The Dirichlet problem for a Petrovskiĭ-elliptic system of second-order equations (Q1972184)

The apparatus of singular integral equations is applied to studying the Dirichlet problem for the system \[ -\Delta u_j + \lambda_j\frac{\partial}{\partial x_j}\sum_{i=1}^n \frac{\partial u_i}{\partial x_i} = 0,\qquad j=1,\dots, n. \] The main results of the article are as follows: Theorem 1. If the parameters \(\lambda_j\) of the system satisfy either the inequalities \(\lambda_j < 1\) or the inequalities \(\lambda_j >2\), then the Dirichlet problem for the system in an arbitrary half-space is solvable for arbitrary differentiable data and the solution is unique. Theorem 2. The Dirichlet problem for the system in an arbitrary convex domain with smooth boundary is of Fredholm type if either all \(\lambda_j<1\) or all \(\lambda_j>2\).