A family of fundamental solutions of elliptic partial differential operators with quaternion constant coefficients

The purpose of this paper is to construct a family of fundamental solutions for elliptic partial differential operators with quaternion constant coefficients. The elements of such family are expressed by means of functions, which depend jointly real analytically on the coefficients of the operators and on the spatial variable. We show some regularity properties in the frame of Schauder spaces for the corresponding single layer potentials. Ultimately, we exploit our construction by showing a real analyticity result for perturbations of the layer potentials corresponding to complex elliptic partial differential operators of order two. Copyright © 2012 John Wiley & Sons, Ltd.     

On Domination Conditions for Systems of Minimal Differential Operators Acting in the Space L ∞ $$\mathbb{R}^n $$

In this paper, we consider the linear space of minimal differential operators with constant coefficients which are dominated by systems of minimal operators acting in spaces with uniform norm. In terms of domination conditions, we prove the quasiellipticity test for a given operator; this criterion generalizes a similar result of De Leeuw and Mirkil to elliptic operators.     

Hypercyclic Sequences of Differential and Antidifferential Operators

In this paper, we provide some extensions of earlier results about hypercyclicity of some operators on the Fréchet space of entire functions of several complex variables. Specifically, we generalize in several directions a theorem about hyper- cyclicity of certain infinite order linear differential operators with constant coefficients and study the corresponding property for certain kinds of “antidifferential” operators which are introduced in the paper. In addition, the existence of hypercyclic functions for certain sequences of differential operators with additional properties, for instance, boundedness or with some nonvanishing derivatives, is established.     

Difference schemes on uniform grids performed by general discrete operators

Our aim is to set the foundations of a discrete vectorial calculus on uniform n-dimensional grids, that can be easily reformulated on general irregular grids. As a key tool we first introduce the notion of tangent space to any grid node. Then we define the concepts of vector field, field of matrices and inner products on the space of grid functions and on the space of vector fields, mimicking the continuous setting. This allows us to obtain the discrete analogous of the basic first order differential operators, gradient and divergence, whose composition define the fundamental second order difference operator. As an application, we show that all difference schemes, with constant coefficients, for first and second order differential operators with constant coefficients can be seen as difference operators of the form for suitable choices of q, and  . In addition, we characterize special properties of the difference scheme, such as consistency, symmetry and positivity in terms of q, and  .     

Differentialoperatoren bei partiellen Differentialgleichungen

Summary In the present paper those formally hyperbolic differential equations are characterized for which solutions can be represented by means of differential operators acting on holomorphic functions. This is done by a necessary and sufficient condition on the coefficients of the differential equation. These operators are determined simultaneously. By it a general procedure is presented to construct differential equations and corresponding differential operators which map holomorphic functions onto solutions of the differential equations. We also discuss the question under which circumstances all the solutions of a differential equation can be represented by differential operators. For the equations characterized previously we determine the Riemann function. Some special classes of differential equations are investigated in detail. Furthermore the possibility of a representation of pseudoanalytic functions and the corresponding Vekua resolvents by differential operators is discussed.

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Herrn Prof. Dr. K. W. Bauer zum 60. Geburtstag gewidmet     

Completeness of eigenfunctions of one class of pencils of differential operators with constant coefficients

In spaces of functions square summable on finite intervals we study the simple completeness of the system of eigen- and associated functions for the pencil of ordinary differential operators generated by a differential expression with constant coefficients (we assume that the roots of the corresponding characteristic equation lie on one and the same ray) and specific semisplitting homogeneous boundary value conditions.     

Necessary Covariance Conditions for a One-Field Lax Pair

We study the covariance with respect to Darboux transformations of polynomial differential and difference operators with coefficients given by functions of one basic field. In the scalar (Abelian) case, the functional dependence is established by equating the Frechet differential (the first term of the Taylor series on the prolonged space) to the Darboux transform; a Lax pair for the Boussinesq equation is considered. For a pair of generalized Zakharov-Shabat problems (with differential and shift operators) with operator coefficients, we construct a set of integrable nonlinear equations together with explicit dressing formulas. Non-Abelian special functions are fixed as the fields of the covariant pairs. We introduce a difference Lax pair, a combined gauge-Darboux transformation, and solutions of the Nahm equations.__________Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 144, No. 1, pp. 122–132, July, 2005.     

A Survey on Explicit Representation Formulae for Fundamental Solutions of Linear Partial Differential Operators

In the present article, we aim at treating the existence of fundamental solutions of linear partial differential operators with constant coefficients from the viewpoint of setting up explicit formulae yielding fundamental solutions.     

Une généralisation du problème des itérés

For given systems of differential operators of constant coefficients we give algebraic conditions to compare their spaces of Gevrey vectors.     

Spectral properties of a fourth-order differential operator with integrable coefficients

The aim of this paper is to study spectral properties of differential operators with integrable coefficients and a constant weight function. We analyze the asymptotic behavior of solutions to a differential equation with integrable coefficients for large values of the spectral parameter. To find the asymptotic behavior of solutions, we reduce the differential equation to a Volterra integral equation. We also obtain asymptotic formulas for the eigenvalues of some boundary value problems related to the differential operator under consideration.     

Estimates for restrictions of monotone operators on the cone of decreasing functions in Orlicz space

We introduce a number of notions related to the Lyapunov transformation of linear differential operators with unbounded operator coefficients generated by a family of evolution operators. We prove statements about similar operators related to the Lyapunov transformation and describe their spectral properties. One of the main results of the paper is a similarity theorem for a perturbed differential operator with constant operator coefficient, an operator which is the generator of a bounded group of operators. For the perturbation, we consider the operator ofmultiplication by a summable operator function. The almost periodicity (at infinity) of the solutions of the corresponding homogeneous differential equation is established.     

An application of the Fourier method of separation of variables to constructing exactly solvable deformations of partial differential operators

As is known, in mathematical physics there are differential operators with constant coefficients whose fundamental solutions can be constructed explicitly; such operators are said to be exactly solvable. In this paper, the problem of adding lower-order terms with variable coefficients to exactly solvable operators in such a way that the new operators (deformations) admit constructing fundamental solutions in explicit form is posed. This problem is directly related to Hadamard’s problem of describing differential operators satisfying the Huygens’ principle. On the basis of the Fourier method of separation of variables and the method of gauge-equivalent operators, an effective method for finding exactly solvable deformations depending on one variable is constructed. An application of such deformations to constructing Huygens’ differential operators associated with the cone of real symmetric positive-definite matrices is suggested. __________ Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 38, Suzdal Conference-2004, Part 3, 2006.     

Differential operators on Hermitian Jacobi forms

We study multilinear differential operators on a space of Hermitian Jacobi forms as well as on a space of Hermitian modular forms of degree 2. First we define a heat operator and construct multilinear differential operators on a space of Hermitian Jacobi forms of degree 2. As a special case of these operators, we also study Rankin-Cohen type differential operators on a space of Hermitian Jacobi forms. And we construct multilinear differential operators on a space of Hermitian modular forms of degree 2 as an application of multilinear differential operators on Hermitian Jacobi forms.     

A non-existence theorem of lacunas for hyperbolic differential operators with constant coefficients

We prove a theorem on non-existence of lacunas of the fundamental solution for hyperbolic differential operators with constant coefficients.     

On a class of coupled Hamiltonian operators and their integrable hierarchies with two potentials

We discuss at first in this paper the Gauge equivalence among several u‐linear Hamiltonian operators and present explicitly the associated Gauge transformation of Bäcklund type among them. We then establish the sufficient and necessary conditions for the linear superposition of the discussed u‐linear operators and matrix differential operators with constant coefficients of arbitrary order to be Hamiltonian, which interestingly shows that the resulting Hamiltonian operators survive only up to the third differential order. Finally, we explore a few illustrative examples of integrable hierarchies from Hamiltonian pairs embedded in the resulting Hamiltonian operators.     

The problem of lacunas and analysis on root systems

A lacuna of a linear hyperbolic differential operator is a domain inside its propagation cone where a proper fundamental solution vanishes identically. Huygens' principle for the classical wave equation is the simplest important example of such a phenomenon. The study of lacunas for hyperbolic equations of arbitrary order was initiated by I. G. Petrovsky (1945). Extending and clarifying his results, Atiyah, Bott and Gårding (1970-73) developed a profound and complete theory for hyperbolic operators with constant coefficients. In contrast, much less is known about lacunas for operators with variable coefficients. In the present paper we study this problem for one remarkable class of partial differential operators with singular coefficients. These operators stem from the theory of special functions in several variables related to finite root systems (Coxeter groups). The underlying algebraic structure makes it possible to extend many results of the Atiyah-Bott-Gårding theory. We give a generalization of the classical Herglotz-Petrovsky-Leray formulas expressing the fundamental solution in terms of Abelian integrals over properly constructed cycles in complex projective space. Such a representation allows us to employ the Petrovsky topological condition for testing regular (strong) lacunas for the operators under consideration. Some illustrative examples are constructed. A relation between the theory of lacunas and the problem of classification of commutative rings of partial differential operators is discussed.

     

How to solve the polynomial ordinary differential equations

This paper discussed how to solve the polynomial ordinary differential equations. At first, we construct the theory of the linear equations about the unknown one variable functions with constant coefficients. Secondly, we use this theory to convert the polynomial ordinary differential equations into the simultaneous first order linear ordinary differential equations with constant coefficients and quadratic equations. Thirdly, we work out the general solution of the polynomial ordinary differential equations which is no longer concerned with the differential. Finally, we discuss the necessary and sufficient condition of the existence of the solution.     

Approximation by solutions of elliptic equations in semilocal spaces

In the present work, we investigate the approximability of solutions of elliptic partial differential equations in a bounded domain Ω by solutions of the same equations in a larger domain. We construct an abstract framework which allows us to deal with such density questions, simultaneously for various norms. More specifically, we study approximations with respect to the norms of semilocal Banach spaces of distributions. These spaces are required to satisfy certain postulates. We establish density results for elliptic operators with constant coefficients which unify and extend previous results. In our density results Ω may possess holes and it is required to satisfy the segment condition. We observe that analogous density results do not hold in spaces where the infinitely smooth functions are not dense. Finally, we provide applications related to the method of fundamental solutions.     

Commuting Differential Operators of Rank 2 with Rational Coefficients

In this paper we find new pairs of self-adjoint commuting differential operators of rank 2 with rational coefficients and prove that any curve of genus 2 written as a hyperelliptic curve is the spectral curve of a pair of commuting differential operators with rational coefficients. We also study the case where curves of genus 3 are the spectral curves of pairs of commuting differential operators of rank 2 with rational coefficients.