首页 | 本学科首页   官方微博 | 高级检索  
相似文献
 共查询到20条相似文献，搜索用时 62 毫秒
1.
 If u is a minimizer of ∫Ω F(|∇u|)dx−∫Ω udμ, then the pointwise estimate
can be reached. This results is obtained for a Young function F with the global Δ2 & ∇2 property. Links to applications to real analysis are given. Received: 26 February 2002 / Revised version: 4 November 2002 Published online: 20 March 2003 Mathematics Subject Classification (2000): 35J60, 31C15, 47B38  相似文献   

2.
Summary.   A turning-point theory is developed for the second-order difference equation
where the coefficients A n and B n have asymptotic expansions of the form
θ≠0 being a real number. In particular, it is shown how the Airy functions arise in the uniform asymptotic expansions of the solutions to this three-term recurrence relation. As an illustration of the main result, a uniform asymptotic expansion is derived for the orthogonal polynomials associated with the Freud weight exp(−x 4 ), xℝ. Received February 21, 2002 / Revised version received April 8, 2002 / Published online October 29, 2002 Mathematics Subject Classification (1991): 41A60, 39A10, 33C45 The work described in this paper was partially supported by a grant from the Research Grants Council of the Hong Kong Special Administrative Region, China (Project No. CityU 1132 | 00P)  相似文献   

3.
Let u be a weak solution of (-△)mu = f with Dirichlet boundary conditions in a smooth bounded domain Ω  Rn. Then, the main goal of this paper is to prove the following a priori estimate:‖u‖ Wω2 m,p(Ω) ≤ C ‖f‖ Lωp (Ω),where ω is a weight in the Muckenhoupt class Ap.  相似文献   

4.
If
denotes the error term in the classical Rankin-Selberg problem, then it is proved that
where Δ1(x) = ∫ x 0 Δ(u)du. The latter bound is, up to ‘ɛ’, best possible. Received: 8 February 2007  相似文献   

5.
We analyze polynomials P n that are biorthogonal to exponentials , in the sense that
Here α>−1. We show that the zero distribution of P n as n→∞ is closely related to that of the associated exponent polynomial
More precisely, we show that the zero counting measures of {P n (−4nx)} n=1∞ converge weakly if and only if the zero counting measures of {Q n } n=1∞ converge weakly. A key step is relating the zero distribution of such a polynomial to that of the composite polynomial
under appropriate assumptions on {Δ n,j }.   相似文献   

6.
We investigate the large time behavior of positive solutions with finite mass for the viscous Hamilton-Jacobi equationu t = Δu + |Δu| p ,t>0,x ∈ ℝ N , wherep≥1 andu(0,.)=u 0≥0,u 0≢0,u 0∈L 1. DenotingI ∞=lim t→∞‖u(t)‖1≤∞, we show that the asymptotic behavior of the mass can be classified along three cases as follows:
–  • ifp≤(N+2)/(N+1), thenI ∞=∞ for allu 0;
–  • if (N+2)/(N+1)<p<2, then bothI ∞=∞ andI ∞<∞ occur;
–  • ifp≥2, thenI ∞<∞ for allu 0.
We also consider a similar question for the equationu t=Δu+u p .  相似文献   

7.
 Suppose that f: ℝ nN →ℝ is a strictly convex energy density of linear growth, f(Z)=g(|Z|2) if N>1. If f satisfies an ellipticity condition of the form
then, following [Bi3], there exists a unique (up to a constant) solution of the variational problem
provided that the given boundary data u 0 W 1 1 (ω;ℝ N ) are additionally assumed to be of class L ∞(ω;ℝ N ). Moreover, if μ<3, then the boundedness of u 0 yields local C 1,α-regularity (and uniqueness up to a constant) of generalized minimizers of the problem
In our paper we show that the restriction u 0L ∞(ω;ℝ N ) is superfluous in the two dimensional case n=2, hence we may prescribe boundary values from the energy class W 1 1 (ω;ℝ N ) and still obtain the above results. Received: 12 February 2002 / Revised version: 7 October 2002 Published online: 14 February 2003 Mathematics Subject Classification (2000): 49N60, 49N15, 49M29, 35J  相似文献   

8.
In this paper we establish some oscillation or nonoscillation criteria for the second order half-linear differential equation
where (i) r,c ∈ C([t 0, ∞), ℝ := (− ∞, ∞)) and r(t) > 0 on [t 0, ∞) for some t 0 ⩾ 0; (ii) Φ(u) = |u|p−2 u for some fixed number p > 1. We also generalize some results of Hille-Wintner, Leighton and Willet.  相似文献   

9.
We consider the superlinear elliptic equation on Sn
where ΔSn is the Laplace-Beltrami operator on S n. We prove that for any k = 1,..., n − 1, there exists p k > 1 such that for 1 < p < p k and ε sufficiently small, there exist at least n−k positive solutions concentrating on a k-dimensional subset of the equator. We also discuss the problem on geodesic balls of S n and establish the existence of positive non-radial solutions. The method extends to Dirichlet problems with more general non-linearities. The proofs are based on the finite-dimensional reduction procedure which was successfully used by the second author in singular perturbation problems.  相似文献   

10.
 For every m≥3, let n=R (L 3 (m)) be the least integer such that for every 2-coloring of the set S={1, 2, …, n}, there exists in S a monochromatic solution to the following system.?
? The main result of this paper is that?
? Moreover, it is shown that, up to a switching of the colors, there exists a unique 2-coloring of the set {1, 2, …, R(L 3 (m)) −1} that avoids a monochromatic solution to the above system. Received: July 29, 1996 Revised: December 21, 1998  相似文献   

11.
We consider the following problem of finding a nonnegative function u(x) in a ball B = B(O, R) ⊂ R n , n ≥ 3:
- Du = V(x)u,     u| ?B = f(x), - \Delta u = V(x)u,\,\,\,\,\,u\left| {_{\partial B} = \phi (x),} \right.  相似文献   

12.
This paper is concerned with the large time behavior of traveling wave solutions to the Cauchy problem of generalized Benjamin–Bona–Mahony–Burgers equations
with prescribed initial data
Here v( > 0), β are constants, u  ±  are two given constants satisfying u + ≠ u − and the nonlinear function f(u) ∈C 2(R) is assumed to be either convex or concave. An algebraic time decay rate to traveling waves of the solutions of the Cauchy problem of generalized Benjamin-Bona-Mahony-Burgers equation is obtained by employing the weighted energy method developed by Kawashima and Matsumura in [6] to discuss the asymptotic behavior of traveling wave solutions to the Burgers equation. revised: May 23 and August 8, 2007  相似文献   

13.
In the paper, the equation
is considered in the scale of the weighted spaces H β s (ℝ n ) (q > 1, a kα ∈ ℂ). We prove that if the expression
does not vanish on the set {ξ ∈ ℝ n ∖ 0, |z| ≤ q β−s+n /2−2m}, then this equation has a unique solution u ∈ H β s+2m (ℝ n ) for every function f ∈ H β s (ℝ n ) provided that β, s ≠ ∈ ℝ, β − s ≠ n/2 + p, and β − s − 2m ≠ − n/2 − p (p = 0, 1, ...). __________ Translated from Trudy Seminara imeni I. G. Petrovskogo, No. 26, pp. 37–55, 2007.  相似文献   

14.
We consider equations of the form L*μ = 0 for bounded measures on _boxclose^d {\mathbb{R}^{d}} , where L is a second order elliptic operator, for example, Lu = Δu + (b,∇u), and the equation is understood as the identity
òLudm = 0 \int {Lud{\mu} = 0}  相似文献   

15.
Summary.   Let ? be the circle [0,J] with the ends identified. We prove long-time existence for the following equation.
Here, =(t,x) is 2-parameter white noise, and we assume that u 0(x) is a continuous function on ?. We show that if g(u) grows no faster than C 0(1+|u|)γ for some γ<3/2, C 0>0, then this equation has a unique solution u(t,x) valid for all times t>0. Received: 27 November 1996 / In revised form: 28 July 1997  相似文献   

16.
   Abstract. Let I be a finite interval, r∈ N and ρ(t)= dist {t, I} , t∈ I . Denote by Δ s + L q the subset of all functions y∈ L q such that the s -difference Δ s τ y(t) is nonnegative on I ,
τ>0 . Further, denote by
, 0≤α<∞ , the classes of functions x on I with the seminorm ||x (r) ρ α ||_ L p ≤ 1 , such that Δ s τ x≥ 0 , τ>0 . For s=0,1,2 , we obtain two-sided estimates of the shape-preserving widths
where M n is the set of all linear manifolds M n in L q , such that dim M n ≤ n , and satisfying
.  相似文献   

17.
In this paper we study the Dirichlet problem in Q T = Ω × (0, T) for degenerate equations of porous medium-type with a lower order term:
The principal part of the operator degenerates in u = 0 according to a nonnegative increasing real function α(u), and the term grows quadratically with respect to the gradient. We prove an existence result for solutions to this problem in the framework of the distributional solutions under the hypotheses that both f and the initial datum u 0 are bounded nonnegative functions. Moreover as further results we get an existence result for the model problem
in the case that the principal part of the operator is of fast-diffusion type, i.e. α(u) = u m , with −1 < m < 0.   相似文献   

18.
Let ℂ[−1,1] be the space of continuous functions on [−,1], and denote by Δ2 the set of convex functions f ∈ ℂ[−,1]. Also, let E n (f) and E n (2) (f) denote the degrees of best unconstrained and convex approximation of f ∈ Δ2 by algebraic polynomials of degree < n, respectively. Clearly, En (f) ≦ E n (2) (f), and Lorentz and Zeller proved that the inverse inequality E n (2) (f) ≦ cE n (f) is invalid even with the constant c = c(f) which depends on the function f ∈ Δ2. In this paper we prove, for every α > 0 and function f ∈ Δ2, that
where c(α) is a constant depending only on α. Validity of similar results for the class of piecewise convex functions having s convexity changes inside (−1,1) is also investigated. It turns out that there are substantial differences between the cases s≦ 1 and s ≧ 2. Dedicated to Jóska Szabados on his 70th birthday  相似文献   

19.
This paper summarized recent achievements obtained by the authors about the box dimensions of the Besicovitch functions given byB(t) := ∞∑k=1 λs-2k sin(λkt),where 1 ＜ s ＜ 2, λk ＞ 0 tends to infinity as k →∞ and λk satisfies λk 1/λk ≥λ＞ 1. The results show thatlimk→∞ log λk 1/log λk = 1is a necessary and sufficient condition for Graph(B(t)) to have same upper and lower box dimensions.For the fractional Riemann-Liouville differential operator Du and the fractional integral operator D-v,the results show that if λ is sufficiently large, then a necessary and sufficient condition for box dimension of Graph(D-v(B)),0 ＜ v ＜ s - 1, to be s - v and box dimension of Graph(Du(B)),0 ＜ u ＜ 2 - s, to be s uis also lim k→∞logλk 1/log λk = 1.  相似文献   

20.
We prove the existence of saddle solutions of nonlinear elliptic equation involving the p-Laplacian
-Dpu=f(u)     \textin  Rn, -\Delta_{p}u=f(u) \quad \text{in}\,\,R^n,  相似文献   

设为首页 | 免责声明 | 关于勤云 | 加入收藏

Copyright©北京勤云科技发展有限公司  京ICP备09084417号