The Dirichlet problem for a Petrovskij elliptic system of an even number of second-order equations (Q1288072)

The article is devoted to studying the system \[ - \Delta u_j + \lambda \frac{\partial}{\partial x_j} \sum_{i = 1}^{k}\left( \frac{\partial u_i}{\partial x_i} + \frac{\partial v_i}{\partial y_i} \right) - \mu \frac{\partial}{\partial y_j} \sum_{i = 1}^{k}\left( - \frac{\partial u_i}{\partial y_i} + \frac{\partial v_i}{\partial x_i} \right) = 0, \] \[ -\Delta v_j + \lambda \frac{\partial}{\partial y_j} \sum_{i = 1}^{k}\left( \frac{\partial u_i}{\partial x_i} + \frac{\partial v_i}{\partial y_i} \right) + \mu \frac{\partial}{\partial x_j} \sum_{i = 1}^{k}\left( - \frac{\partial u_i}{\partial y_i} + \frac{\partial v_i}{\partial x_i} \right) = 0. \] The author concludes that this system is a Petrovskij elliptic system for \(\lambda \neq 1\) and \(\mu \neq 1\). If \(\lambda \neq 1\), \(\mu \neq 1\), and \((\lambda - 2)(\mu - 2)(2 - \lambda - \mu) > 0\) then the Dirichlet problem with continuously differentiable boundary data is always solvable in the half-space of the space \({\mathbb R}^{2k}\) of the variables \((x_1,\dots,x_k,y_1,\dots,y_k)\) and the solution is unique in the class of functions bounded at infinity. In the case of an arbitrary bounded domain, the Dirichlet problem enjoys the Fredholm property.