M. Faierman

AN OSCILLATION THEOREM FOR A TWO-PARAMETER SYSTEM OF DIFFERENTIAL EQUATIONS

Abstract The oscillation theorem for two simultaneous Sturm-Liouville systems in two parameters is well known when the coefficients of the differential equations are subjected to the usual definiteness condition. However, in practical applications the usual definiteness condition may fail to hold, and hence in this paper we consider the oscillation theorem under another important definiteness condition.

The Essential Spectrum of a System of Singular Ordinary Differential Operators of Mixed Order. Part II: The Generalization of Kako's Problem

A system of ordinary differential equations of mixed order on an interval (0, r0) is considered, where some coefficients are singular at 0. Special cases have been dealt with by Kako, where the essential spectrum of an operator associated with a linearized MHD model was calculated, and more recently by Hardt, Mennicken and Naboko. In both papers this operator is a selfadjoint extension of an operator on sufficiently smooth functions. The approach in the present paper is different in that a suitable operator associated with the given system of ordinary differential equations is explicitly defined as the closure of an operator defined on sufficiently smooth functions. This closed operator can be written as a sum of a selfadjoint operator and a bounded operator. It is shown that its essential spectrum is a nonempty compact subset of ℂ, and formulas for the calculation of the essential spectrum in terms of the coefficients are given.

Eigenfunction expansions associated with a two-parameter system of differential equations

In this work techniques from the theory of partial differential equations are used to prove the uniform convergence of the eigenfunction expansion associated with a left definite two-parameter system of ordinary differential equations.

On the essential spectrum of the linearized Navier-Stokes operator

We determine the essential spectrum of the linearized Navier-Stokes operator with physical boundary conditions. In contrast to other approaches we do not make use of pseudo-differential operators. We establish a direct proof using only some fundamental results for matrix operators.

On the Essential Spectrum of a Differentially Rotating Star

Natural oscillations of a differentially rotating star are governed by the linearized Euler equations. Separation of variables leads to a family L k,0 (k ∈ Z) of mixed order partial differential operators. It is shown that for k ¬= 0 their closures L k have nonempty essential spectrum. Indeed, it is shown that the essential spectrum of L k coincides with the essential spectrum of a bounded operator. Some parts of the essential spectrum are calculated explicitly. It is still an open problem if there are more points in the essential spectrum.

EXPANSIONS IN EIGENFUNCTIONS OF A TWO-PARAMETER SYSTEM OF DIFFERENTIAL EQUATIONS

Abstract Techniques from the theory of partial differential equations are used to prove the uniform convergence of the eigenfunction expansion associated with a two-parameter system of ordinary differential equations of the second order under left definiteness and semi-definitness assumptions.

Eigenvalue asymptotics for an elliptic boundary problem involving an indefinite weight

We are concerned here with the eigenvalue asymptotics for a non-selfadjoint elliptic boundary problem involving an indefinite weight function which vanishes on a set of positive measure. The asymptotic behaviour of the eigenvalues is well known for the case of second order operators. However for higher order operators, results have only been established under the restriction that the order of the operator exceeds the dimension of the underlying Euclidean space in which the problem is set. In this paper we establish the eigenvalue asymptotics for the case of higher order operators without any such restriction.

ON AN ELLIPTIC BOUNDARY VALUE PROBLEM ARISING IN MAGNETOHYDRODYNAMICS

Abstract Results are derived concerning the spectral properties of an elliptic boundary value problem arising in the mathematical theory of magnetohydrodynamics which are of basic importance for the further development of this theory.

An elliptic boundary value problem occurring in magnetohydrodynamics

We derive results concerning the spectral properties of an elliptic boundary value problem arising in the mathematical theory of magnetohydrodynamics which are of basic importance for the further development of this theory.

Necessity of Parameter-ellipticity for Multi-order Systems of Differential Equations

In this paper we investigate parameter-ellipticity conditions for multi-order systems of differential equations on a bounded domain.Unde r suitable assumptions on smoothness and on the order structure of the system, it is shown that parameter-dependent a priori estimates imply the conditions of parameter-ellipticity, i.e., interior ellipticity, conditions of Shapiro- Lopatinskii type, and conditions of Vishik-Lyusternik type.T he mixed-order systems considered here are of general form; in particular, it is not assumed that the diagonal operators are of the same order.Th is paper is a continuation of an article by the same authors where the sufficiency was shown, i.e., a priori estimates for the solutions of parameter-elliptic multi-order systems were established.