The Dirichlet Problem for an Elliptic System of Second-Order Equations with Constant Real Coefficients in the Plane

A solution of the Dirichlet problem for an elliptic system of equations with constant coefficients and simple complex characteristics in the plane is expressed as a double-layer potential. The boundary-value problem is solved in a bounded simply connected domain with Lyapunov boundary under the assumption that the Lopatinskii condition holds. It is shown how this representation is modified in the case of multiple roots of the characteristic equation. The boundary-value problem is reduced to a system of Fredholm equations of the second kind. For a Hölder boundary, the differential properties of the solution are studied.