Fundamental solutions of a class of first-order linear elliptic systems

Abstract

The Cauchy–Riemann type systems in ℝ n+1 are systems of first-order partial differential equations whose solutions satisfy the Laplace equation. The systems can be written in the form , where E 0 = I m is the identity matrix, E i (i ≥ 1) are m × m constant matrices satisfying the condition E i E j + E j E i = − 2δ ij I m . We introduce a more general condition for E i , they are variable coefficients and satisfy the condition E i (x) · E j (x) + E j (x) · E i (x) = − 2a ij (x)I m , where [a ij ] is a symmetric and positive matrix. Each 𝒞2-solution of the system now becomes a solution of a second-order elliptic system with the leading part . In this article, we introduce the Levi functions and construct fundamental solutions of the systems ‘in the large’.