CHAPTER 11 - Boundary value problems for weakly elliptic systems of differential equations

Abstract

This chapter elaborates boundary value problems for weakly elliptic systems of differential equations. In a bounded domain with boundary, the chapter considers a class of systems of linear partial differential equations which are called weakly elliptic because the systems of this class possess the properties of elliptic systems, such as hypoellipticity, analyticity of all the generalized solutions for the systems with analytic coefficients, and analytic right-hand side. The determination of weak ellipticity of a given system reduces to solving a linear system of algebraic equations and to calculating the rank of matrices. This chapter uses some finite number of members of the complete symbol of a system which have different orders of homogeneity. It is important that the ellipticity of the system is developed without dependence on the choice of the initial orders of rows and columns. The chapter gives three equivalent definitions of weak ellipticity for pseudodifferential operators. The first definition is suitable for constructing regularizers and proving the theorem on solvability. Verification that some operator belongs to the class of weakly elliptic systems is made by means of two other definitions.