Integral Operators in Sobolev Spaces on Domains with Boundary

Properties of integral operators with weak singularities arc investigated. It is assumed that G ? ?ⁿ is a bounded domain. The boundary δG should be smooth concerning the Sobolev trace theorem. It will be proved that the integral operators $\int {_G \frac{{f\left(\Theta \right)}}{{x - y|^{n - 1} }}u\left(\nu \right)d\partial G_\nu }$ and $ \int {_{\partial G} \frac{{f\left(\Theta \right)}}{{|x - y|^{n - 1} }}u\left(y \right)d\partial G_y }$ maps W_p^k(G) into W_p^k+1(G) and W_p^k?1(G) into W_p^k/p(G), respectively, and are bounded. Here θ ∈ S ? ?ⁿ, where S is the unit sphere. Furthermore, f possesses bounded first order derivatives and is bounded on S. Then applications to first order systems are discussed.     

Fredholm One-Dimensional Boundary-Value Problems in Sobolev Spaces

Ukrainian Mathematical Journal - For systems of ordinary differential equations on a compact interval, we study the character of solvability of the most general linear boundary-value problems in...     

General Regularizing Functionals for Solving Ill-Posed Problems in Lebesgue Spaces

We study sufficient conditions for general integral functionals in Lebesgue spaces to possess regularizing properties required for solving nonlinear ill-posed problems. We select special classes of such functionals: uniformly convex and quasiuniformly convex (in the extended sense). We give a series of examples and, in particular, a functional that can be used in a generalized version of the maximum entropy method in Lebesgue spaces.     

Grand Sobolev Spaces on Metric Measure Spaces

Siberian Mathematical Journal - We study the properties of the so-called grand Sobolev spaces on a metric measure space. The introduction of the spaces is motivated by the available...     

Pseudo-uniform Convexity in Hp and Some Extremal Problems on Sobolev Spaces

We extend Newman and Keldysh theorems to the behavior of sequences of functions in H^p (μ) which explain geometric properties of discs in these spaces. Through Keldysh's theorem we obtain asymptotic results for extremal polynomials in Sobolev spaces.     

Radon Transform on Sobolev Spaces

Siberian Mathematical Journal - The Radon transform  $ R $ maps a function  $ f $ on  $ {??}^{n} $ to the family of the integrals of  $ f $ over...     

A Note on Sobolev Spaces

In the theory of averaging and other branches of mathematical physics, a crucial role is played by Sobolev spaces connected with a Borel measure defined in the Euclidean space ℝ^d. These spaces are defined as closures of the sets of smooth functions in an appropriate norm. In this paper, we propose another (dual) definition of Sobolev spaces and give an example of using this definition.__________Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 10, Suzdal Conference-4, 2003.     

Operators on Sobolev Type Spaces

In this paper, we introduce the work done on Hardy-Sobolev spaces and Fock-Sobolev spaces and their operators and operator algebras, including the study of boundedness, compactness, Fredholm property, index theory, spectrum and essential spectrum, norm and essential norm, Schatten-p classes, and the $C^∗$ algebras generated by them.     

拓扑序空间中的广义向量值均衡问题

利用一个KKM型定理,在拓扑序空间中,建立了关于广义向量值均衡问题解的某些新的存在性定理.     

Positive Functionals on some Spaces of Fractions

The aim of this work is to study norm preserving extensions of positive functionals on some spaces of fractions. The main result is stated in the case of unitary complex Banach algebras with involution. Moreover, we deal with C*-algebras and the commutative case as well. An application is also given.     

The Dirichlet Energy Integral and Variable Exponent Sobolev Spaces with Zero Boundary Values

We define and study variable exponent Sobolev spaces with zero boundary values. This allows us to prove that the Dirichlet energy integral has a minimizer in the variable exponent case. Our results are based on a Poincaré-type inequality, which we prove under a certain local jump condition for the variable exponent.     

Sobolev Spaces with Zero Boundary Values on Metric Spaces

We generalize the definition of the first order Sobolev spaces with zero boundary values to an arbitrary metric space endowed with a Borel regular measure. We show that many classical results extend to the metric setting. These include completeness, lattice properties and removable sets.     

拓扑向量空间中的广义向量F-隐补问题

该文在拓扑向量空间中引入了一类新的向量F-隐补问题(简记为:(GVF-ICP))和两类(强和弱)向量F-隐变分不等式问题(分别简记为:(GVF-IVI)_1和(GVF-IVI)_2).在齐性条件下得到了(GVF-ICP)与(GVF-IVI)_1的等价关系,同时在适当假设下利用KKM定理得到了(GVF-IVI)_2的新的解的存在性定理.     

Equivalence of Sobolev Spaces

We repair the proof of equivalence of certain L²-Sobolev spaces on manifolds with bounded curvature of all orders from [4]. The results are extended to generalized compatible Dirac operators, fractional order Sobolev spaces and weighted Sobolev spaces. A certain way of doing coordinate free computations is presented.     

Continuous Essential Selections and Integral Functionals

Given a strictly positive measure, we characterize inner semicontinuous solid convex-valued mappings for which continuous functions which are selections almost everywhere are selections. This class contains continuous mappings as well as fully lower semicontinuous closed convex-valued mappings that arise in variational analysis and optimization of integral functionals. The characterization allows for extending existing results on convex conjugates of integral functionals on continuous functions. We also give an application to integral functionals on left continuous functions of bounded variation.     

More on a Variational Property of Integral Functionals

In this note, we improve Theorem 1 of Ref. 1 by removing some superfluous assumptions and pointing out a more precise conclusion.     

Ultracomplete Topological Spaces

In 1987 V. I. Ponomarev and V. V. Tkachuk characterized strongly complete topological spaces as those spaces which have countable character in their Stone-Cech compactification. On the other hand, in 1998 S. Romaguera intro- duced the notion of cofinally Cech complete spaces and he showed that a metriz- able space admits a cofinally complete metric (otherwise called ultracomplete metric), a term introduced independently by N.R. Howes in 1971 and Á. Csáaszár in 1975, if and only if it is cofinally Cech complete. In this paper we show that these two notions are equivalent and in this way answer a question raised in [16] about giving an internal characterization for strongly complete topological spaces.