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1.
Let Ω be a bounded Lipschitz domain. Define B
0,1
1,
r
(Ω) = {f∈L
1 (Ω): there is an F∈B
0,1
1 (ℝ
n
) such that F|Ω = f} and B
0,1
1
z
(Ω) = {f∈B
0,1
1 (ℝ
n
) : f = 0 on ℝ
n
\}. In this paper, the authors establish the atomic decompositions of these spaces. As by-products, the authors obtained the
regularity on these spaces of the solutions to the Dirichlet problem and the Neumann problem of the Laplace equation of ℝ
n
+.
Received June 8, 2000, Accepted October 24, 2000 相似文献
2.
Robert S. Strichartz 《Journal d'Analyse Mathématique》2006,99(1):333-353
For certain Cantor measures μ on ℝn, it was shown by Jorgensen and Pedersen that there exists an orthonormal basis of exponentialse
2πiγ·x for λεΛ. a discrete subset of ℝn called aspectrum for μ. For anyL
1 functionf, we define coefficientsc
γ(f)=∝f(y)e
−2πiγiy
dμ(y) and form the Mock Fourier series ∑λ∈Λcλ(f)e
2πiλ·x
. There is a natural sequence of finite subsets Λn increasing to Λ asn→∞, and we define the partial sums of the Mock Fourier series by
We prove, under mild technical assumptions on μ and Λ, thats
n(f) converges uniformly tof for any continuous functionf and obtain the rate of convergence in terms of the modulus of continuity off. We also show, under somewhat stronger hypotheses, almost everywhere convergence forfεL
1.
Research supported in part by the National Science Foundation, Grant DMS-0140194. 相似文献
3.
N. Yu. Dodonov 《Journal of Mathematical Sciences》2007,144(6):4592-4611
Approximation aggregates of summatory type are constructed for functions f: Q → X, where Q is a convex polygon in ℝ2 and X is a real Banach space. An assertion providing an accuracy estimate for an uniform approximation in terms of the modulus
of continuity of the second order for f is proved. Differential properties of such aggregates are studied. Bibliography: 7 titles.
__________
Translated from Problemy Matematicheskogo Analiza, No. 35, 2007, pp. 59–77 相似文献
4.
V. D. Zalizko 《Ukrainian Mathematical Journal》2007,59(1):28-44
The Jackson inequality
relates the value of the best uniform approximation E
n
(f) of a continuous 2π-periodic function f: ℝ → ℝ by trigonometric polynomials of degree ≤ n − 1 to its third modulus of continuity ω
3(f, t). In the present paper, we show that this inequality is true if continuous 2π-periodic functions that change their convexity
on [−π, π) only at every point of a fixed finite set consisting of an even number of points are approximated by polynomials
coconvex to them.
__________
Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 59, No. 1, pp. 29–43, January, 2007. 相似文献
5.
A. P. Oskolkov 《Journal of Mathematical Sciences》1998,91(2):2840-2859
Existence theorems are proved for the solutions of the first and second initial boundary-value problems for the equations
of Kelvin-Voight fluids and for the penalized equations of Kelvin-Voight fluids in the smoothness classes W
∞
r
(ℝ+;W
2
2+k
(Ω)), W
2
r
(ℝ+;W
2
2+k
(Ω)) and S
2
r
(ℝ+;W
2
2+k
(Ω)) (r=1,2; k=0,1,2, …) under the condition that the right-hand sides f(x,t) belong to the classes W
∞
r-1
(ℝ+;W
2
k
(Ω)), W
2
r-1
(ℝ+;W
2
k
(Ω)) and S
2
r-1
(ℝ+;W
2
k
(Ω)), respectively, and for the solutions of the first and second T-periodic boundary-value problems for the same equations
in the smoothness classes W
∞
r−1
(ℝ; W
2
2+k
(Ω)) and W
2
r−1
(0, T; W
2
2+k
(Ω)) (r=1,2, k=0,1,2…) under the condition that f(x,t) are T-periodic and belong to the spaces W
∞
r−1
(ℝ+; W
2
k
(Ω)) and W
2
r−1
(0,T; W
2
k
(Ω)), respectively. It is shown that as ɛ→0, the smooth solutions {vɛ} of the perturbed initial boundary-value and T-periodic boundary-value problems for the penalized equations of Kelvin-Voight
fluids converge to the corresponding smooth solutions (v,p) of the initial boundary-value and T-periodic boundary-value problems
for the equations of Kelvin-Voight fluids. Bibliography: 27 titles.
Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 230, 1995, pp. 214–242.
Translated by T. N. Surkova. 相似文献
6.
Martin Fuchs 《manuscripta mathematica》1991,72(1):131-140
Given a smooth domain Ω in ℝ
m+1 with compact closure and a smooth integrable functionh: ℝ
m+1→ℝ satisfyingh(x)≥H
∂Ω
(x) on ∂Ω whereH
∂ω denotes the mean curvature of ∂Ω calculated w.r.t. the interior unit normal we show that there is a setA⊂ℝ
m+1 with the properties
andH
∂A=h on ∂A. 相似文献
7.
Alexander Koldobsky 《Israel Journal of Mathematics》2011,185(1):277-292
We say that a random vector X = (X
1, …, X
n
) in ℝ
n
is an n-dimensional version of a random variable Y if, for any a ∈ ℝ
n
, the random variables Σa
i
X
i
and γ(a)Y are identically distributed, where γ: ℝ
n
→ [0,∞) is called the standard of X. An old problem is to characterize those functions γ that can appear as the standard of an n-dimensional version. In this paper, we prove the conjecture of Lisitsky that every standard must be the norm of a space that
embeds in L
0. This result is almost optimal, as the norm of any finite-dimensional subspace of L
p
with p ∈ (0, 2] is the standard of an n-dimensional version (p-stable random vector) by the classical result of P. Lèvy. An equivalent formulation is that if a function of the form f(‖ · ‖
K
) is positive definite on ℝ
n
, where K is an origin symmetric star body in ℝ
n
and f: ℝ → ℝ is an even continuous function, then either the space (ℝ
n
, ‖·‖
K
) embeds in L
0 or f is a constant function. Combined with known facts about embedding in L
0, this result leads to several generalizations of the solution of Schoenberg’s problem on positive definite functions. 相似文献
8.
O. P. Filatov 《Mathematical Notes》2000,67(3):365-371
For the class II(ℝ
m
) of continuous almost periodic functionsf: ℝ
m
→ ℝ, we consider the problem of the existence of the limit
where the least upper bound is taken over all solutions (in the sense of Carathéodory) of the generalized differential equation
{ie365-1} εG, γ(0)=a
0. We establish that if the compact setG ⊂ ℝ
m
is not contained in a subspace of ℝ
m
of dimensionm−1 (i.e., if it is nondegenerate), then the limit exists uniformly in the initial vectora
0 ε ℝ
m
. Conversely, if for any functionf ε π(ℝ
m
), the limit exists uniformly in the initial vectora
0 ε ℝ
m
, then the compact setG is nondegenerate. We also prove that there exists an extremal solution for which a limit of the maximal mean uniform in the
initial conditions is realized.
Translated fromMatematicheskie Zametki, Vol. 67, No. 3, pp. 433–440, March, 2000. 相似文献
(1) |
9.
I. V. Filimonova 《Journal of Mathematical Sciences》2007,143(4):3415-3428
One considers a semilinear parabolic equation u
t
= Lu − a(x)f(u) or an elliptic equation u
tt
+ Lu − a(x)f(u) = 0 in a semi-infinite cylinder Ω × ℝ+ with the nonlinear boundary condition
, where L is a uniformly elliptic divergent operator in a bounded domain Ω ∈ ℝn; a(x) and b(x) are nonnegative measurable functions in Ω. One studies the asymptotic behavior of solutions of such boundary-value problems
for t → ∞.
__________
Translated from Trudy Seminara imeni I. G. Petrovskogo, No. 26, pp. 368–389, 2007. 相似文献
10.
E. A. Sevost’yanov 《Ukrainian Mathematical Journal》2009,61(7):1151-1157
The paper is devoted to investigations in the field of space mappings. We prove that open discrete mappings f ∈ W
1,n
loc such that their outer dilatation K
O
(x, f) belongs to L
n−1
loc and the measure of the set B
f
of branching points of f is equal to zero have finite length distortion. In other words, the images of almost all curves γ in the domain D under the considered mappings f : D → ℝ
n
, n ≥ 2, are locally rectifiable, f possesses the (N)-property with respect to length on γ, and, furthermore, the (N)-property also holds in the inverse direction for liftings of curves. The results obtained generalize the well-known Poletskii
lemma proved for quasiregular mappings. 相似文献
11.
Let Ω
ϕ
r
={f:f
(r-1) abs. cont. on [0,1], ‖qr(D)f‖p≤1, f(2K+σ) (0)=f(2K+σ)=0, (k)=0,...,l-1}. where
, and I is an identical operator. Denote Kolmogorov, linear, Geelfand and Bernstein n-widths of Ω
ϕ
r
in Lp byd
n
(Ω
ϕ
r
;L
p
),δ
n
(Ω
ϕ
r
;L
p
),d
n
(Ω
p
r
;L
p
) andb
n
(Ω
p
r
;L
p
), respectively. In this paper, we find a method to get an exact estimation of these n-widths. Related optimal subspaces and
an optimal linear operator are given. For another subset
, similar results are also derrived. 相似文献
12.
We prove that two basic questions on outer measure are undecidable. First we show that consistently every sup-measurable functionf: ℝ2 → ℝ is measurable. The interest in sup-measurable functions comes from differential equations and the question for which
functionsf: ℝ2 → ℝ the Cauchy problemy′=f(x,y), y(x0)=y0 has a unique almost-everywhere solution in the classAC
t(ℝ) of locally absolutely continuous functions on ℝ. Next we prove that consistently every functionf: ℝ → ℝ is continuous on some set of positive outer Lebesgue measure. This says that in a strong sense the family of continuous
functions (from the reals to the reals) is dense in the space of arbitrary such functions.
For the proofs we discover and investigate a new family of nicely definable forcing notions (so indirectly we deal with nice
ideals of subsets of the reals—the two classical ones being the ideal of null sets and the ideal of meagre ones).
Concerning the method, i.e., the development of a family of forcing notions, the point is that whereas there are many such
objects close to the Cohen forcing (corresponding to the ideal of meagre sets), little has been known on the existence of
relatives of the random real forcing (corresponding to the ideal of null sets), and we look exactly at such forcing notions.
The first author thanks The Hebrew University of Jerusalem for support during his visits to Jerusalem and the KBN (Polish
Committee of Scientific Research) for partial support through grant 2P03A03114.
The research of the second author was partially supported by the Israel Science Foundation. Publication 736. 相似文献
13.
N. A. Chalkina 《Moscow University Mathematics Bulletin》2011,66(6):231-234
Sufficient conditions for the existence of an inertial manifold are found for the equation u
tt
+ 2γu
t
− Δu = f(u, u
t
), u = u(x, t), x ∈ Ω ⋐ ℝ
N
, u|
∂Ω = 0, t > 0 under the assumption that the function f satisfies the Lipschitz condition. 相似文献
14.
Abdelmajid Siai 《Potential Analysis》2006,24(1):15-45
Let Ω be an open bounded set in ℝN, N≥3, with connected Lipschitz boundary ∂Ω and let a(x,ξ) be an operator of Leray–Lions type (a(⋅,∇u) is of the same type as the operator |∇u|p−2∇u, 1<p<N). If τ is the trace operator on ∂Ω, [φ] the jump across ∂Ω of a function φ defined on both sides of ∂Ω, the normal derivative
∂/∂νa related to the operator a is defined in some sense as 〈a(⋅,∇u),ν〉, the inner product in ℝN, of the trace of a(⋅,∇u) on ∂Ω with the outward normal vector field ν on ∂Ω. If β and γ are two nondecreasing continuous real functions everywhere
defined in ℝ, with β(0)=γ(0)=0, f∈L1(ℝN), g∈L1(∂Ω), we prove the existence and the uniqueness of an entropy solution u for the following problem,
in the sense that, if Tk(r)=max {−k,min (r,k)}, k>0, r∈ℝ, ∇u is the gradient by means of truncation (∇u=DTku on the set {|u|<k}) and
, u measurable; DTk(u)∈Lp(ℝN), k>0}, then
and u satisfies,
for every k>0 and every
.
Mathematics Subject Classifications (2000) 35J65, 35J70, 47J05. 相似文献
15.
A. A. Kovalevskii 《Ukrainian Mathematical Journal》1996,48(9):1402-1422
By using special local characteristics of domains Ω
s
⊂Ω,s=12,..., we establish necessary and sufficient conditions for the γ-convergence of sequences of integral functionalsI
λs
:W
k,m
(Ω
s
)→ℝ, λ⊂Ω to interal functionals defined on W
k,m
(Ω).
Institute of Applied Mathematics and Mechanics, Ukrainian Academy of Sciences, Donetsk. Translated from Ukrainskii Matematicheskii
Zhurnal, Vol. 48, No. 9, pp. 1236–1254, September, 1996. 相似文献
16.
Let Ω be a bounded co.nvex domain in Rn(n≥3) and G(x,y) be the Green function of the Laplace operator -△ on Ω. Let hrp(Ω) = {f ∈ D'(Ω) :(E)F∈hp(Rn), s.t. F|Ω = f}, by the atom characterization of Local Hardy spaces in a bounded Lipschitz domain, the bound of f→(△)2(Gf) for every f ∈ hrp(Ω) is obtained, where n/(n 1)<p≤1. 相似文献
17.
The aim of this paper is to put the foundations of a new theory of functions, called holomorphic Cliffordian, which should
play an essential role in the generalization of holomorphic functions to higher dimensions. Let ℝ0,2m+1 be the Clifford algebra of ℝ2m+1 with a quadratic form of negative signature, be the usual operator for monogenic functions and Δ the ordinary Laplacian. The holomorphic Cliffordian functions are functionsf: ℝ2m+2 → ℝ0,2m+1, which are solutions ofDδ
m
f = 0.
Here, we will study polynomial and singular solutions of this equation, we will obtain integral representation formulas and
deduce the analogous of the Taylor and Laurent expansions for holomorphic Cliffordian functions.
In a following paper, we will put the foundations of the Cliffordian elliptic function theory. 相似文献
18.
The fractional Laplacian
(-\triangle)g/2(-\triangle)^{\gamma/2} commutes with the primary coordination transformations in the Euclidean space ℝ
d
: dilation, translation and rotation, and has tight link to splines, fractals and stable Levy processes. For 0 < γ < d, its inverse is the classical Riesz potential I
γ
which is dilation-invariant and translation-invariant. In this work, we investigate the functional properties (continuity,
decay and invertibility) of an extended class of differential operators that share those invariance properties. In particular,
we extend the definition of the classical Riesz potential I
γ
to any non-integer number γ larger than d and show that it is the unique left-inverse of the fractional Laplacian
(-\triangle)g/2(-\triangle)^{\gamma/2} which is dilation-invariant and translation-invariant. We observe that, for any 1 ≤ p ≤ ∞ and γ ≥ d(1 − 1/p), there exists a Schwartz function f such that I
γ
f is not p-integrable. We then introduce the new unique left-inverse I
γ, p
of the fractional Laplacian
(-\triangle)g/2(-\triangle)^{\gamma/2} with the property that I
γ, p
is dilation-invariant (but not translation-invariant) and that I
γ, p
f is p-integrable for any Schwartz function f. We finally apply that linear operator I
γ, p
with p = 1 to solve the stochastic partial differential equation
(-\triangle)g/2 F = w(-\triangle)^{\gamma/2} \Phi=w with white Poisson noise as its driving term w. 相似文献
19.
V. Vyalov 《Journal of Mathematical Sciences》2008,150(1):1771-1786
We study the local smoothness of solutions to the magnetohydrodynamic equations
where Ω is a domain in ℝ3, QT = Ω × (−T, 0), v: QT → ℝ3 is the velocity, p: QT → ℝ is the pressure, and H: QT → ℝ3 is the stress of the magnetic field. An analog of the known Caffarelli-Kohn-Nirenberg theorem is established. Conditions
of ε-regularity are derived. Bibliography: 8 titles.
__________
Translated from Problemy Matematicheskogo Analiza, No. 36, 2007, pp. 3–13. 相似文献
20.
Zeng Jian LOU Shou Zhi YANG Dao Jin SONG 《数学学报(英文版)》2005,21(4):949-954
We give a decomposition of the Hardy space Hz^1(Ω) into "div-curl" quantities for Lipschitz domains in R^n. We also prove a decomposition of Hz^1(Ω) into Jacobians det Du, u ∈ W0^1,2 (Ω,R^2) for Ω in R^2. This partially answers a well-known open problem. 相似文献