On first order elliptic systems

The aim of this paper is the study of first-order stationary systems of PDEs of the form Sigma(k) A(k)partial derivative U-k + KU = 0 with K (sic) 0 on Omega = R-d and Omega subset of R-d bounded. We prove that the classical assumption K > 0 is not necessary for the well-posedness of the system and is violated in the particular case of the first-order Poisson problem. In the case Omega = R-d, we use Fourier analysis for the existence and uniqueness of solutions. For Omega subset of R-d bounded, we use a complex analog of the Banach-Necas-Babuska theorem to obtain the existence and uniqueness of a solution in a setting that encompasses both Friedrichs' systems and the first order reduction of the Poisson problem. The techniques used to prove the classical inf-sup conditions are inspired by harmonic analysis arguments that are consistent with the case Omega = R-d. In order to illustrate our approach, we study in detail the reduction of the Poisson equation to a first-order system.