On boundary value problems for the Cauchy-Riemann and Bitsadze systems of equations

Abstract

In this paper, we consider boundary value problems for elliptic systems of equations of orders 1 and 2, which are known as the Cauchy‐Riemann and Bitsadze generalized systems of equations. The author’s interest in these systems arose from the theory of partial differential equations; more specifically, it was caused by the following observation. In the theory of ordinary differential equations, the settings of well-posed problems for one second-order equation and a system of secondorder equations virtually coincide. In the theory of partial differential equations, the situation is different. For one second-order elliptic equation, the Dirichlet problem is well-posed if either some constraints are imposed on the coefficients of the equation or the domain in which the problem is considered is small. For second-order systems of equations that are elliptic in the sense of Petrovskii but not strongly elliptic, there exist examples [1] in which the homogeneous Dirichlet problem in a disk of arbitrarily small radius has infinitely many linearly independent solutions. The matrix form of this system of equations is