Spectral Theory of 2 × 2 Elliptic Systems of the Second Order

Abstract

The article deals with the investigation of the spectral properties of the Dirichlet problem for 2x2 elliptic system of the second order. The need to study the properties of the solvability of boundary value problems for kxk systems of linear differential equations of order n, elliptical, hyperbolic, parabolic, and mixed types appears whenever there is, for example, a problem similar to the one once explored by A.V. Bitsadze. In particular, he pointed to the circumstance in which the Dirichlet problem for elliptic 2x2 system of linear differential equations in partial derivatives was incorrect, which was quite surprising at the time. Important applications of the theory of systems of linear partial differential and the problems associated with the study of the properties of the solvability of boundary value problems formulated to stimulate research relevant spectral problems. Spectral theory of closed differential operators generated by boundary value problems for systems of linear differential equations in partial derivatives started to develop recently. We studied at the same time as the asymptotic behaviour of the own values and the location of the spectrum on the complex plane, and the basic properties of systems composed of vector-functions. Investigation of the structure of the spectrum and the possibility of expanding solutions sets of vector-functions is now one of the main directions in the study of problems of the spectral theory of boundary value problems for systems of linear differential equations in partial derivatives. The carried out research is based on the modified method of model operators.