On first order elliptic systems

The aim of this paper is the study of first-order stationary systems of PDEs of the form \(\sum _k A_k\partial _k U + KU=0\) ${\sum }_k{A}_k{\partial }_{k}U+KU=0$ with \(K\ngtr 0\) $K\ngtr 0$ on \(\Omega = {\mathbb {R}}^d\) $\mathrm{\Omega }={\mathbb{R}}^d$ and \(\Omega \subset {\mathbb {R}}^d\) $\mathrm{\Omega }\subset {\mathbb{R}}^d$ bounded. We prove that the classical assumption \(K>0\) $K>0$ 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 \(\Omega = {\mathbb {R}}^d,\) $\mathrm{\Omega }={\mathbb{R}}^d,$ we use Fourier analysis for the existence and uniqueness of solutions. For \(\Omega \subset {\mathbb {R}}^d\) $\mathrm{\Omega }\subset {\mathbb{R}}^d$ bounded, we use a complex analog of the Banach–Nečas–Babuška 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 \(\Omega ={\mathbb {R}}^d.\) $\mathrm{\Omega }={\mathbb{R}}^d.$ In order to illustrate our approach, we study in detail the reduction of the Poisson equation to a first-order system.