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On first order linear PDE systems all of whose solutions are harmonic functions. (English) Zbl 1123.35011
Summary: We study the first order linear system \(u_{\overline z}+\overline v_w=0\), \(u_{\overline w}-\overline v_z=0\) in a domain \(\Omega\subset \mathbb{C}^2\) (first considered by G. Cimmino [Rend. Sem. Mat. Univ. Padova 12, 89–113 (1941; Zbl 0026.40601 and JFM 67.0329.01)]. 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. We derive (by a Hilbert space technique compatibility relations among the free terms and boundary data in the boundary value problem \(u_{\overline z}+\overline v_w=f\), \(u_{\overline w}-\overline v_w=f\), \(u_{\overline w}-\overline v_z=g\) in \(\Omega\), and \(u= \varphi\), \(v=\psi\) on \(\partial\Omega\). If \(F=(u,v):\Omega\to\mathbb{C}^2\) is a solution such that \(\sup_{\varepsilon>0}\int_{\partial \Omega_\varepsilon} |F(z,w)|^p\,d\sigma_\varepsilon(z,w)<\infty\) for some \(p\geq 2\) then we show that \(F\) admits nontangential limits at almost every \((\zeta,\omega)\in\partial \Omega\).

MSC:
35F05 Linear first-order PDEs
35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
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