Existence and uniqueness theorems for cusped prismatic shells in the Nth hierarchical model

We study the well posedness of boundary value problems for elastic cusped prismatic shells in the Nth approximation of I. Vekua's hierarchical models under (all reasonable) boundary conditions at the cusped edge and given displacements at the non‐cusped edge and stresses at the upper and lower faces of the shell. Copyright © 2008 John Wiley & Sons, Ltd.     

On a Mathematical Problem of Cusped Double-Layered Plates

The present paper is devoted to the system of degenerate partial differential equations arise from the investigation of elastic two layered prismatic shells. Harmonic vibration of cusped double-layered plates is consider. The weak setting of the BVPs in the case of the zero approximation of hierarchical models is considered. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Nonlocal boundary value problem for a system of singular differential equations and applications to cusped prismatic shells

The paper deals with a system of singular partial differential equations of the first and second order arising in the zero approximation of I.Vekua's hierarchical models of prismatic shells, when the thickness of the shell varies as a power function of one argument and vanishes at the cusped edge of the shell. For this system of special type a nonlocal boundary value problem in a half-plane is solved in the explicit form. The boundary value problem under consideration corresponds to stress-strain state of the cusped prismatic shell [1,2] under the action of concentrated forces and moments. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Investigation of hyperbolic systems with order degeneration arising in I. Vekua's hierarchical models

We study the well-posedness of an initial-boundary value problem corresponding to the zeroth approximation of I. Vekua's hierarchical models for elastic cusped prismatic shells. The mathematical model is described by a two-dimensional order-degenerating hyperbolic system. We formulate the problem in the weak setting and prove the uniqueness and existence theorems. We show that the sequence of corresponding explicit Galerkin approximations converges to the exact solution in an appropriate weighted Lebesgue space.     

Some degenerate elliptic systems and applications to cusped plates

We study the tension‐compression vibration of an elastic cusped plate under (all reasonable) boundary conditions at the cusped edge and given displacements at the non‐cusped edge and stresses at the upper and lower faces of the plate. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Cusped elastic beams under the action of stresses and concentrated forces

A dynamical problem in the (0, 0) approximation of elastic cusped prismatic beams is investigated when stresses are applied at the face surfaces and the ends of the beam. Two types of cusped ends are considered when the beam cross-section turns into either a point or a straight line segment. Correspondingly, at the cusped end either a force concentrated at the point or forces concentrated along the straight line segment is applied. We prove the exists and uniqueness theorems in appropriate weighted Sobolev spaces.     

A cusped prismatic shell-like body under the action of concentrated moments

The elastic equilibrium problem of a cusped prismatic shell-like body, when its projection is a half-plane x ₂ ≥ 0, under the action of a concentrated moment is solved in the explicit form within the framework of the zero approximation of I.Vekua’s hierarchical models of prismatic shells. The thickness of the prismatic shell-like body is proportional to the coordinate x ₂ raised to a non-negative exponent. When the exponent equals to zero, the above solution contains the well-known solution of the classical Carothers’ problem [1] in the case of an elastic half-plane (see also [2], §39).      

A cusped prismatic shell-like body under the action of concentrated moments

The elastic equilibrium problem of a cusped prismatic shell-like body, when its projection is a half-plane x ₂ ≥ 0, under the action of a concentrated moment is solved in the explicit form within the framework of the zero approximation of I.Vekua’s hierarchical models of prismatic shells. The thickness of the prismatic shell-like body is proportional to the coordinate x ₂ raised to a non-negative exponent. When the exponent equals to zero, the above solution contains the well-known solution of the classical Carothers’ problem [1] in the case of an elastic half-plane (see also [2], §39).     

On a dynamical hierarchical model for prismatic shells in the theory of elastic mixtures

The present paper is devoted to the design of a hierarchy of two‐dimensional models for dynamical problems within the theory of multicomponent linearly elastic mixtures in the case of prismatic shells with thickness which may vanish on some part of its boundary. The hierarchical model is obtained by a semidiscretization of the three‐dimensional problem in the transverse direction. In suitable weighted Sobolev spaces we investigate the well‐posedness of the two‐dimensional problems, prove pointwise convergence of the sequence of approximate solutions restored from the solutions of the reduced problems to the exact solution of the original problem and estimate the rate of convergence. Copyright © 2005 John Wiley & Sons, Ltd.     

Korn inequalities for shells with zero Gaussian curvature

We consider shells with zero Gaussian curvature, namely shells with one principal curvature zero and the other one having a constant sign. Our particular interests are shells that are diffeomorphic to a circular cylindrical shell with zero principal longitudinal curvature and positive circumferential curvature, including, for example, cylindrical and conical shells with arbitrary convex cross sections. We prove that the best constant in the first Korn inequality scales like thickness to the power 3/2 for a wide range of boundary conditions at the thin edges of the shell. Our methodology is to prove, for each of the three mutually orthogonal two-dimensional cross-sections of the shell, a “first-and-a-half Korn inequality”—a hybrid between the classical first and second Korn inequalities. These three two-dimensional inequalities assemble into a three-dimensional one, which, in turn, implies the asymptotically sharp first Korn inequality for the shell. This work is a part of mathematically rigorous analysis of extreme sensitivity of the buckling load of axially compressed cylindrical shells to shape imperfections.     

Approximation of tricomi problem with Neumann boundary condition

Summary The Tricomi problem with Neumann boundary condition is reduced to a degenerate problem in the elliptic region with a non-local boundary condition and to a Cauchy problem in the hyperbolic region. A variational formulation is given to the elliptic problem and a finite element approximation is studied. Also some regularity results in weighted Sobolev spaces are discussed.     

Regularity and a priori error analysis of a Ventcel problem in polyhedral domains

We consider the regularity of a mixed boundary value problem for the Laplace operator on a polyhedral domain, where Ventcel boundary conditions are imposed on one face of the polyhedron and Dirichlet boundary conditions are imposed on the complement of that face in the boundary. We establish improved regularity estimates for the trace of the variational solution on the Ventcel face and use them to derive a decomposition of the solution into a regular and a singular part that belongs to suitable weighted Sobolev spaces. This decomposition, in turn, via interpolation estimates both in the interior as well as on the Ventcel face, allows us to perform an a priori error analysis for the finite element approximation of the solution on anisotropic graded meshes. Numerical tests support the theoretical analysis. Copyright © 2016 John Wiley & Sons, Ltd.     

Reproducing kernel approximation in weighted Bergman spaces: Algorithm and applications

In this paper, we present algorithms of preorthogonal adaptive Fourier decomposition (POAFD) in weighted Bergman spaces. POAFD, as has been studied, gives rise to sparse approximations as linear combinations of the corresponding reproducing kernels. It is found that POAFD is unavailable in some weighted Hardy spaces that do not enjoy the boundary vanishing condition; as a result, the maximal selections of the parameters are not guaranteed. We overcome this difficulty with two strategies. One is to utilize the shift operator while the other is to perform weak POAFD. In the cases when the reproducing kernels are rational functions, POAFD provides rational approximations. This approximation method may be used to 1D signal processing. It is, in particular, effective to some Hardy H^p space functions for p not being equal to 2. Weighted Bergman spaces approximation may be used in system identification of discrete time‐varying systems. The promising effectiveness of the POAFD method in weighted Bergman spaces is confirmed by a set of experiments. A sequence of functions as models of the weighted Hardy spaces, being a wider class than the weighted Bergman spaces, are given, of which some are used to illustrate the algorithm and to evaluate its effectiveness over other Fourier type methods.     

Dynamical two-dimensional models of solid-fluid interaction

The present paper is devoted to the construction and investigation of two-dimensional hierarchical models for solid-fluid interaction. Applying the variational approach, the three-dimensional initial-boundary value problem is reduced to a sequence of two-dimensional problems and the existence and uniqueness of their solutions in suitable functional spaces is proved. The convergence of the sequence of vector-functions of three space variables to the solution of the original problem is proved and under additional regularity conditions the rate of approximation is estimated. Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 51, Differential Equations and Their Applications, 2008.     

Existence and Multiplicity of solutions for a quasilinear elliptic system on unbounded domains involving nonlinear boundary conditions

We prove two existence results for the nonlinear elliptic boundary value system involving $p$-Laplacian over an unbounded domain in $R^N$ with noncompact boundary. The proofs are based on variational methods applied to weighted spaces.     

Shape derivatives in differential forms I: an intrinsic perspective

We treat Zolésio’s velocity method of shape calculus using the formalism of differential forms, in particular, the notion of Lie derivative. This provides a unified and elegant approach to computing even higher-order shape derivatives of domain and boundary integrals and avoids the tedious manipulations entailed by classical vector calculus. Hitherto unknown expressions for shape Hessians can be derived with little effort. The perspective of differential forms perfectly fits second-order boundary value problems (BVPs). We illustrate its power by deriving the shape derivatives of solutions to second-order elliptic BVPs with Dirichlet, Neumann and Robin boundary conditions. A new dual mixed variational approach is employed in the case of Dirichlet boundary conditions.     

加权Lipschitz空间上的Littlewood-Paley算子

本文研究了加权Lipschitz空间上的Littlewood-Paley算子.,证明了一个加权Lipschitz 函数在Littlewood-Paley算子下的象或者几乎处处等于无穷或者仍是一个加权Lipschitz函数.     

On a vibration problem of antiplane strain (shear) of orthotropic non-homogeneous prismatic shell-like bodies

Vibration problem of an antiplane strain (shear) of orthotropic non-homogeneous prismatic shell-like bodies is considered when the shear moduli depending on the body projection (i.e. on a domain lying in the plane of interest) variables may vanish on a part of the boundary of the projection.     

Jacobi Polynomials,Weighted Sobolev Spaces and Approximation Results of Some Singularities

In this paper, we give some polynomial approximation results in a class of weighted Sobolev spaces, which are related to the Jacobi operator. We further give some embeddings of those weighted Sobolev spaces into usual ones and into spaces of continuous functions, in order to use the above approximation results in the p‐version (or the spectral method) of some finite or boundary element methods. Finally, two typical examples of the polynomial approximation of some singularities of boundary value problems in polygonal or polyhedral domains are presented.     

Strengthened Cauchy-Schwarz inequality for biorthogonal wavelets in Sobolev spaces

We prove a strengthened Cauchy-Schwarz inequality for one-dimensional biorthogonal wavelets. The functional frame is given by a class of Hilbert spaces, defined in terms of weighted Fourier transforms, which contain as relevant examples the standard Sobolev spaces H(s) as well as their homogeneous version. Intended applications concern multilevel and hierarchical methods for numerical approximation of partial differential equations.