共查询到20条相似文献,搜索用时 0 毫秒
1.
Hermann König 《Journal of Functional Analysis》1977,24(1):32-51
For an open set Ω ? N, 1 ? p ? ∞ and λ ∈ +, let denote the Sobolev-Slobodetzkij space obtained by completing in the usual Sobolev-Slobodetzkij norm (cf. A. Pietsch, “r-nukleare Sobol. Einbett. Oper., Ellipt. Dgln. II,” Akademie-Verlag, Berlin, 1971, pp. 203–215). Choose a Banach ideal of operators , 1 ? p, q ? ∞ and a quasibounded domain Ω ? N. Theorem 1 of the note gives sufficient conditions on λ such that the Sobolev-imbedding map exists and belongs to the given Banach ideal : Assume the quasibounded domain fulfills condition Ckl for some l > 0 and 1 ? k ? N. Roughly this means that the distance of any to the boundary ?Ω tends to zero as for , and that the boundary consists of sufficiently smooth ?(N ? k)-dimensional manifolds. Take, furthermore, 1 ? p, q ? ∞, p > k. Then, if μ, ν are real positive numbers with λ = μ + v ∈ , μ > λ S(; p,q:N) and v > N/l · λD(;p,q), one has that belongs to the Banach ideal . Here λD(;p,q;N)∈+ and λS(;p,q;N)∈+ are the D-limit order and S-limit order of the ideal , introduced by Pietsch in the above mentioned paper. These limit orders may be computed by estimating the ideal norms of the identity mappings lpn → lqn for n → ∞. Theorem 1 in this way generalizes results of R. A. Adams and C. Clark for the ideals of compact resp. Hilbert-Schmidt operators (p = q = 2) as well as results on imbeddings over bounded domains.Similar results over general unbounded domains are indicated for weighted Sobolev spaces.As an application, in Theorem 2 an estimate is given for the rate of growth of the eigenvalues of formally selfadjoint, uniformly strongly elliptic differential operators with Dirichlet boundary conditions in , where Ω fulfills condition C1l.For an open set Ω in N, let denote the Sobolev-Slobodetzkij space obtained by completing in the usual Sobolev-Slobodetzkij norm, see below. Taking a fixed Banach ideal of operators and 1 ? p, q ? ∞, we consider quasibounded domains Ω in N and give sufficient conditions on λ such that the Sobolev imbedding operator exists and belongs to the Banach ideal. This generalizes results of C. Clark and R. A. Adams for compact, respectively, Hilbert-Schmidt operators (p = q = 2) to general Banach ideals of operators, as well as results on imbeddings over bounded domains. Similar results over general unbounded domains may be proved for weighted Sobolev spaces. As an application, we give an estimate for the rate of growth of the eigenvalues of formally selfadjoint, uniformly strongly elliptic differential operators with Dirichlet boundary conditions in , where Ω is a quasibounded open set in N. 相似文献
2.
Hermann König 《Journal of Functional Analysis》1978,29(1):74-87
Let Ω ? N be an open set with and some l > 0 satisfying an additional regularity condition. We give asymptotic estimates for the approximation numbers αn of Sobolev imbeddings over these quasibounded domains Ω. Here denotes the Sobolev space obtained by completing under the usual Sobolev norm. We prove , where . There are quasibounded domains of this type where γ is the exact order of decay, in the case p ? q under the additional assumption that either 1 ? p ? q ? 2 or 2 ? p ? q ? ∞. This generalizes the known results for bounded domains which correspond to l = ∞. Similar results are indicated for the Kolmogorov and Gelfand numbers of Ip,qm. As an application we give the rate of growth of the eigenvalues of certain elliptic differential operators with Dirichlet boundary conditions in , where Ω is a quasibounded domain of the above type. 相似文献
3.
Rmi Arcangli María Cruz Lpez de Silanes Juan Jos Torrens 《Journal of Approximation Theory》2009,161(1):198-212
Given a function u belonging to a suitable Beppo–Levi or Sobolev space and an unbounded domain Ω in , we prove several Sobolev-type bounds involving the values of u on an infinite discrete subset A of Ω. These results improve the previous ones obtained by Madych and Potter [W.R. Madych, E.H. Potter, An estimate for multivariate interpolation, J. Approx. Theory 43 (1985) 132–139] and Madych [W.R. Madych, An estimate for multivariate interpolation II, J. Approx. Theory 142 (2006) 116–128]. 相似文献
4.
Judy Munshower 《Journal of Mathematical Analysis and Applications》2008,338(1):111-123
This paper studies the operator dd∗+d∗d acting on q-forms on an unbounded domain with smooth boundary, where d is the exterior derivative and d∗ is the adjoint of d calculated using the Sobolev space topology. The domain of d∗ is determined and an expression for d∗ is obtained. The operator dd∗+d∗d gives rise to a boundary value problem. Global regularity is obtained using weighted norms and global existence is obtained by using the theory of compact operators. 相似文献
5.
J. A. S. Martins 《Annali di Matematica Pura ed Applicata》1977,115(1):271-294
Summary Piecewise polynomial and Fourier approximation of functions in the Sobolev spaces
on unbounded domains Θ ⊂ Rn are applied to the study of the type of compact embeddings into appropriate Lebesgue and Orlicz spaces. It is shown that
if Θ and s satisfy certain conditions, the embeddings
, m/n+1/q−1/p>0 and
, Φ being an Orlicz function subordinate to both φ(t)=|t|p
exp |t|n/(n−m) and Φσ(t)=exp |t|σ−1, σ ⩾ 1, m/n>1/p, are of type ls. One result dealing with multiplications maps from
into Lq(Θ) is also obtained.
Entrata in Redazione il 14 ottobre 1976. 相似文献
6.
Mohan Namasivayam 《Annali di Matematica Pura ed Applicata》1986,144(1):157-172
Summary
Integral representations of junctions in the anisotropic Sobolev spaces
on unbounded domains are used in the study of the embeddings of these spaces into Lebesgue spaces. Estimates of entropy numbers of the embedding
are obtained, where k, p and q satisfy certain conditions and where is a certain type of quasibounded domain, 相似文献
7.
Vladimir I. Bogachev Andrey Yu. Pilipenko Alexander V. Shaposhnikov 《Journal of Mathematical Analysis and Applications》2014
We study extensions of Sobolev and BV functions on infinite-dimensional domains. Along with some positive results we present a negative solution of the long-standing problem of existence of Sobolev extensions of functions in Gaussian Sobolev spaces from a convex domain to the whole space. 相似文献
8.
Our objective is to study regularity of superharmonic functions of a nonlinear potential theory on metric measure spaces. In particular, we are interested in the local integrability properties of a superharmonic function and its derivative. We show that every superharmonic function has a weak upper gradient and provide sharp local integrability estimates. In addition, we study absolute continuity of a superharmonic function. 相似文献
9.
《复变函数与椭圆型方程》2012,57(6):441-451
This article deals with weighted boundary limits of monotone Sobolev functions on bounded s-John domains in a metric space. 相似文献
10.
Sofia Melo 《Bulletin of the Brazilian Mathematical Society》2014,45(1):91-116
We study the existence of minimal graphs in with prescribed boundary data, possibly infinite. We give necessary and sufficient conditions on the “lengths” of the sides of the inscribed polygons in an unbounded domain in ?2, that yield solutions to the minimal surface equation with prescribed boundary data. 相似文献
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12.
John L. Lewis 《Arkiv f?r Matematik》1987,25(1-2):255-264
13.
We present several examples of homogeneous domains in complex space that do not have bounded realisations. The domains are equivalent to tubes over affinely homogeneous domains in real space. 相似文献
14.
We present several examples of homogeneous domains in complex space that do not have bounded realisations. The domains are equivalent to tubes over affinely homogeneous domains in real space. 相似文献
15.
Javad Namazi 《Journal of Mathematical Analysis and Applications》2004,290(2):553-562
Let 1<p<∞, and k,m be positive integers such that 0(k−2m)pn. Suppose ΩRn is an open set, and Δ is the Laplacian operator. We will show that there is a sequence of positive constants cj such that for every f in the Sobolev space Wk,p(Ω), for all xΩ except on a set whose Bessel capacity Bk−2m,p is zero. 相似文献
16.
A. V. Bukhvalov 《Journal of Mathematical Sciences》1989,47(6):2908-2909
Necessary and sufficient conditions are derived for the equality WℓE(X)*=WℓE'(X*), where E is a symmetric space, X a Banach space, ℓ > 0 is an integer.
Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR,
Vol. 159, pp. 119–120, 1987. 相似文献
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