Singular Integral Operators with Coefficients of a Special Structure Related to Operator Equalities

In our previous works we have constructed operator equalities which transform scalar singular integral operators with shift to matrix characteristic singular integral operators without shift and found some of their applications to problems with shift. In this article the operator equalities are used for the study of matrix characteristic singular integral operators. Conditions for the invertibility of the singular integral operators with orientation preserving shift and coefficients with a special structure generated by piecewise constant functions, t, t ⁻¹, were found. Conditions for the invertibility of the matrix characteristic singular integral operators with four-valued piecewise constant coefficients of a special structure were likewise obtained. Submitted: June 15, 2007. Revised: October 25, 2007. Accepted: November 5, 2007.     

On algebras of singular integral operators with shift in Hölder weighted spaces

We consider algebras of singular integral operators with shift and piecewise Hölder coefficients in a Hölder weighted space on a Lyapunov contour. For this algebra, we construct the similarity isomorphism to the algebra of singular integral operators with piecewise Hölder coefficients in a Hölder space with “canonical” weight on the circle. We construct the symbol calculus, formulate necessary and sufficient conditions for the Fredholm property, and give the formula for the index of Fredholm operators.Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 9, Suzdal Conference-3, 2003.     

Algebra of singular integral operators with a Carleman backward slowly oscillating shift

In this paper we study the Banach algebra of singular integral operators with piecewise continuous coefficients and a Carleman orientation-reversing slowly oscillating shift on the Lebesgue space with a power weight on the unit circle. The slow oscillation of the shift derivative, in contrast to the classic assumption on its piecewise continuity, leads to the appearance of massive local spectra for the considered operators. Applying localization techniques and the theory of Mellin pseudodifferential and associated limit operators, we construct a symbol calculus for the above-mentioned operator algebra and find a Fredholm criterion and an index formula for the operators in this algebra in terms of their symbols.Partially supported by CONaCYT grant, Cátedra Patrimonial, No. 990017-EX and by CONACYT project 32726-E, México.Partially supported by F. C. T. grant Praxis XXI/2/2.1/MAT/441/94, Portugal.     

First-order differential-operator equation with variable domains of piecewise smooth operators

We prove the existence, uniqueness, and smoothness of weak solutions of a first-order differential-operator equation with variable domains of nonself-adjoint piecewise smooth operators for which one has the corresponding majorant operators. We analyze the well-posedness and smoothness of weak solutions of three new mixed problems with piecewise smooth (in time) coefficients in the equations of finite and infinite order and in the boundary conditions.     

On the well-posedness of the cauchy problem and the mixed problem for some class of hyperbolic systems and equations with constant coefficients and characteristics of variable multiplicity

We study the well-posedness of the mixed problem for hyperbolic equations with constant coefficients and with characteristics of variable multiplicity. We single out a class of higher-order hyperbolic operators with constant coefficients and with characteristics of variable multiplicity, for which we obtain a generalization of the Sakamoto conditions for the well-posedness of the mixed problem in L ₂.     

Problème de Cauchy pour certaines équations du type de Schrödinger

We consider a 2-evolution operator in the sense of Petrowsky, and we assume that the characteristic roots of the principal polynomial with constant coefficients are real and of constant multiplicities. Then we give sufficient conditions so that the Cauchy problem both for the future and for the past is well-posed in the Sobolev spaces. Our conditions are analogous to the Levi conditions and to the decomposition conditions of operators in the hyperbolic case for Kowalewskian operators.     

Spectral properties of k p schrödinger operators in one space dimension

In the physics of layered semiconductor devices the k · p method in combination with the envelope function approach is a well established tool for band structure calculations. We perform a rigorous mathematical analysis of spectral properties for the corresponding spatially one dimensional k · p Schrödinger operators;thereby encompassing a wide class of such operators. This class covers many of the k · p operators prevalent in solid state physics. It includes k · p Schrödinger operators with piecewise constant coefficients, a prerequisite for dealing with the important case of semiconductor hetero-structures. In particular, we address the question of persistence of a spectral gap over the wave vector range. We also introduce a regularization of the problem which gives rise to a consistent discretization of k · p operators with jumping coefficients and describe design patterns for the numerical treatment of k · p operators.     

Mixed multidimensional integral operators with piecewise constant kernels and their representations

We consider the algebra of mixed multidimensional integral operators. In particular, Fredholm integral operators belong to this algebra. For the piecewise constant kernels, we provide an explicit representation of the algebra as a direct product of simple matrix algebras. This representation allows us to compute the inverse operators and to find the spectrum explicitly. Moreover, explicit traces and determinants of such operators are also constructed. Generally speaking, the analysis of integral operators is reduced to the analysis of matrices.     

Boundary observation and exact control of multilayered piezoelectric body

This paper considers the transmission problem for the system of piezoelectricity having piecewise constant coefficients. Under suitable geometric conditions imposed on the domain and the interfaces where the coefficients have a jump discontinuity, results on boundary observation and exact controllability are established. Copyright © 2003 John Wiley & Sons, Ltd.     

An Approximation Theory for Operators Generated by Shifts

This paper develops a theory about the applicability of approximation methods to operators that can be represented as a function of a shift. An axiomatic approach is used, in which a small set of conditions that involve the operator and the method are proved to guarantee the applicability. All concrete methods we know for the underlying operators are subjected to this approach. We also study the behavior of the singular values and related questions. New results concerning Galerkin approximation methods for Mellin operators with piecewise continuous symbol are given to illustrate the application of the theory.     

On the boundary value problem for the Sturm–Liouville equation with the discontinuous coefficient

We consider a boundary value problem for the Sturm–Liouville equation with piecewise‐constant leading coefficient. We prove that some integral representations for the solutions of the considered equation can be obtained by using classical transformation operators for the Sturm–Liouville operator at the end points of a finite interval. We also investigate the spectral characteristics of the boundary value problem, prove the completeness and expansion theorem. Copyright © 2012 John Wiley & Sons, Ltd.     

The overdetermined Cauchy problem in some classes of ultradifferentiable functions

In this paper we study the Cauchy problem for overdetermined systems of linear partial differential operators with constant coefficients in some spaces of ultradifferentiable functions of class (M _p ). We show that evolution is equivalent to the validity of a Phragmén-Lindel?f principle for entire and plurisubharmonic functions on some irreducible affine algebraic varieties, and make applications in different situations. We find necessary and sufficient conditions for well posedness, and relate the hyperbolicity of a given system to that of its principal part. Received: January 19, 1999?Published online: May 10, 2001     

Exact boundary controllability in problems of transmission for the system of electromagneto‐elasticity

This paper considers transmission problem for the system of electromagneto‐elasticity having piecewise constant coefficients in a bounded domain. The result on exact boundary controllability is obtained provided the interfaces, where the coefficients have a jump discontinuity, are all star‐shaped with respect to one and the same point and the coefficients satisfy a certain monotonicity conditions. Copyright © 2001 John Wiley & Sons, Ltd.     

On ϵ-Collocation for Pseudodifferential Equations on a Closed Curve

This paper is devoted to the approximate solution of one-dimensional pseudodifferential equations on a closed curve via spline collocation methods with variable collocation points and represents a continuation of [11]. We give necessary and sufficient conditions ensuring the L²-convergence for operators with smooth and piecewise continuous coefficients.     

On the stability of solutions of certain systems of differential equations with piecewise constant argument

We obtain some sufficient conditions for the existence of the solutions and the asymptotic behavior of both linear and nonlinear system of differential equations with continuous coefficients and piecewise constant argument.     

On strongly elliptic singular integral operators with piecewise continuous coefficients

The concept of strongly elliptic operators is one of the main tools for approximating the solution of boundary integral equations by finite element methods (see [4-10]). In the present paper necessary and sufficient conditions for the strong ellipticity of singular integral operators with piecewise continuous matrix coefficients on a closed or open Ljapunov curve are obtained.     

Algebras of Singular Integral Operators with Piecewise Continuous Coefficients on Reflexive Orlicz Spaces

We consider singular integral operators with piecewise continuous coefficients on reflexive Orlicz spaces L_m(σ) which are generalizations of the Lebesgue spaces L_P(σ), 1 < p < ∞. We suppose that σ belongs to a large class of Carleson curves, including curves with corners and cusps as well as curves that look locally like two logarithmic spirals scrolling up at the same point. For the singular integral operator associated with the Riemann boundary value problem with a piecewise continuous coefficient G, we establish a Fredholm criterion and an index formula in terms of the essential range of G complemented by spiralic horns depending on the Boyd indices of L_M(σ) and contour properties. Our main result is a symbol calculus for the closed algebra of singular integral operators with piecewise continuous matrix - valued coefficients on L_Mⁿ(σ).     

The solvability of the initial problem for a degenerate linear hybrid system with variable coefficients

We consider a linear hybrid system with variable coefficients and known mode switching moments under the assumption that matrices at the derivative of the desired vector function are identically degenerate. We obtain the necessary and sufficient conditions for the existence of a piecewise smooth solution (either continuous or not in its definition domain) for the initial problem. We study an equivalent form of a nonstationary system of linear ordinary differential equations that is not resolved with respect to the derivative and is identically degenerate in its definition domain. We propose a constructive algorithm for obtaining such a form even if the rank of the matrix at the derivative is not constant.     

Approximation by means of piecewise linear functions

In the present paper we use piecewise linear functions in order to obtain representations and estimates for the remainder in approximating continuous functions by positive linear operators. Applications of these results for Bernstein and Stancu’s operators are also presented. In addition, we give some partial results concerning the best constant problem for Bernstein operators with respect to the second order modulus of continuity.     

Boundary value problem for a second-order elliptic equation in the exterior of an ellipse

We consider a boundary value problem for a second-order linear elliptic differential equation with constant coefficients in a domain that is the exterior of an ellipse. The boundary conditions of the problem contain the values of the function itself and its normal derivative. We give a constructive solution of the problem and find the number of solvability conditions for the inhomogeneous problem as well as the number of linearly independent solutions of the homogeneous problem. We prove the boundary uniqueness theorem for the solutions of this equation.