Elliptic Systems of Differential Equations

Abstract

We consider the most general linear elliptic system of two first-order partial differential equations in two unknown functions. After the considerations in Section 11–2.6 and with suitable hypotheses on the coefficients, we may assume that the system is in the integrable normal form (II-2.65): $\begin{gathered} U_{x_1 }^1 - U_{x_2 }^1 + A_1^1 (x)U^1 + A_2^1 (x)U^2 + C^1 (x) = 0, \hfill \\ U_{x_2 }^1 - U_{x_1 }^2 + A_1^2 (x)U^1 + A_2^2 (x)U^2 + C^2 (x) = 0, \hfill \\ \end{gathered}$ where U ¹(x) and U ²(x) are the unknown functions.