) andf(x) ≥ 0,x ∈I then, for anyn ≥k ? 1, there exists a polynomialP n =P n (x) of degree ≤n such thatP n (x) ≥ 0,x ∈I, and |f(x) ?P n (x)| ≤c(k)ω k (f;n ?2 +n ?1 √1 ?x 2),x ∈I. 相似文献
3.
Xiaojing Yang 《Mathematische Nachrichten》2004,268(1):102-113
In this paper, the boundedness of all solutions of the nonlinear differential equation (φp(x′))′ + αφp(x+) – βφp(x–) + f(x) = e(t) is studied, where φp(u) = |u|p–2 u, p ≥ 2, α, β are positive constants such that = 2w–1 with w ∈ ?+\?, f is a bounded C5 function, e(t) ∈ C6 is 2πp‐periodic, x+ = max{x, 0}, x– = max{–x, 0}. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim) 相似文献
4.
《Comptes Rendus de l'Academie des Sciences Series IIA Earth and Planetary Science》1999,328(4):291-296
The purpose of this paper is to prove the existence of a solution for a nonlinear parabolic equation in the form ut - div(a(t, x, u, Du)) = H(t, x, u, Du) - div(g(t, x)) in QT =]0,T[×Ω, Ω ⊂ RN, with an initial condition u(0) = u0, where u0 is not bounded, |H(t,x, u, ξ)⩽ β|ξ|p + f(t,x) + βeλ1|u|f, |g|p/(p-1) ∈ Lr(QT) for some r = r{N) ⩾ 1, and - div(a(t,x,u, Du)) is the usual Leray-Lions operator. 相似文献
5.
S. R. Hayrapetyan 《Journal of Contemporary Mathematical Analysis (Armenian Academy of Sciences)》2010,45(4):221-238
The paper considers a class of regular, hypoelliptic in x
1, two-dimensional operators P(D) = P(D
1,D
2) in rather wide strip Ω
H
= {x = (x
1; x
2) ∈ $
\mathbb{E}
$
\mathbb{E}
2, |x
1| < H, x
2 ∈ $
\mathbb{E}
$
\mathbb{E}
1}. It is proved the infinite differentiability in Ω
H
of those generalized solutions of the equation P(D)
u
= 0, for which D
2
j
u ∈ L
2(Ω
H
), j = 0, …, ord
x2
P. 相似文献
6.
The linear equation Δ2u = 1 for the infinitesimal buckling under uniform unit load of a thin elastic plate over ?2 has the particularly interesting nonlinear generalization Δg2u = 1, where Δg = e?2u Δ is the Laplace‐Beltrami operator for the metric g = e2ug0, with g0 the standard Euclidean metric on ?2. This conformal elliptic PDE of fourth order is equivalent to the nonlinear system of elliptic PDEs of second order Δu(x)+Kg(x) exp(2u(x)) = 0 and Δ Kg(x) + exp(2u(x)) = 0, with x ∈ ?2, describing a conformally flat surface with a Gauss curvature function Kg that is generated self‐consistently through the metric's conformal factor. We study this conformal plate buckling equation under the hypotheses of finite integral curvature ∫ Kg exp(2u)dx = κ, finite area ∫ exp(2u)dx = α, and the mild compactness condition K+ ∈ L1(B1(y)), uniformly w.r.t. y ∈ ?2. We show that asymptotically for |x|→∞ all solutions behave like u(x) = ?(κ/2π)ln |x| + C + o(1) and K(x) = ?(α/2π) ln|x| + C + o(1), with κ ∈ (2π, 4π) and . We also show that for each κ ∈ (2π, 4π) there exists a K* and a radially symmetric solution pair u, K, satisfying K(u) = κ and maxK = K*, which is unique modulo translation of the origin, and scaling of x coupled with a translation of u. © 2001 John Wiley & Sons, Inc. 相似文献
7.
Jiaqing Pan 《Applications of Mathematics》2013,58(6):657-671
This work is concerned with the inverse problem of determining initial value of the Cauchy problem for a nonlinear diffusion process with an additional condition on free boundary. Considering the flow of water through a homogeneous isotropic rigid porous medium, we have such desire: for every given positive constants K and T 0, to decide the initial value u 0 such that the solution u(x, t) satisfies $\mathop {\sup }\limits_{x \in H_u (T_0 )} |x| \geqslant K$ , where H u(T 0) = {x, ?N: u(x, T 0) > 0}. In this paper, we first establish a priori estimate u t ? C(t)u and a more precise Poincaré type inequality $\left\| \phi \right\|_{L^2 (B_\varrho )}^2 \leqslant \varrho \left\| {\nabla \phi } \right\|_{L^2 (B_\varrho )}^2 $ , and then, we give a positive constant C 0 and assert the main results are true if only $\left\| {u_0 } \right\|_{L^2 (\mathbb{R}^N )} \geqslant C_0 $ . 相似文献
8.
V. V. Karachik 《Siberian Advances in Mathematics》2008,18(2):103-117
Let u(x) be a function analytic in some neighborhood D about the origin, $ \mathcal{D} Let u(x) be a function analytic in some neighborhood D about the origin, ⊂ ℝ
n
. We study the representation of this function in the form of a series u(x) = u
0(x) + |x|2
u
1(x) + |x|4
u
2(x) + …, where u
k
(x) are functions harmonic in . This representation is a generalization of the well-known Almansi formula.
Original Russian Text ? V. V. Karachik, 2007, published in Matematicheskie Trudy, 2007, Vol. 10, No. 2, pp. 142–162. 相似文献
9.
W. Kratz 《Rendiconti del Circolo Matematico di Palermo》1987,36(3):457-473
The following limit theorem on Hamiltonian systems (resp. corresponding Riccati matrix equations) is shown: Given(N, N)-matrices,A, B, C andn ∈ {1,…, N} with the following properties:A and kemelB(x) are constant, rank(I, A, …, A n?1) B(x)≠N,B(x)∈C n(R), andB(x)(A T)j-1 C(x)∈C n-j(R) forj=1, …, n. Then \(\mathop {\lim }\limits_{x \to x_0 } \eta _1^T \left( x \right)V\left( x \right)U^{ - 1} \left( x \right)\eta _2 \left( x \right) = d_1^T \left( {x_0 } \right)U\left( {x_0 } \right)d_2 \) forx 0∈R, whenever the matricesU(x), V(x) are a conjoined basis of the differential systemU′=AU + BV, V′=CU?A TV, and whenever ηi(x)∈R N satisfy ηi(x 0)=U(x 0)d i ∈ imageU(x 0) η′i-Aηni(x) ∈ imageB(x),B(x)(η′i(x)-Aηi(x)) ∈C n-1 R fori=1,2. 相似文献
10.
B. A. Khudaikuliev 《Mathematical Notes》2012,92(5-6):820-829
This paper deals with the behavior of the nonnegative solutions of the problem $$- \Delta u = V(x)u, \left. u \right|\partial \Omega = \varphi (x)$$ in a conical domain Ω ? ? n , n ≥ 3, where 0 ≤ V (x) ∈ L1(Ω), 0 ≤ ?(x) ∈ L1(?Ω) and ?(x) is continuous on the boundary ?Ω. It is proved that there exists a constant C *(n) = (n ? 2)2/4 such that if V 0(x) = (c + λ 1)|x|?2, then, for 0 ≤ c ≤ C *(n) and V(x) ≤ V 0(x) in the domain Ω, this problem has a nonnegative solution for any nonnegative boundary function ?(x) ∈ L 1(?Ω); for c > C *(n) and V(x) ≥ V 0(x) in Ω, this problem has no nonnegative solutions if ?(x) > 0. 相似文献
11.
The linear non-autonomous evolution equation , with the initial datum u(0) = x, in the space C([0, T], E), where E is a Banach space and {A(t)} is a family of infinitesimal generators of bounded analytic semigroups is considered; the domains D(A(t)) are supposed constant in t and possibly not dense in E. Maximal regularity of the strict and classical solutions, i.e., regularity of u′ and A(·)u(·) with values in the interpolation spaces DA(0)(θ, ∞) and DA(0)(θ) between D(A(0)) and E, is studied. A characterization of such spaces in a concrete case is also given. 相似文献
12.
V. N. Margaryan H. G. Ghazaryan 《Journal of Contemporary Mathematical Analysis (Armenian Academy of Sciences)》2018,53(1):6-15
A linear differential operator P(x, D) = P(x1,... x n , D1,..., D n ) = ∑αγα(x)Dα with coefficients γα(x) defined in E n is called formally almost hypoelliptic in E n if all the derivatives DνξP(x, ξ) can be estimated by P(x, ξ), and the operator P(x, D) has uniformly constant power in En. In the present paper, we prove that if P(x, D) is a formally almost hypoelliptic operator, then all solutions of equation P(x, D)u = 0, which together with some of their derivatives are square integrable with a specified exponential weight, are infinitely differentiable functions. 相似文献
13.
14.
15.
Luis Silvestre 《纯数学与应用数学通讯》2007,60(1):67-112
Given a function φ and s ∈ (0, 1), we will study the solutions of the following obstacle problem:
- u ≥ φ in ?n,
- (??)su ≥ 0 in ?n,
- (??)su(x) = 0 for those x such that u(x) > φ(x),
- lim|x| → + ∞ u(x) = 0.
16.
O. Yu. Khachay 《Differential Equations》2008,44(2):282-285
We consider the Cauchy problem for the nonlinear differential equation where ? > 0 is a small parameter, f(x, u) ∈ C ∞ ([0, d] × ?), R 0 > 0, and the following conditions are satisfied: f(x, u) = x ? u p + O(x 2 + |xu| + |u|p+1) as x, u → 0, where p ∈ ? \ {1} f(x, 0) > 0 for x > 0; f u 2(x, u) < 0 for (x, u) ∈ [0, d] × (0, + ∞); Σ 0 +∞ f u 2(x, u) du = ?∞. We construct three asymptotic expansions (external, internal, and intermediate) and prove that the matched asymptotic expansion approximates the solution uniformly on the entire interval [0, d].
相似文献
$$\varepsilon \frac{{du}}{{dx}} = f(x,u),u(0,\varepsilon ) = R_0 ,$$
17.
G. A. Kalyabin 《Proceedings of the Steklov Institute of Mathematics》2014,284(1):161-167
Explicit upper and lower estimates are given for the norms of the operators of embedding of , n ∈ ?, in L q (dµ), 0 < q < ∞. Conditions on the measure µ are obtained under which the ratio of the above estimates tends to 1 as n → ∞, and asymptotic formulas are presented for these norms in regular cases. As a corollary, an asymptotic formula (as n → ∞) is established for the minimum eigenvalues λ1, n, β , β > 0, of the boundary value problems (?d 2/dx 2) n u(x) = λ|x| β?1, x ∈ (?1, 1), u (k)(±1) = 0, k ∈ {0, 1, ..., n ? 1}. 相似文献
18.
J. Bourgain 《Israel Journal of Mathematics》1992,77(1-2):1-16
We study the almost everythere convergence to the initial dataf(x)=u(x, 0) of the solutionu(x, t) of the two-dimensional linear Schrödinger equation Δu=i? t u. The main result is thatu(x, t) →f(x) almost everywhere fort → 0 iff ∈H p (R2), wherep may be chosen <1/2. To get this result (improving on Vega’s work, see [6]), we devise a strategy to capture certain cancellations, which we believe has other applications in related problems. 相似文献
19.
We consider the problem
- u t=u xx+e u whenx ∈ ?,t > 0,
- u(x, 0) =u 0(x) whenx ∈ ?,
20.
B. I. Golubov 《Mathematical Notes》2006,79(1-2):196-214
For functions from the Lebesgue space L(?+), we introduce the modified strong dyadic integral J α and the fractional derivative D (α) of order α > 0. We establish criteria for their existence for a given function f ∈ L(?+). We find a countable set of eigenfunctions of the operators D (α) and J α, α > 0. We also prove the relations D (α)(J α(f)) = f and J α(D (α)(f)) = f under the condition that $\smallint _{\mathbb{R}_ + } f(x)dx = 0$ . We show the unboundedness of the linear operator $J_\alpha :L_{J_{_\alpha } } \to L(\mathbb{R}_ + )$ , where L J α is its natural domain of definition. A similar assertion is proved for the operator $D^{(\alpha )} :L_{D^{(\alpha )} } \to L(\mathbb{R}_ + )$ . Moreover, for a function f ∈ L(?+) and a given point x ∈ ?+, we introduce the modified dyadic derivative d (α)(f)(x) and the modified dyadic integral j α(f)(x). We prove the relations d (α)(J α(f))(x) = f(x) and j α(D (α)(f)) = f(x) at each dyadic Lebesgue point of the function f. 相似文献