## Tsukuba Journal of Mathematics

- Tsukuba J. Math.
- Volume 30, Number 1 (2006), 149-170.

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

Sorin Dragomir and Ermanno Lanconelli

#### 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 $(\xi, \omega) \in \partial\Omega$.

#### Article information

**Source**

Tsukuba J. Math., Volume 30, Number 1 (2006), 149-170.

**Dates**

First available in Project Euclid: 30 May 2017

**Permanent link to this document**

https://projecteuclid.org/euclid.tkbjm/1496165034

**Digital Object Identifier**

doi:10.21099/tkbjm/1496165034

**Mathematical Reviews number (MathSciNet)**

MR2248289

**Zentralblatt MATH identifier**

1123.35011

#### Citation

Dragomir, Sorin; Lanconelli, Ermanno. On first order linear PDE systems all of whose solutions are harmonic functions. Tsukuba J. Math. 30 (2006), no. 1, 149--170. doi:10.21099/tkbjm/1496165034. https://projecteuclid.org/euclid.tkbjm/1496165034