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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
##### Keywords:
Morera type theorem; Hilbert space technique
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