On first order linear PDE systems all of whose solutions are harmonic functions

Abstract

We study the first order linear system $u_{\tilde{z}} + \bar{v}_{w} = 0, u_{\bar{w}} - \bar{v}_{z} = 0$ in a domain $\Omega \subset \mathbb{C}^{2}$ (first considered by G. Cimmino, [3]). We prove a Morera type theorem, emphasizing the analogy to the Cauchy-Riemann system, and a representation formula yielding a result on removable singularities of solutions to (2). We derive (by a Hilbert space technique outlined in [5]) compatibility relations among the free terms and boundary data in the boundary value problem $u_{\tilde{z}}+ \bar{v}_{w} = f, u_{\bar{w}} - \bar{v}_{z} = g$ in $\Omega$, and $u = \varphi,v = \psi$ on $\partial \Omega$. If $F = (u, v) : \Omega \to \mathbb{C}^{2}$ is a solution to (2) such that $\sup_{\varepsilon \gt O}$\int_{\partial \Omega_{\varepsilon}}|F(z, w)|^{p} d\sigma_{\varepsilon}(z, w} \lt \infty$ for some $p \geq 2$, then we show that $F$ admits non tangential limits at almost every $(\xi, \omega) \in \partial\Omega$.