Space Of Scalar Functions Research Articles

On Isomorphism for the Space of Solenoidal Vector Fields and Its Application to the Incompressible Flows

We consider the space of solenoidal vector fields in an unbounded domain $\Omega\subset \mathbb{R}^n$ whose boundary is given as a Lipschitz graph. It is shown that the space of solenoidal vector fields is isomorphic to the $n-1$ product space of the space of scalar functions in some natural topology such as $L^2(\Omega)$. As an application, we introduce a systematic reduction of the equations describing the motion of incompressible flows. This gives a new perspective of the derivation of Ukai's solution formula for the Stokes equations in the half space and provides a key step for the generalization of Ukai's approach to the Stokes semigroup in the case of the curved boundary.

Interpolation and Embeddings of Weighted Tent Spaces

Given a metric measure space X, we consider a scale of function spaces T^{p,q}_s(X), called the weighted tent space scale. This is an extension of the tent space scale of Coifman, Meyer, and Stein. Under various geometric assumptions on X we identify some associated interpolation spaces, in particular certain real interpolation spaces. These are identified with a new scale of function spaces, which we call Z-spaces, that have recently appeared in the work of Barton and Mayboroda on elliptic boundary value problems with boundary data in Besov spaces. We also prove Hardy–Littlewood–Sobolev-type embeddings between weighted tent spaces.

Аналитическая задача Коши в классе функций с интегральной метрикой по пространственной и временной переменным

The article considers the complex Cauchy problem for general systems of linear differential equations in Lebesgue's Banach spaces with the weight of analytic functions with integral metrics. The functions from the solution spaces can admit power law-type singularities in integral sense when approaching the cone lateral boundary. The necessary and sufficient conditions under which the stated Cauchy problem is a locally well- posed one in the specified scale of functional spaces are derived. It turns out that these conditions for the orders of differential operators in case of extended singularities of solutions over the cone lateral boundary are identical with the Leray-Volevich conditions. In case of one equation containing the Kovalevskaya inequalities, these conditions are less restrictive than they are in the case of extended singularities of the solution over the cylinder lateral surface. Thus, the inequalities either exactly describe the structure of systems of linear differential equations for which the Cauchy problem is locally well-posed in the specified class, or they describe the scale of functional spaces in which the specified Cauchy problem is locally a well-posed one. In the course of proving the main theorem, new lemmas about the integral properties of analytic functions have been obtained, which may be of interest for the theory of functions of complex variable. These lemmas show an estimate for the norm of derivative from an analytic function through the norm of the function itself; they also show an estimate of the norm of integral with a parameter from the analytic function through the norm of the initial function, as well as other properties.

Triebel--Lizorkin type spaces with variable exponents

In this article, the authors first introduce the Triebel--Lizorkin-type space $F_{p(\cdot),q(\cdot)}^{s(\cdot),\phi}(\mathbb R^n)$ with variable exponents, and establish its $\varphi$-transform characterization in the sense of Frazier and Jawerth, which further implies that this new scale of function spaces is well defined. The smooth molecular and the smooth atomic characterizations of $F_{p(\cdot),q(\cdot)}^{s(\cdot),\phi}(\mathbb R^n)$ are also obtained, which are used to prove a trace theorem of $F_{p(\cdot),q(\cdot)}^{s(\cdot),\phi}(\mathbb R^n)$. The authors also characterize the space $F_{p(\cdot),q(\cdot)}^{s(\cdot),\phi}(\mathbb R^n)$ via Peetre maximal functions.

An Algebraic Multigrid Method for the Finite Element Analysis of Large Scattering Problems

An efficient iterative solver is proposed for the linear matrix equation that arises in the analysis of large scattering problems by the frequency-domain finite-element method. A Krylov method is preconditioned by a technique that approximately solves equivalent problems in two auxiliary spaces: a space of scalar functions and a space of piecewise linear, “nodal,” vector functions. On each space the traditional algebraic multigrid method is employed. Further, the “shifted Laplace” idea is used to improve the performance of the solver as the frequency increases. Results are reported for a waveguide cavity filter and three free-space scatterers: a conducting sphere, a metallic frequency selective surface, and a metamaterial lens made of split-ring resonators containing dielectric and metal.

Methods of evaluation and scale of functional spaces

Linear equations of mathematical physics with constant coefficients have fed calculational mathematics since the 18th century. The area of nonlinear equations with variable coefficients arose due to gas-hydrodynamic problems in the 20th century. Now, one of the methods of research for properties of solutions of such equations and, accordingly, applied problems is the use of calculations on modern computers. The capacities of computers and their efficiency have increased in the 21st century and allowed progress to be made in solving applied problems, except for cases of methodical errors in calculations. One of the basic sources of such methodical errors is the “uncontrollable” machine accuracy of calculations. One of the methods of solving such numerical problems is a suitable localization of the problem and the choice of an adequate basis of the necessary functional space. Below, we state new results in this area of mathematical and applied research.

The curvature of semidirect product groups associated with two-component Hunter–Saxton systems

In this paper, we study two-component versions of the periodic Hunter–Saxton equation and its μ-variant. Considering both equations as a geodesic flow on the semidirect product of the circle diffeomorphism group with a space of scalar functions on we show that both equations are locally well posed. The main result of this paper is that the sectional curvature associated with the 2HS is constant and positive and that 2µHS allows for a large subspace of positive sectional curvature. The issues of this paper are related to some of the results for 2CH and 2DP presented in Escher et al (2011 J. Geom. Phys. 61 436–52).

Field on Poincaré Group and Quantum Description of Orientable Objects

We propose an approach to the quantum-mechanical description of relativistic orientable objects. It generalizes Wigner's ideas concerning the treatment of nonrelativistic orientable objects (in particular, a nonrelativistic rotator) with the help of two reference frames (space-fixed and body-fixed). A technical realization of this generalization (for instance, in 3+1 dimensions) amounts to introducing wave functions that depend on elements of the Poincare group $G$. A complete set of transformations that test the symmetries of an orientable object and of the embedding space belongs to the group $\Pi =G\times G$. All such transformations can be studied by considering a generalized regular representation of $G$ in the space of scalar functions on the group, $f(x,z)$, that depend on the Minkowski space points $x\in G/Spin(3,1)$ as well as on the orientation variables given by the elements $z$ of a matrix $Z\in Spin(3,1)$. In particular, the field $f(x,z)$ is a generating function of usual spin-tensor multicomponent fields. In the theory under consideration, there are four different types of spinors, and an orientable object is characterized by ten quantum numbers. We study the corresponding relativistic wave equations and their symmetry properties.

The coupled problem of a solid oscillating in a viscous fluid under the action of an elastic force

The torsional oscillations are studied of a solid of revolution under the action of elastic torque inside a container with a viscous incompressible fluid. We prove the asymptotic stability of the static equilibrium. We use the two approaches: the direct Lyapunov and linearization methods. The global asymptotic stability is established using a one-parameter family of Lyapunov functionals. Then small oscillations are studied of the fluid-solid system. The linearized operator of the problem of a solid oscillating in a fluid can be realized as an operator matrix obtained by appending two scalar rows and two columns to the Stokes operator. This operator is therefore a two-dimensional bordering of the Stokes operator and inherits many properties of the latter; in particular, the spectrum is discrete. The eigenvalue problem for the linearized operator is reduced to solving a dispersion equation. Inspection of the equation shows that all eigenvalues lie inside the right (stable) half-plane. Basing on this, we justify the linearization. Using an abstract theorem of Yudovich, we prove the asymptotic stability in a scale of function spaces, the infinite differentiability of solutions, and the decay of all their derivatives in time.

Elliptic Operators in a Refined Scale of Functional Spaces

We study the theory of elliptic boundary-value problems in a refined two-sided scale of the Hormander spaces H s , ϕ, where s ∈ R and ϕ is a functional parameter slowly varying at +∞. For the Sobolev spaces H s , the function ϕ(|ξ|) ≡ 1. We establish that the considered operators possess the Fredholm property, and solutions are globally and locally regular.

Interpolation properties of a scale of spaces

A scale of function spaces is considered which proved to be of considerable importance in analysis. Interpolation properties of these spaces are studied by means of the real interpolation method. The main result consists in demonstrating that this scale is interpolated in a way different from that for $L^{ p }$ spaces, namely, the interpolation space is not from this scale.

Boundary Equations in Dynamic Problems of the Theory of Elasticity

We study various nonstationary boundary equations that arise when the mixed problems of the theory of elasticity and thermoelasticity are solved by means of dynamic surface potentials. In the review, the results obtained are presented on their unique solvability in the scales of function spaces of the Sobolev type.

A Class of Periodic Function Spaces and Interpolation on Sparse Grids

We present a unified approach to error estimates of periodic interpolation on equidistant, full, and sparse grids for functions from a scale of function spaces which includes L 2-Sobolev spaces, the Wiener algebra and the Korobov spaces.

Simple Cubature Formulas with High Polynomial Exactness

We study cubature formulas for d -dimensional integrals with arbitrary weight function of tensor product form. We present a construction that yields a high polynomial exactness: for fixed degree, the number of knots depends on the dimension in an order-optimal way. The cubature formulas are universal: the order of convergence is almost optimal for two different scales of function spaces. The construction is simple: a small number of arithmetical operations is sufficient to compute the knots and the weights of the formulas.

Pseudodifferential operators on manifolds with edges

We define pseudodifferential operators on manifolds with singularities of smooth edge type and construct a calculus of such operators. We prove that a pseudodifferential operator is invariant with respect to a certain natural class of diffeomorphisms of the manifold. We introduce a scale of function spaces (weighted analogs of the Sobolev classes) and establish theorems on boundedness of pseudodifferential operators in this scale. Bibliography: 8 titles.

Totally characteristic pseudo-differential operators in Besov-Lizorkin-Triebel spaces

A full symbolic calculus for totally characteristic pseudo-differential operators acting in general scales of function spaces with conormal asymptotics of several types will be developed. By using modified methods we will show that the results of S. Rempel and B.-W. Schulze for full asymptotics in Sobolev spaces can be generalized for Besov-Lizorkin-Triebel spaces, in particular, Holder spaces, with other types of asymptotics.