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1.
The Agmon-Miranda maximum principle for the polyharmonic equations of all orders is shown to hold in Lipschitz domains in
ℝ3. In ℝn,n≥4, the Agmon-Miranda maximum principle andL
p-Dirichlet estimates for certainp>2 are shown to fail in Lipschitz domains for these equations. In particular if 4≤n≤2m+1 theL
p Dirichlet problem for Δ
m
fails to be solvable forp>2(n−1)/(n−3).
Supported in part by the NSF. 相似文献
2.
Yu. A. Aminov 《Journal of Mathematical Sciences》1999,94(2):1141-1144
Immersions of domains of the n-dimensional Lobachevski space Ln in the (2n−1)-dimensional Euclidean space E2n−1 are studied. It is shown that the problem of isometric immersion of domains of Ln in E2n−1 is reduced to the study of a certain system of nonlinear partial differential equations, yielding the sine-Gordon equation
as one of the special cases.
Published inZapiski Nauchnykh Seminarov POMI, Vol. 234, 1996, pp. 11–16. 相似文献
3.
B. Bojarski 《Proceedings of the Steklov Institute of Mathematics》2006,255(1):65-81
We prove that a function f is in the Sobolev class W
loc
m,p
(ℝ
n
) or W
m,p
(Q) for some cube Q ⊂ ℝ
n
if and only if the formal (m − 1)-Taylor remainder R
m−1
f(x,y) of f satisfies the pointwise inequality |R
m−1
f(x,y)| ≤ |x − y|
m
[a(x) + a(y)] for some a ε L
p
(Q) outside a set N ⊂ Q of null Lebesgue measure. This is analogous to H. Whitney’s Taylor remainder condition characterizing the traces of smooth
functions on closed subsets of ℝ
n
.
Dedicated to S.M. Nikol’skiĭ on the occasion of his 100th birthday
The main results and ideas of this paper were presented in the plenary lecture of the author at the International Conference
and Workshop Function Spaces, Approximation Theory and Nonlinear Analysis dedicated to the centennial of Sergei Mikhailovich Nikol’skii, Moscow, May 24–28, 2005. 相似文献
4.
We investigate Besov spaces and their connection with trigonometric polynomial approximation inL
p[−π,π], algebraic polynomial approximation inL
p[−1,1], algebraic polynomial approximation inL
p(S), and entire function of exponential type approximation inL
p(R), and characterizeK-functionals for certain pairs of function spaces including (L
p[−π,π],B
s
a(L
p[−π,π])), (L
p(R),s
a(Lp(R))),
, and
, where 0<s≤∞, 0<p<1,S is a simple polytope and 0<α<r.
This project is supported by the National Science Foundation of China. 相似文献
5.
M. Ivette Gomes 《Annals of the Institute of Statistical Mathematics》1984,36(1):71-85
Summary Let {X
n}n≧1 be a sequence of independent, identically distributed random variables. If the distribution function (d.f.) ofM
n=max (X
1,…,X
n), suitably normalized with attraction coefficients {αn}n≧1(αn>0) and {b
n}n≧1, converges to a non-degenerate d.f.G(x), asn→∞, it is of interest to study the rate of convergence to that limit law and if the convergence is slow, to find other d.f.'s
which better approximate the d.f. of(M
n−bn)/an thanG(x), for moderaten. We thus consider differences of the formF
n(anx+bn)−G(x), whereG(x) is a type I d.f. of largest values, i.e.,G(x)≡Λ(x)=exp (-exp(−x)), and show that for a broad class of d.f.'sF in the domain of attraction of Λ, there is a penultimate form of approximation which is a type II [Ф
α(x)=exp (−x−α), x>0] or a type III [Ψ
α(x)= exp (−(−x)α), x<0] d.f. of largest values, much closer toF
n(anx+bn) than the ultimate itself. 相似文献
6.
We prove L p Poincaré inequalities with suitable dimension free constants for functions on the discrete cube {?1, 1} n . As well known, such inequalities for p an even integer allow to recover an exponential inequality hence the concentration phenomenon first obtained by Bobkov and Götze. We also get inequalities between the L p norms of $ \left\vert \nabla f\right\vert We prove L
p
Poincaré inequalities with suitable dimension free constants for functions on the discrete cube {−1, 1}
n
. As well known, such inequalities for p an even integer allow to recover an exponential inequality hence the concentration phenomenon first obtained by Bobkov and
G?tze. We also get inequalities between the L
p
norms of and moreover L
p
spaces may be replaced by more general ones. Similar results hold true, replacing functions on the cube by matrices in the
*-algebra spanned by n fermions and the L
p
norm by the Schatten norm C
p
. 相似文献
7.
Hei-Chi Chan 《数学学报(英文版)》2011,27(4):625-634
In this paper, we study a certain partition function a(n) defined by Σ
n≥0
a(n)q
n
:= Π
n=1(1 − q
n
)−1(1 − q
2n
)−1. We prove that given a positive integer j ≥ 1 and a prime m ≥ 5, there are infinitely many congruences of the type a(An + B) ≡ 0 (mod m
j
). This work is inspired by Ono’s ground breaking result in the study of the distribution of the partition function p(n). 相似文献
8.
Edith Adan-Bante 《Archiv der Mathematik》2005,85(4):297-303
Let G be a finite p-group, where p is a prime number, and a ∈ G. Denote by Cl(a) = {gag−1| g ∈ G} the conjugacy class of a in G. Assume that |Cl(a)| = pn. Then Cl(a) Cl(a−1) = {xy | x ∈ Cl(a), y ∈ Cl(a−1)} is the union of at least n(p − 1) + 1 distinct conjugacy classes of G.
Received: 16 December 2004 相似文献
9.
Artūras Dubickas 《Israel Journal of Mathematics》2009,170(1):95-111
Let f
1 and f
2 be two positive numbers of the field , and let f
n+2 = f
n+1 + f
n
for each n ≥ 1. Let us denote by {x} the fractional part of a real number x. We prove that, for each ξ ∉ K, the inequality {ξf
n
} > 2/3 holds for infinitely many positive integers n. On the other hand, we prove a result which implies that there is a transcendental number ξ such that {ξf
n
} < 39/40 for each n ≥ 1. Moreover, it is shown that, for every a > 1, there is an interval of positive numbers that contains uncountably many numbers ξ such that {a
n
} 6 min 2/(a − 1), (34a
2 − 32a + 7)/(9(2a − 1)2) for each n > 1. Here, the minimum is strictly smaller than 1 for each a > 1. In contrast, by an old result of Weyl, for any a > 1, the sequence {ξa
n
}, n = 1, 2, ..., is uniformly distributed in [0, 1] (and so everywhere dense in [0, 1]) for almost all real numbers ξ. 相似文献
10.
J. J. Grobler 《Israel Journal of Mathematics》1988,64(1):32-38
LetA be a unital Banach lattice algebra and leta εA
+ satisfy ‖a ‖≦1. Then either ‖a
n+1 −a
n ‖=2 for alln≧0 or else ‖a
n+1 −a
n ‖ → 0 asn → ∞. Cyclicity of the peripheral spectrum ofa is also established. 相似文献
11.
Robert Černý 《Central European Journal of Mathematics》2012,10(2):590-602
Let Ω ⊂ ℝ
n
, n ≥ 2, be a bounded domain and let α < n − 1. Motivated by Theorem I.6 and Remark I.18 of [Lions P.-L., The concentration-compactness principle in the calculus of
variations. The limit case. I, Rev. Mat. Iberoamericana, 1985, 1(1), 145–201] and by the results of [Černy R., Cianchi A.,
Hencl S., Concentration-Compactness Principle for Moser-Trudinger inequalities: new results and proofs, Ann. Mat. Pura Appl.
(in press), DOI: 10.1007/s10231-011-0220-3], we give a sharp estimate of the exponent concerning the Concentration-Compactness
Principle for the embedding of the Orlicz-Sobolev space W
01
L
n
log
α
L(Ω) into the Orlicz space corresponding to a Young function that behaves like exp t
n/(n−1−α) for large t. We also give the result for the case of the embedding into double and other multiple exponential spaces. 相似文献
12.
For a domainU on a certaink-dimensional minimal submanifold ofS
n orH
n, we introduce a “modified volume”M(U) ofU and obtain an optimal isoperimetric inequality forU k
k
ω
k
M (D)
k-1
≤Vol(∂D)
k
, where ω
k
is the volume of the unit ball ofR
k
. Also, we prove that ifD is any domain on a minimal surface inS
+
n
(orH
n, respectively), thenD satisfies an isoperimetric inequality2π A≤L
2+A2 (2π A≤L2−A2 respectively). Moreover, we show that ifU is ak-dimensional minimal submanifold ofH
n, then(k−1) Vol(U)≤Vol(∂U).
Supported in part by KME and GARC 相似文献
13.
We obtain the best approximation in L
1(ℝ), by entire functions of exponential type, for a class of even functions that includes e
−λ|x|, where λ>0, log |x| and |x|
α
, where −1<α<1. We also give periodic versions of these results where the approximating functions are trigonometric polynomials of bounded
degree. 相似文献
14.
We study the asymptotic behaviour of the certain class of non-Newtonian incompressible fluids-power law fluids in the whole
space when the external force is zero. Assuming that initial data belong toL
1∩L
2 we prove thatL
2 decay in time ist
−1/4.
Sunto Noi studiamo il comportamento asintotico in tutto lo spazio di una classe di fluidi incomprimibili non-Newtoniani con una legge ?fluids-power? in assenza di forze esterne. Assumendo che i dati iniziali appartengano aL 1∩L 2 noi proviamo che il decadimento nel tempo inL 2 èt −1/4.相似文献
15.
Florian Luca 《Monatshefte für Mathematik》2005,146(3):239-256
In [2], it was shown that if a and b are multiplicatively independent integers and ɛ > 0, then the inequality gcd (an − 1,bn − 1) < exp(ɛn) holds for all but finitely many positive integers n. Here, we generalize the above result. In particular, we show that if f(x),f1(x),g(x),g1(x) are non-zero polynomials with integer coefficients, then for every ɛ > 0, the inequality
holds for all but finitely many positive integers n. 相似文献
16.
I. J. Schoenberg 《Israel Journal of Mathematics》1971,10(3):364-372
In 1936 the author showed that the function sin(π(x+1)/4) is the entire function of least exponential type (=π/4) among all entire functionsf(z) with the property thatf
(n)(z) vanishes somewhere in the real interval [−1, 1] (n=0, 1,2,…). Now more precise results of this kind are obtained by working within the class ∞[−1, 1].
For Paul Montel on his 95th birthday 相似文献
17.
In this paper the Cauchy problem for a class of nonhomogeneous Navier-Stokes equations in the infinite cylinderS
T
=ℝn x [0,T) is considered. We construct a unique local solution inL
q([0,T);L
p
(ℝ
n
)) for a class of nonhomogeneous Navier-Stokes equations provided that initial data are inL
r
(ℝ
n
), wherer>1 is an exponent determined by the structure of nonlinear terms andp,q are such that 2/q=n(1/r−1/p). Meanwhile under suitable conditions we also obtain thatu(t)≠L
q([0,∞];L
p
(ℝ
n
)) provided that initial data are sufficiently small.
This work is supported by the National Natural Sciences Foundation of China and the Foundation of LNM Laboratory of Institute
of Mechanics of the Chinese Academy of Sciences. 相似文献
18.
We give necessary conditions and sufficient conditions for sequences of reproducing kernels (kΘ(·, λn))n ≥ 1 to be overcomplete in a given model space KΘp where Θ is an inner function in H∞, p ∈ (1, ∞), and where (λn)n ≥ 1 is an infinite sequence of pairwise distinct points of
Under certain conditions on Θ we obtain an exact characterization of overcompleteness. As a consequence we are able to describe
the overcomplete exponential systems in L2 (0, a). 相似文献
19.
D. V. Gorbachev 《Mathematical Notes》2000,68(2):159-166
We consider extremum problems for entire functions of exponential spherical type related to important extremum problems on
the optimal point (the Chernykh point) in the sharp jackson inequality in the spaceL
2(ℝ
n
) and the connection between codes and designs on the torusT
n
.
Translated fromMatematicheskie Zametki, Vol. 68, No. 2, pp. 179–187, August, 2000. 相似文献
20.
The Marcinkiewicz-Zygmund inequality and the Bernstein inequality are established on ∮2m(T,R)∩L2(R) which is the space of polynomial splines with irregularly distributed nodes T={tj}j∈Z, where {tj}j∈Z is a real sequence such that {eitξ}j∈Z constitutes a Riesz basis for L2([-π,π]). From these results, the asymptotic relation E(f,Bπ,2)2=lim E(f,∮2m(T,R)∩L2(R))2 is proved, where Bπ,2 denotes the set of all functions from L2(R) which can be continued to entire functions of exponential type ≤π, i.e. the classical Paley-Wiener class. 相似文献