Tubular Domains Research Articles

Theorems of Hardy-Littlewood type for signed measures on a cone

It is known that the positivity condition plays an important role in theorems of Hardy-Littlewood type. In the multi-dimensional case this condition can be relaxed significantly by replacing it with the condition of sign-definiteness on trajectories along which asymptotic properties are investigated. A number of theorems are proved in this paper that demonstrate this effect. Our main tool is a theorem on division of tempered distributions by a homogeneous polynomial, preserving the corresponding quasi-asymptotics. The results obtained are used to study the asymptotic behaviour at a boundary point of holomorphic functions in tubular domains over cones.

Center manifolds in equations from hydrodynamics

Elliptic partial differential equations on tubular domains arise in hydrodynamics and in other applied settings. The center manifold approach to the analysis of small solutions has as one of its steps the study of a reduced equation of differential-integral type leading to a system of ordinary differential equations, generally of finite order. Here we give existence and regularity results for reduced equations which are adapted to the Holder space setting and which are suitable for application to quasi-linear problems. Applications to flow problems are given.

Inertial manifold theory for a class of reaction-diffusion equations on thin tubular domains

Inertial manifold theory for a class of reaction-diffusion equations on thin tubular domains

AC loss analysis and domain structure in magnetostrictive amorphous wires

Low-field ac susceptibilities of rotating water quenched amorphous Fe 77.5Si 7.5B 15 wires have been measured as functions of frequency and position. Anomalous eddy-current losses have been detected for different positions at the remanence state of samples with different lengths, related to the domain structures. An eddy-current loss model has been developed which combines the effects of magnetization rotations in a tubular shell, the displacements of a tubular domain wall in the core region, and the displacements of the shell-core boundary. Using this model, the ac losses have been analyzed, concluding that the radius of the boundary is 83% of the wire radius and the domain division makes a roughly parabolic axial position dependence of the average magnetization in the core, no matter if the wire is in the stable state or during magnetization reversal. Thus, the magnetization reversal should be controlled by domain wall pinning. In addition, magnetic relaxation and magnetomechanical coupling play important roles in the technical magnetization of such wires.

Eigenvalue Comparison for Tubular Domains

S. Y. Cheng’s eigenvalue comparison theorem for geodesic balls is generalized to certain tubular domains. This gives application to the infinitesimal volume comparison theory developed by E. Heintze and H. Karcher as well as A. Gray.

Eigenvalue comparison for tubular domains

S. Y. Cheng’s eigenvalue comparison theorem for geodesic balls is generalized to certain tubular domains. This gives application to the infinitesimal volume comparison theory developed by E. Heintze and H. Karcher as well as A. Gray.

Existence of stable stationary solutions of scalar reaction-diffusion equations in thin tubular domains

Scalar reaction-diffusion equations on thin tabular domain is studied. It is shown by using a comparison technique that there exists a stable stationary solution if an associated one-dimensional equation has a stable stationary solution

Jessen theorems for holomorphic, almost-periodic functions in tubular domains

Jessen theorems for holomorphic, almost-periodic functions in tubular domains

Stationary solutions of the Navier-Stokes equations in periodic tubes

Let Ω be a tubular domain in Rn, n=2,3, with a Lipschitz boundary δΩ, invariant with respect to a translation by the vector $$\vec \ell \in R^n$$ . It is proven that, for any prescribed real number ρo, there exists at least one solution $$\left\{ {\vec \upsilon ,\rho } \right\}$$ of the nonhomogeneous boundary-value problem for a stationary Navier-Stokes system with a periodic $$\vec \upsilon$$ and pressure ρ, having the drop ρo over the period. (The exterior forces and the boundary values of the velocity field are assumed to be periodic.) In addition, one proves the existence of a “critical” nonnegative number ρ*, depending only on the geometry of the domain Ω, the viscosity coefficient, the exterior forces and the boundary values of $$\vec \upsilon$$ , such that for ¦ρ0¦>ρ* “the fluid flows along the direction of the decrease of the pressure.”

Tempered Ultradistributions as Boundary Values of Analytic Functions

The spaces ${S_{{a_k}}}$, ${S^{{b_q}}}$ and $S_{{a_{k}}}^{{b_q}}$ were introduced by I. M. Gel’fand as a generalization of the test function spaces of type $S$; the elements of the corresponding dual spaces are called tempered ultradistributions. It is shown that a function which is analytic in a tubular radial domain and satisfies a certain nonpolynomial growth condition has a distributional boundary value in the weak topology of the tempered ultradistribution space $(S_{{b_{k}}}^{{a_{q}}})\prime$, which is the space of Fourier transforms of elements in $(S_{{a_{k}}}^{{b_{q}}})\prime$. This gives rise to a representation of the Fourier transform of an element $U \in (S_{{a_{k}}}^{{b_{q}}})\prime$ having support in a certain convex set as a weak limit of the analytic function. Converse results are also obtained. These generalized Paley-Wiener-Schwartz theorems are established by means of a number of new lemmas concerning $S_{{a_{k}}}^{{b_{q}}}$ and its dual. Finally, in the appendix the equality $S_{{a_k}}^{{b_q}} = {S_{{a_k}}} \cap {S^{{b_q}}}$ is proved.

Tempered ultradistributions as boundary values of analytic functions

The spaces S a k {S_{{a_k}}} , S b q {S^{{b_q}}} and S a k b q S_{{a_{k}}}^{{b_q}} were introduced by I. M. Gel’fand as a generalization of the test function spaces of type S S ; the elements of the corresponding dual spaces are called tempered ultradistributions. It is shown that a function which is analytic in a tubular radial domain and satisfies a certain nonpolynomial growth condition has a distributional boundary value in the weak topology of the tempered ultradistribution space ( S b k a q ) ′ (S_{{b_{k}}}^{{a_{q}}})\prime , which is the space of Fourier transforms of elements in ( S a k b q ) ′ (S_{{a_{k}}}^{{b_{q}}})\prime . This gives rise to a representation of the Fourier transform of an element U ∈ ( S a k b q ) ′ U \in (S_{{a_{k}}}^{{b_{q}}})\prime having support in a certain convex set as a weak limit of the analytic function. Converse results are also obtained. These generalized Paley-Wiener-Schwartz theorems are established by means of a number of new lemmas concerning S a k b q S_{{a_{k}}}^{{b_{q}}} and its dual. Finally, in the appendix the equality S a k b q = S a k ∩ S b q S_{{a_k}}^{{b_q}} = {S_{{a_k}}} \cap {S^{{b_q}}} is proved.

Cauchy and Poisson integral representations for ultradistributions of compact support and distributional boundary values

SynopsisUltradistributions of compact support are represented as the boundary values of Cauchy and Poisson integrals corresponding to tubular radial domains Tc' =ℝn + iC', C'⊂⊂C, where C is an open, connected, convex cone. The Cauchy integral of is shown to be an analytic function in TC' which satisfies a certain boundedness condition. Analytic functions which satisfy a specified growth condition in TC' have a distributional boundary value which can be used to determine an distribution.

Analytic functions having distributional boundary values in W′-spaces

AbstractIt is shown that the functions which are analytic in tubular radial domains and satisfy certain growth conditions have distributional boundary values in the weak topology of (WΩ)′-space. Representation of analytic functions in terms of distributional boundary values are given. Converse results are also obtained. An analytic decomposition theorem is proved. The main theorems are established by means of a number of lemmas concerning WM, WΩ spaces and their dual spaces. Several new lemmas are proved for K{Mp} spaces from which results for WM-spaces can be easily deduced.

Solution of boundary-value problems in the theory of analytic functions of several variables in Vladimirov algebras

A solution is given for the Riemann problem for tubular domains in Vladimirov algebras in closed form by means of an integral representation of Bochner-Vladimirov type which is constructed here. In particular, the Schwartz problem is solved. The statement of the Hilbert problem in Vladimirov algebras is examined and its solution is given by a reduction to the Riemann problem, and in one case by a reduction to the Schwartz problem.

Representation of distributions with compact support

Distributions having compact support are represented as the boundary value of Cauchy and Poisson integrals corresponding to tubular radial domains TC in ℂn where C is an open convex cone. The Cauchy integral of U e ɛ′ is shown to be an analytic function in TC which satifies a certain boundedness condition. All analytic functions in TC having this boundedness condition have a distributional boundary value which can be used to determine an ɛ′ distribution. The results are extended to vector valued distributions.

Many-dimensional operators of convolution type in spaces of weight-integrable functions

We show that a convolution operator in the weight space Lp is similar to a generalized convolution operator in Lp. We obtain necessary and sufficient conditions for an operator of convolution type, acting in a weight space, to have the Noether property in a cone. These conditions say, in effect, that the operator symbol must not degenerate on the hull of some tubular domain associated with the weight and the cone.

On the plate paradox of sapondzhyan and babuška

We consider flows of an incompressible Navier–Stokes fluid in a tubular domain with Navier’s slip boundary condition imposed on the impermeable wall. We focus on several implementational issues associated with this type of boundary conditions within the framework of the standard Taylor-Hood mixed finite element method and present the computational results for flows in a tubular domain of finite length with one inlet and one outlet. In particular, we present the details regarding variants of the Nitsche method concerning the incorporation of the impermeability condition on the wall. We also find that the manner in which the normal to the boundary is numerically implemented influences the nature of the computational results. As a benchmark, we set up steady flows in a tube of finite length and compare the computational results with the analytical solutions. Finally, we identify various quantities of interest, such as the dissipation, wall shear stress, vorticity, pressure drop, and provide their precise mathematical definitions. We document how well these quantities are computationally approximated in the case of the benchmark.Although the geometry of the benchmark is simple, the correct computational results require careful selection of numerical methods and surprisingly non-trivial computational resources. Our goal is to test, using the setting with a known analytical solution, a robust computational tool that would be suitable for computations on real complex geometries that have relevance to problems in engineering and medicine. The model parameters in our computations are chosen based on flows in large arteries.

Generalized Cauchy and Poisson Integrals and Distributional Boundary Values

Let C be an open connected cone, and let $O(C)$ denote its convex envelope. Cauchy and Poisson integrals of distributions in $\mathcal{D}'_{Lp} $, $1 < p \leqq 2$, corresponding to tubular radial domains $T^{O(C)} = \mathbb{R}^n + iO(C)$ are defined; and properties of these integrals are obtained. The boundary values of these integrals are obtained in the distributional sense on the distinguished boundary of $T^{O(C)} $ Functions which are analytic and have a specified growth condition in $T^{O(C)} $ are related to the Cauchy and Poisson integrals of their distributional boundary values. The results concerning these functions extend some well-known theorems concerning the Hardy $H^p (T^{O(C)} )$-spaces to our distributional setting. Further, functions which are analytic in disconnected tubular cones are considered; and in particular conditions are obtained under which such a function has an analytic extension to the convex envelope of the tubular cone.

ON CAUCHY-BOCHNER REPRESENTATIONS

In this paper the classical Cauchy-Bochner integral representation is generalized to a more extensive class of functions holomorphic in tubular domains TC over a cone C.

PLURISUBHARMONIC FUNCTIONS IN TUBULAR RADIAL DOMAINS. II

For plurisubharmonic functions in a convex radial tubular domain of growth of order p with respect to the imaginary part, we study various growth indexes. The results of this study are then applied to establish a connection between the growth of the Fourier-Laplace transform of a generalized function and the properties of its decrease.