Sense Of Douglis Research Articles

Essential spectrum of non-self-adjoint singular matrix differential operators

The purpose of this paper is to study the essential spectrum of non-self-adjoint singular matrix differential operators in the Hilbert space L2(R)⊕L2(R) induced by matrix differential expressions of the form(0.1)(τ11(⋅,D)τ12(⋅,D)τ21(⋅,D)τ22(⋅,D)), where τ11, τ12, τ21, τ22 are respectively m-th, n-th, k-th and 0 order ordinary differential expressions with m=n+k being even. Under suitable assumptions on their coefficients, we establish an analytic description of the essential spectrum. It turns out that the points of the essential spectrum either have a local origin, which can be traced to points where the ellipticity in the sense of Douglis and Nirenberg breaks down, or they are caused by singularity at infinity.

On Coincidence of λ- and μ-Holomorphic Functions on the Boundary of a Domain and Applications to Elliptic Boundary Value Problems

We establish that λ- and μ-holomorphic functions can coincide on the boundary of a domain only if the complex numbers λ and μ lie in the same half-plane Im z > 0 or Im z < 0. We describe applications to the uniqueness question for the Dirichlet problem for second order elliptic systems and the Schwartz problem for vector-valued functions that are analytic in the sense of Douglis.

A generalization of Douglis–Nirenberg elliptic operators

It is well known that the composition of two elliptic differential operators of orders ω and ω′ is again an elliptic operator of order ω+ω′. This fact is not true for the composition of two multi–order systems elliptic in the sense of Douglis and Nirenberg. We consider a new more general class of multi–order pseudodifferential operators. This class of multi–order essentially elliptic operators is closed with respect to the composition, coordinate and basis transformation. Moreover, every multi–order essentially elliptic operator has a parametrix belonging to the class and therefore is a Fredholm operator. Bibliography: 5 titles.

Ellipticity Conditions of the Shallow Water Balance Equations for Atmospheric Data

Abstract Nonlinear normal mode initialization equations, which provide required balance relations for atmospheric data, are considered in the generalized case of shallow water equations in arbitrary orthogonal coordinates. Using the concept of ellipticity in the sense of Douglis–Nirenberg, the conditions of well posedness of boundary value problems for balance equations are derived in the cases of constrained streamfunction, constrained potential vorticity, and constrained pressure fields.

Hyperanalytic Functions and Their Applications

A survey on the theory of hyperanalytic functions in the sense of Douglis is presented. Some applications of hyperanalytic functions are also given, in particular, to the description of solutions of elliptic systems in the plane. Special attention is devoted to some systems arising in the plane elasticity theory and linearized Stokes system of hydromechanics.

COMPLETE SCALE OF ISOMORPHISMS FOR ELLIPTIC PSEUDODIFFERENTIAL BOUNDARY-VALUE PROBLEMS

The aim of the paper is to establish a complete scale of isomorphisms for elliptic pseudodifferential boundary value problems generated by the operators of the Boutet de Monvel algebra. The result is an extension of the corresponding known results on elliptic differential boundary value problems. Because for any elliptic pseudodifferential boundary value problem there exists a parametrix belonging to the Boutet de Monvel algebra, the proof presented is much shorter than the known proof for differential problems. Systems that are elliptic in the sense of Douglis and Nirenberg are also considered.

A douglis-nirenberg elliptic operator in biot theory

The Biot equations of acoustics in porous ealstic media are written as a matrix operator which turr~s out to be elliptic in the sense of Douglis and Nirenberg. The fundamental solution is calculated and properties of solutions to the Biot equation are analyzed. Similarities and contrasts to the theory of thermoelasticity are pointed out.

Asymptotics of the Spectrum of Douglis - Nirenberg Elliptic Operators on a Compact Manifold

Pseudo- differential systems on closed manifolds, elliptic in the sense of DOUGLIS and Nirenberg. are considered. It is proved that the system is similar to an diagonal operator up to an operator of order - ∞, and the similarity transformation preserves ellipticity and parameter -ellipticity. The similarity transformation may be chosen so that it preserves even self adjointness up to an operator of order less than the lowest order of the diagonal entry. These results are applied to prove the eigenvalue asymptotics with the sharp estimate of the remainder in the self adjoint case as well as a rough asymptotics in the non-self-adjoint case. Another application is the eigenvalue asymptotics with the sharp estimate of the remainder for general self- adjoint elliptic boundary value problems with the spectral parameter appearing linearly in boundary conditions.

On the definition of ellipticity for systems of partial differential equations

One of the fundamental problems in the theory of partial differential equations is that of classifying equations and systems by type. A specific problem associated with the definition of ellipticity is that when a higherorder equation or system is reduced to a first order system, ellipticity may be destroyed. One approach to that problem was introduced by Douglis and Nirenberg [4], who gave a definition of ellipticity for systems which involved assigning weights to each of the equations and dependent variables and then defining the principal part of the system in terms of those weights. The advantage of the definition of ellipticity given in [4] is that ellipticity can be preserved while a higher-order equation or system is reduced to an equivalent first order system. The disadvantage is that the definition is not invariant under nonsingular changes of variables. An alternative approach suggested by Protter [S] is to reduce the original equation or system to an overdetermined first order system, and then define ellipticity in a natural and invariant way for such systems. In the present article, we show that any determined or overdetermined system with smooth coefficients which is elliptic in the sense of Douglis and Nirenberg can be reduced to an overdetermined first order system which is elliptic in the sense of Protter, and that any overdetermined first order system with smooth coefficients which is elliptic in the sense of Protter can be converted to a determined second order system which is elliptic in the sense of Douglis and Nirenberg, or under any reasonable definition of ellipticity. The conversion to a second order system allows the application of the regularity results of [3, 4, 6, 71; in fact, second order systems are treated in

Boundary values of generalized solutions of systems that are elliptic in the sense of Douglis and Nirenberg

Boundary values of generalized solutions of systems that are elliptic in the sense of Douglis and Nirenberg

A theorem about the complete set of isomorphisms for systems elliptic in the sense of Douglis and Nirenberg

i. Theorems about the complete set of isomorphisms for homogeneous elliptic systems and ~ystems elliptic in the sense of Petrovskii with normal boundary conditions are established in [3, 4], respecLive].y; Kovalenko [4] has also investigated the problem formally adjoint to the problem for systems elliptic in the sense of Petrovskii. Additional restrictions are made in [3-6] that are not connected with the important aspects. Our method is different from that used in [3-6]; it is a further development of the method used in [7], and in it we use the connection given in [7] between the theorem about the complete set of isomorphisms (see [i, 2, 8]) and elliptic problems with coboundary operators (see [9-11]). This method allows us to prove a theorem about the complete set of isomorphisms for general systems elliptic in the sense of Douglis and Nirenberg, under only a necessary restriction about the ellipticity of the boundary problems being considered.