Operators Of Boundary Value Problems Research Articles

Boundary Problems for Three-Dimensional Dirac Operators and Generalized MIT Bag Models for Unbounded Domains

We consider operators of boundary value problems for 3D- Dirac operators in unbounded domains with the uniformly regular boundary. We give effective conditions of self-adjointness of operators under consideration and a description of their essential spectra. We also give applications to operators of the MIT bag problems for unbounded domains

The nonlocal problem for the differential-operator equation of the even order with the involution

In this paper, the problem with boundary nonself-adjoint conditions for a differential-operator equations of the order $2n$ with involution is studied. Spectral properties of operator of the problem is investigated.
 By analogy of separation of variables the nonlocal problem for the differential-operator equation of the even order is reduced to a sequence $ \{L_{k}\}_{k=1}^{\infty}$ of operators of boundary value problems for ordinary differential equations of even order. It is established that each element $L_{k}$, of this sequence, is an isospectral perturbation of the self-adjoint operator $L_{0,k}$ of the boundary value problem for some linear differential equation of order 2n.
 We construct a commutative group of transformation operators whose elements reflect the system $V(L_{0,k})$ of the eigenfunctions of the operator $L_{0,k}$ in the system $V(L_{k})$ of the eigenfunctions of the operators $L_{k}$. The eigenfunctions of the operator $L$ of the boundary value problem for a differential equation with involution are obtained as the result of the action of some specially constructed operator on eigenfunctions of the sequence of operators $L_{0,k}.$
 The conditions under which the system of eigenfunctions of operator $L$ the studied problem is a Riesz basis is established.

Lp -theory of boundary integral operators for domains with unbounded smooth boundary

Abstract The paper is devoted to the L p ${L^{p}}$ -theory of boundary integral operators for boundary value problems described by anisotropic Helmholtz operators with variable coefficients in unbounded domains with unbounded smooth boundary. We prove the invertibility of boundary integral operators for Dirichlet and Neumann problems in the Bessel-potential spaces H s , p ⁢ ( ∂ ⁡ D ) ${H^{s,p}(\partial D)}$ , p ∈ ( 1 , ∞ ) ${p\in(1,\infty)}$ , and the Besov spaces B p , q s ⁢ ( ∂ ⁡ D ) ${B_{p,q}^{s}(\partial D)}$ , p , q ∈ [ 1 , ∞ ] ${p,q\in[1,\infty]}$ . We prove also the Fredholmness of the Robin problem in these spaces and give the index formula.

A posteriori estimates of inverse operators for boundary value problems in linear elliptic partial differential equations

This paper presents constructive a posteriori estimates of inverse operators for boundary value problems in linear elliptic partial differential equations (PDEs) on a bounded domain. This type of estimate plays an important role in the numerical verification of the solutions for boundary value problems in nonlinear elliptic PDEs. In general, it is not easy to obtain the a priori estimates of the operator norm for inverse elliptic operators. Even if we can obtain these estimates, they are often over estimated. Our proposed a posteriori estimates are based on finite-dimensional spectral norm estimates for the Galerkin approximation and expected to converge to the exact operator norm of inverse elliptic operators. This provides more accurate estimates, and more efficient verification results for the solutions of nonlinear problems.

One possible generalization of the concept of Sobolev space

A concept of co-sheaf of Sobolev spaces, which generalizes the concept of Sobolev space, is introduced. A generalization of the Rellich-Friedrichs theorem that covers operators of boundary value problems in unbounded domains, in particular, waveguides, is established.

Stability of the Inverse Operators of Boundary Value Problems in Smooth Expanding Domains

We study the problem of uniform boundedness of inverse operators of boundary value problems in expanding domains with conical points on the boundary. We give applications to the problem of approximation of solutions of boundary value problems in large domains by solutions of equations on R n . Moreover we establish a connection between boundary value problems with a parameter in bounded domains and boundary value problems in expanding domains what allows us to give necessary and sufficient conditions for the invertibility of boundary value problems for large values of the parameter in domains with conical points on the boundary.

AN ALGEBRA OF BOUNDARY VALUE PROBLEMS FOR A CLASS OF PSEUDODIFFERENTIAL OPERATORS OF VARIABLE ORDER

An algebra is constructed containing the operators of boundary value problems for pseudodifferential operators which are formally hypoelliptic in the sense of Hormander and elliptic at points of the boundary; parametrices for these boundary value problems are also constructed. The Fredholm property is demonstrated in a scale of weighted spaces, based on Lp, 1<p<∞, and the index is computed.

Difference analogues of orthogonal decompositions, basic differential operators and some boundary problems of mathematical physics. I

In recent times, in connection with the development of the theory of generalized functions and the theory of generalized solutions of boundary value problems for differential equations, the approach to the construction of difference operators, which approximate the differential problem, has essentially changed. First of all we have in mind the following: 1. (1) the rejection, because of results obtained, of the attempt to approximate (in more or less explicit form) a differential equation, with continuity conditions and boundary conditions, which must be done locally and independently of each other, by systems of finite difference equations; 2. (2) the development of numerical methods for a group of non-stationary problems, in which the form of the difference schemes is determined, not by the requirements of the best local approximation, but by conditions of stability and simplicity of calculation; 3. (3) the construction and basis of difference operators which possess the integral properties of differential operators of boundary value problems. We shall dwell only on the last group of methods. Difference equations for whose investigation integral characteristics were used, have been considered, for example, in works [1]–[13]. For the study of the properties of difference operators, which keep the integral properties of the differential problems, we can successfully use functional methods of investigation. It is well known with what success functional methods are applied to the study of problems of mathematical physics, in particular the method of orthogonal projections, which form a considerable part of the theory of generalized solutions (see [14]–[17]). A significant advance in the theory of difference equations has been marked by the fact that functional methods of directly investigating their inner properties (see [18]–[20]) have been applied to equations on a network. In the present work we put forward and systematically apply a method of orthogonal projections for network vector-functions. It permits us 1. (1)to investigate the structural properties of subspaces of a space of network functions, 2. (2) to compose difference equations for some problems of mathematical physics, 3. (3) to solve the question of difference analogues of boundary conditions by using the fact that the solution is a member of a definite subspace of functions, 4. (4) to investigate the properties of the solution of difference equations, 5. (5) to give optimal evaluations for the solutions of difference equations by means of projectional operators on the right-hand sides of these equations. The difference equations obtained for boundary value problems of types I, II and III, for self-conjugate elliptic equations of the second order, possess a regular closure of the computational algorithm; with symmetrization of these systems of equations they are converted into known systems of equations of a divergent type. We have called the difference equations obtained “difference analogues” of the problems considered, wishing thereby to stress the fact that in conclusions obtained from them we did not make use of the idea of local approximation of the differential operator by a difference operator, but obtained these equations by using functional (integral) methods. Local approximation was used only in replacing the first derivative of a function by a centrally divided difference. With the selected approach to the construction of difference operators we shall consider as a natural form of approximation an approximation, in the weak sense, which characterizes the nearness of two functionals, with which we may meet not only the case of local smallness of the discrepancy in the approximation of the differential operator by a difference, but also the case of its rapid oscillation. Both cases are equally well represented in the chosen method of evaluation of the errors of solutions. For clearness and simplicity of exposition we have limited ourselves to the case of three-dimensional space and have considered problems involving the Laplace operator.