The Nonclassical Boundary-contact Problems of Elasticity for Homogeneous Anisotropic Media

The existence and uniqueness of solutions of the nonclassical boundary-contact problems (i.e., problems with a contact on some part of the boundaries) of elasticity for homogeneous anisotropic media are investigated in Besov and Bessel potential spaces using methods of potential theory and the theory of pseudodifferential equations on manifolds with boundary. The smoothness of the solutions obtained is studied.     

Some Boundary — Contact Problems of the Elasticity Theory with Mixed Boundary Conditions Outside the Contact Surface

n — Dimensional (n ≥ 2) boundary-contact problems of statics of the elasticity theory for homogeneous anisotropic media are investigated when the contact of two bounded domains occurs from the outside on some part of boundaries with mixed boundary conditions. Theorems on the existence and uniqueness of solutions of boundary-contact problems in Besov and Bessel potential spaces are obtained. The smoothness of solutions is studied in closed domains occupied by elastic media.     

Strongly elliptic second-order systems with boundary conditions on a nonclosed Lipschitz surface

We consider boundary value problems and transmission problems for strongly elliptic second-order systems with boundary conditions on a compact nonclosed Lipschitz surface S with Lipschitz boundary. The main goal is to find conditions for the unique solvability of these problems in the spaces H ^s , the simplest L ₂-spaces of the Sobolev type, with the use of potential type operators on S. We also discuss, first, the regularity of solutions in somewhat more general Bessel potential spaces and Besov spaces and, second, the spectral properties of problems with spectral parameter in the transmission conditions on S, including the asymptotics of the eigenvalues.     

Dirichlet-Neumann-impedance boundary value problems arising in rectangular wedge diffraction problems

Boundary value problems originated by the diffraction of an electromagnetic (or acoustic) wave by a rectangular wedge with faces of possible different kinds are analyzed in a Sobolev space framework. The boundary value problems satisfy the Helmholtz equation in the interior (Lipschitz) wedge domain, and are also subject to different combinations of boundary conditions on the faces of the wedge. Namely, the following types of boundary conditions will be under study: Dirichlet-Dirichlet, Neumann-Neumann, Neumann-Dirichlet, Impedance-Dirichlet, and Impedance-Neumann. Potential theory (combined with an appropriate use of extension operators) leads to the reduction of the boundary value problems to integral equations of Fredholm type. Thus, the consideration of single and double layer potentials together with certain reflection operators originate pseudo-differential operators which allow the proof of existence and uniqueness results for the boundary value problems initially posed. Furthermore, explicit solutions are given for all the problems under consideration, and regularity results are obtained for these solutions.

     

An operator approach for an oblique derivative boundary‐transmission problem

We consider a boundary‐transmission problem for the Helmholtz equation, in a Bessel potential space setting, which arises within the context of wave diffraction theory. The boundary under consideration consists of a strip, and certain conditions are assumed on it in the form of oblique derivatives. Operator theoretical methods are used to deal with the problem and, as a consequence, several convolution type operators are constructed and associated to the problem. At the end, the well‐posedness of the problem is shown for a range of non‐critical regularity orders of the Bessel potential spaces, which include the finite energy norm space. In addition, an operator normalization method is applied to the critical orders case. Copyright © 2004 John Wiley & Sons, Ltd.     

Mixed Boundary Value Problems for the Helmholtz Equation in a Quadrant

The main objective is the study of a class of boundary value problems in weak formulation where two boundary conditions are given on the half-lines bordering the first quadrant that contain impedance data and oblique derivatives. The associated operators are reduced by matricial coupling relations to certain boundary pseudodifferential operators which can be analyzed in detail. Results are: Fredholm criteria, explicit construction of generalized inverses in Bessel potential spaces, eventually after normalization, and regularity results. Particular interest is devoted to the impedance problem and to the oblique derivative problem, as well.     

Wave diffraction by a strip with first and second kind boundary conditions: the real wave number case

We prove unique existence of solution for a class of plane wave diffraction problems by a strip with first and second kind boundary conditions. This is done in a Bessel potential spaces framework, and for a real (noncomplex) wave number. At the end, results about the regularity (and data dependence) of the solution are exhibited upon the initial setting and the boundary parameters. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Index transforms with Weber-type kernels

New index transforms with Weber-type kernels, consisting of products of Bessel functions of the first and second kind, are investigated. Mapping properties and inversion formulas are established for these transforms in Lebesgue spaces. The results are applied to solve a boundary value problem on the wedge for a fourth-order partial differential equation.     

Finite Interval Convolution Operators with Transmission Property

We study convolution operators in Bessel potential spaces and (fractional) Sobolev spaces over a finite interval. The main purpose of the investigation is to find conditions on the convolution kernel or on a Fourier symbol of these operators under which the solutions inherit higher regularity from the data. We provide conditions which ensure the transmission property for the finite interval convolution operators between Bessel potential spaces and Sobolev spaces. These conditions lead to smoothness preserving properties of operators defined in the above-mentioned spaces where the kernel, cokernel and, therefore, indices do not depend on the order of differentiability. In the case of invertibility of the finite interval convolution operator, a representation of its inverse is presented in terms of the canonical factorization of a related Fourier symbol matrix function.     

Special vekua equations with singular coefficients

Special equations of Vekua-type with singular coefficients are considered. As a first step we study the influence of the coefficients of model equations on the choice of the function spaces for its solutions and on the boundary conditions. As an application we sketch the consideration of boundary value problems for Vekua equations with variable coefficients having a strong singularity at z =0     

The Dirichlet,Neumann and Mixed Boundary Value Problems of the Theory of Elasticity in n-Dimensional Domains with Boundaries Containing Closed Cuspidal Edges

The Dirichlet, Neumann and mixed problems of statics of the theory of elasticity for anisotropic homogeneous media are studied in n-dimensional (n ≧ 2) domains with boundaries containing closed cuspidal edges. Theorems of the existence and uniqueness of solutions of these problems in the Besov and Bessel potential spaces are obtained. Smoothness of solutions in a closed domain occupied by elastic medium is investigated.     

A globally convergent semi-smooth Newton method for control-state constrained DAE optimal control problems

We investigate a semi-smooth Newton method for the numerical solution of optimal control problems subject to differential-algebraic equations (DAEs) and mixed control-state constraints. The necessary conditions are stated in terms of a local minimum principle. By use of the Fischer-Burmeister function the local minimum principle is transformed into an equivalent nonlinear and semi-smooth equation in appropriate Banach spaces. This nonlinear and semi-smooth equation is solved by a semi-smooth Newton method. We extend known local and global convergence results for ODE optimal control problems to the DAE optimal control problems under consideration. Special emphasis is laid on the calculation of Newton steps which are given by a linear DAE boundary value problem. Regularity conditions which ensure the existence of solutions are provided. A regularization strategy for inconsistent boundary value problems is suggested. Numerical illustrations for the optimal control of a pendulum and for the optimal control of discretized Navier-Stokes equations conclude the article.     

非局部初值问题解的存在性定理

近来非局部问题的研究日见增多,但涉及带非线性边界条件的初值问题文献 较少.本文目的在于证明一个半线性方程的齐次边值问题和一个非线性边界条件问题 解的存在性.主要使用半群,分数次幂函数空间,广义Poincare算子等工具.     

Three-dimensional contact problems for an elastic wedge with a coating

Three-dimensional contact problems for an elastic wedge, one face of which is reinforced with a Winkler-type coating with different boundary conditions on the other face of the wedge, are investigated. A power-law dependence of the normal displacement of the coating on the pressure is assumed. The contact area, the pressure in this region, and the relation between the force and the indentation of a punch are determined using the method of non-linear boundary integral equations and the method of successive approximations. The results of calculations are analysed for different values of the aperture angle of the wedge, the relative distance of the punch from the edge of the wedge, the ratio of the radii of curvature of the punch (an elliptic paraboloid), and the non-linearity factors of the coating. The results obtained are compared with the solutions of similar problems for a wedge without a coating.     

On wave diffraction by a half‐plane with different face impedances

The impedance wave diffraction problem by a half‐plane screen is revisited in view of its well‐posedness upon different impedance and wave parameters. The problem is analysed with the help of potential and pseudo‐differential operators. Seven conditions between the impedance and wave numbers are found under which the problem will be well‐posed in Bessel potential spaces. In addition, an improvement of the regularity of the solutions is shown for the previous seven conditions. Copyright © 2006 John Wiley & Sons, Ltd.     

On Relations Between Bessel Potential Spaces and Riesz Potential Spaces

We present a relation between the Bessel potential spaces and the Riesz potential spaces. The ideas of the proof are to characterize each potential spaces and to give a correspondence between individual Bessel potentials and Riesz potentials.     

Edge quantisation of elliptic operators

The ellipticity of operators on a manifold with edge is defined as the bijectivity of the components of a principal symbolic hierarchy s = (s_y,s_ù){\sigma=(\sigma_\psi,\sigma_\wedge)} , where the second component takes values in operators on the infinite model cone of the local wedges. In the general understanding of edge problems there are two basic aspects: Quantisation of edge-degenerate operators in weighted Sobolev spaces, and verifying the ellipticity of the principal edge symbol s_ù{\sigma_\wedge} which includes the (in general not explicitly known) number of additional conditions of trace and potential type on the edge. We focus here on these questions and give explicit answers for a wide class of elliptic operators that are connected with the ellipticity of edge boundary value problems and reductions to the boundary. In particular, we study the edge quantisation and ellipticity for Dirichlet–Neumann operators with respect to interfaces of some codimension on a boundary. We show analogues of the Agranovich–Dynin formula for edge boundary value problems.     

Mixed problems in a Lipschitz domain for strongly elliptic second-order systems

We consider mixed problems for strongly elliptic second-order systems in a bounded domain with Lipschitz boundary in the space ℝ ⁿ . For such problems, equivalent equations on the boundary in the simplest L ₂-spaces H ^s of Sobolev type are derived, which permits one to represent the solutions via surface potentials. We prove a result on the regularity of solutions in the slightly more general spaces H _p ^s of Bessel potentials and Besov spaces B _p ^s . Problems with spectral parameter in the system or in the condition on a part of the boundary are considered, and the spectral properties of the corresponding operators, including the eigenvalue asymptotics, are discussed.     

Mixed boundary value problems and parametrices in the edge calculus

Mixed elliptic boundary value problems are characterised by conditions which have a jump along an interface of codimension 1 on the boundary. We study such problems in weighted edge spaces and show the Fredholm property and the existence of parametrices under additional conditions of trace and potential type on the interface. We develop a new method for computing the interface conditions in terms of the index of boundary value problems in weighted spaces on infinite cones, combined with structures from the calculus of boundary value problems on a manifold with edges. This will be illustrated by the Zaremba problem and other mixed problems for the Laplace operator. The approach itself is completely general.     

The coupling of bem and fem for the two-dimensional viscous flow problem

In this paper we propose a numerical scheme for treating the problem of sJow viscous flow past an obstacle in the plane. This scheme is a combination of boundary element and finite element methods. By introducing an auxiliary boundary curve, we divide the region under consideration into two subregions, an inner and an outer region. In the inner region, we employ a finite element method (FEM) for solving a system of simplified field equations with proper natural boundary conditions. In the outer region, the solution is expressed in the form of a simple-layer potential with density function satisfying a system of modified integral equations of the first kind. The latter are solved by a boundary element method (BEM). Both solutions are matched on the common auxiliary boundary curve. Error estimates in suitable function spaces are derived in terms of the mesh widths as well as the small parameters, the Reynolds numbers