共查询到20条相似文献,搜索用时 15 毫秒
1.
We explore spectral duality in the context of measures in ℝ
n
, starting with partial differential operators and Fuglede’s question (1974) about the relationship between orthogonal bases
of complex exponentials in L
2(Ω) and tiling properties of Ω, then continuing with affine iterated function systems. We review results in the literature
from 1974 up to the present, and we relate them to a general framework for spectral duality for pairs of Borel measures in
ℝ
n
, formulated first by Jorgensen and Pedersen. 相似文献
2.
Andreas Fr?hlich 《Annali dell'Universita di Ferrara》2000,46(1):11-19
We consider weights of Muckenhoupt classA
q, 1<q<∞. For a bounded Lipschitz domain Ω⊂ℝn we prove a compact embedding and a Poincaré inequality in weighted Sobolev spaces. These technical tools allow us to solve
the weak Neumann problem for the Laplace equation in weighted spaces on ℝn, ℝn
+, on bounded and on exterior domains Ω with boundary of classC
1, which will yield the Helmholtz decomposition ofL
ω
q(Ω)n for general ω∈A
q. This is done by transferring the method of Simader and Sohr [4] to the weighted case. Our result generalizes a result of
Farwig and Sohr [2] where the Helmholtz decomposition ofL
ω
p(Ω)n is proved for an exterior domain and weights of Muckenhoupt class without singularities or degeneracies in a neighbourhood
of ϖΩ.
Sunto In questo lavoro consideriamo dei pesi della classe di MuckenhouptA q, 1<q<∞. Per un dominio limitato lipschitziano Ω⊂ℝn, dimostriamo una immersione compatta ed una disuguaglianza di Poincaré in spazi di Sobolev con peso. Questa tecnica ci consente di risolvere il problema debole di Neumann per l’equazione di Laplace in spazi pesati in ℝn, ℝn + in domini limitati ed in domini esterni con frontiera di classeC 1, che conduce alla decomposizione di Helmholtz diL ω q(Ω)n per un qualsiasi ω∈A q. Il risultato è ottenuto trasferendo il metodo di Simader e Sohr [4] al caso pesato. Quello qui presente estende un risultato di Farwig e Sohr [2] dove la decomposizione di Helmholtz diL ω q(Ω)n è dimostrata per domini esterni e pesi della classe di Muckenhoupt privi di singolarità in un intorno di ϖΩ.相似文献
3.
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
4.
We study the existence result of solutions for the nonlinear degenerated elliptic problem of the form, -div(a(x, u,△↓u)) = F in Ω, where Ω is a bounded domain of R^N, N≥2, a :Ω×R×R^N→R^N is a Carathéodory function satisfying the natural growth condition and the coercivity condition, but they verify only the large monotonicity. The second term F belongs to W^-1,p′(Ω, w^*). The existence result is proved by using the L^1-version of Minty's lemma. 相似文献
5.
A. P. Oskolkov 《Journal of Mathematical Sciences》1997,83(2):320-326
In this paper, we study some nonlocal problems for the Kelvin-Voight equations (1) and the penalized Kelvin-Voight equations
(2): the first and second initial boundary-value problems and the first and second time periodic boundary problems. We prove
that these problems have global smooth solutions of the classW
∞
1
(ℝ+;W
2
2+k
(Ω)),k=1,2,...;Ω⊂ℝ3. Bibliography: 25 titles.
Dedicated to N. N. Uraltseva on her jubilee
Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 221, 1995, pp. 185–207.
Translated by N. A. Karazeeva. 相似文献
6.
We investigate the Caucy problem for linear elliptic operators withC
∞-coefficients at a regular domain ℝ ⊂ ℝ, which is a classical example of an ill-posed problem. The Cauchy data are given at
the manifold Γ⊂∂Ω and our goal is to obtain a stability estimate inH
4(Ω). 相似文献
7.
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
8.
Area,coarea, and approximation in <Emphasis Type="Italic">W</Emphasis><Superscript>1,1</Superscript>
David Swanson 《Arkiv f?r Matematik》2007,45(2):381-399
Let Ω⊂ℝ
n
be an arbitrary open set. We characterize the space W
1,1
loc(Ω) using variants of the classical area and coarea formulas. We use these characterizations to obtain a norm approximation
and trace theorems for functions in the space W
1,1(ℝ
n
). 相似文献
9.
Lower semicontinuity for polyconvex functionals of the form ∫Ω
g(detDu)dx with respect to sequences of functions fromW
1,n
(Ω;ℝ
n
) which converge inL
1 (Ωℝ
n
) and are uniformly bounded inW
1,n−1 (Ω;ℝ
n
), is proved. This was first established in [5] using results from [1] on Cartesian Currents. We give a simple direct proof
which does not involve currents. We also show how the method extends to prove natural, essentially optimal, generalizations
of these results.
Supported by MURST, Gruppo Nazionale 40%
Partially supported by Australian Research Council 相似文献
10.
The goal of this work is to study the inhomogeneous Dirichlet problem for the Stokes system in a Lipschitz domain Ω ⊆ ℝ
n
, n⩾2. Our main result is that this problem is well posed in Besov-Triebel-Lizorkin spaces, provided that the unit normal ν to Ω has small mean oscillation. 相似文献
11.
Stephen J. Gardiner 《Journal d'Analyse Mathématique》1996,68(1):95-106
Let Ω be an open set in ℝ
n
andE be a relatively closed subset of Ω. Further, letC
e(E) be the collection of real-valued continuous functions onE which extend continuously to the closure ofE in ℝ
n
. We characterize those pairs (Ω,E) which have the following property: every function inC
e(E) which is harmonic onE
0 can be uniformly approximated onE by functions which are harmonic on Ω and whose restrictions toE belong toC
e(E). 相似文献
12.
M. V. Korobkov 《Siberian Mathematical Journal》2009,50(5):874-886
We find necessary and sufficient conditions for a curve in ℝ
m×n
to be the gradient range of a C
1-smooth function υ: Ω ⊂ ℝ
n
→ ℝ
m
. We show that this curve has tangents in a weak sense; these tangents are rank 1 matrices and their directions constitute
a function of bounded variation. We prove also that in this case v satisfies an analog of Sard’s theorem, while the level
sets of the gradient mapping ▿υ: Ω → ℝ
m×n
are hyperplanes. 相似文献
13.
Pu Zhang 《数学学报(英文版)》2010,26(9):1709-1722
Let μΩ^mb be the commutator generalized by μΩ, the n-dimensional Marcinkiewicz integral, and b ∈ BMO(R^n). The author establishes the weighted weak LlogL-type estimates for μΩ^mb when Ω satisfies a kind of Dini conditions, which improves the known result essentially. 相似文献
14.
Sergio Polidoro 《Annali dell'Universita di Ferrara》1991,37(1):131-150
Riassunto Viene provata l’esistenza di soluzioni positive per il problema di Dirichlet quasilineareMu+f(u)=0 in Ω,u=0 su βΩ, con Ω aperto di ℝ
N
, limitato o non limitato, verificante opportune proprietà geometriche. QuiM è un operatore ellittico in forma di divergenza, non degenere o degenere, come ilp-laplaciano o l’operatore di curvatura media.
We prove the existence of positives solutions for the quasilinear Dirichlet problemMu+f(u)=0 in Ω,u=0 on βΩ, were Ω is a bounded or unbounded open subset of ℝ N , satisfying suitable geometrical properties. HereM is a non degenerate or degenerate elliptic operator in divergence form, like thep-laplacian or the mean curvature operator.相似文献
15.
WANG Meng CHEN Jiecheng & FAN Dashan Department of Mathematics Zhejiang University 《中国科学A辑(英文版)》2006,49(1):98-108
We study certain square functions on product spaces Rn × Rm, whose integral kernels are obtained from kernels which are homogeneous in each factor Rn and Rm and locally in L(log L) away from Rn × {0} and {0} × Rm by means of polynomial distortions in the radial variable. As a model case, we obtain that the Marcinkiewicz integral operator is bounded on Lp(Rn × Rm)(P > 1) for Ω∈ e Llog L(Sn-1 × Sm-1) satisfying the cancellation condition. 相似文献
16.
Kentaro Hirata 《Potential Analysis》2009,30(2):165-177
In an unbounded domain Ω in ℝ
n
(n ≥ 2) with a compact boundary or Ω = ℝ
n
, we investigate the existence of limits at infinity of positive superharmonic functions u on Ω satisfying a nonlinear inequality like as
where Δ is the Laplacian and c > 0 and p > 0 are constants. The result is applicable to positive solutions of semilinear elliptic equations of Matukuma type.
This work was partially supported by Grant-in-Aid for Young Scientists (B) (No. 19740062), Japan Society for the Promotion
of Science. 相似文献
17.
Isabel M. C. Salavessa 《Bulletin of the Brazilian Mathematical Society》2010,41(4):495-530
On a Riemannian manifold $
\bar M^{m + n}
$
\bar M^{m + n}
with an (m + 1)-calibration Ω, we prove that an m-submanifold M with constant mean curvature H and calibrated extended tangent space ℝH ⋇ TM is a critical point of the area functional for variations that preserve the enclosed Ω-volume. This recovers the case described
by Barbosa, do Carmo and Eschenburg, when n = 1 and Ω is the volume element of $
\bar M
$
\bar M
. To the second variation we associate an Ω-Jacobi operator and define Ω-stability. Under natural conditions, we show that
the Euclidean m-spheres are the unique Ω-stable submanifolds of ℝ
m+n
. We study the Ω-stability of geodesic m-spheres of a fibred space form M
m+n
with totally geodesic (m + 1)-dimensional fibres. 相似文献
18.
Frank Pacard 《manuscripta mathematica》1993,79(1):161-172
For scalar non-linear elliptic equations, stationary solutions are defined to be critical points of a functional with respect
to the variations of the domain. We consideru a weak positive solution of −Δu=u
α in -Δu=u
α in Ω ⊂ ℝ
n
, which is stationary. We prove that the Hausdorff dimension of the singular set ofu is less thann−2α+1/α−1, if α≥n+2/n−2. 相似文献
19.
Lisheng Shu Rulong Xie 《分析论及其应用》2007,23(3):201-212
Let μmΩ,b be the higher order commutator generated by Marcinkiewicz integral μΩ and a BMO(Rn) function b(x). In this paper, we will study the continuity ofμΩ and μmΩ,b on homogeneous Morrey-Herz spaces. 相似文献
20.
In this paper, the authors consider the behaviors of a class of parametric Marcinkiewicz integrals μ
Ω
ρ
, μ
Ω,λ
*,ρ
and μ
Ω,S
ρ
on BMO(ℝ
n
) and Campanato spaces with complex parameter ρ and the kernel Ω in Llog+
L(S
n−1). Here μ
Ω,λ
*,ρ
and μ
Ω,S
ρ
are parametric Marcinkiewicz functions corresponding to the Littlewood-Paley g
λ
*-function and the Lusin area function S, respectively. Under certain weak regularity condition on Ω, the authors prove that if f belongs to BMO(ℝ
n
) or to a certain Campanato space, then [μ
Ω,λ
*,ρ
(f)]2, [μ
Ω,S
ρ
(f)]2 and [μ
Ω
ρ
(f)]2 are either infinite everywhere or finite almost everywhere, and in the latter case, some kind of boundedness are also established. 相似文献