A remark on the Moisil-Theodorescu system
The Moisil-Theodorescu system is \[ -\partial f_ 2/\partial x_ 1- \partial f_ 3/\partial x_ 2+\partial f_ 4/\partial x_ 3=0,\quad \partial f_ 1/\partial x_ 1+\partial f_ 3/\partial x_ 3-\partial f_ 4/\partial x_ 2=0 \] \[ \partial f_ 1/\partial x_ 2-\partial f_ 2/\partial x_ 3+\partial f_ 4/\partial x_ 1=0,\quad \partial f_ 1/\partial x_ 3+\partial f_ 2/\partial x_ 2-\partial f_ 3/\partial x_ 1=0. \] Let \({\mathcal D}\subset R^ 3\) be an open set. The left side of the system forms a differential operator P on \((C^ 1({\mathcal D}))^ 4\). The problem is to find \(f=(f_ 1,f_ 2,f_ 3,f_ 4),\) such that \(Pf=0\) in \({\mathcal D}\), \(f_ 1=\phi.\) A necessary condition for the existence of a solution is that \(\phi\) be a harmonic function on \({\mathcal D}\). The following statements are proved in the paper: 1. If \({\mathcal D}\) is open in \(R^ 3\), int \(\bar {\mathcal D}={\mathcal D}\), and if the considered problem is solvable for an arbitrary in \({\mathcal D}\) harmonic function \(\phi\), then \(R^ 3\setminus {\mathcal D}\) has no compact components of connectedness. 2. If \({\mathcal D}\) is a bounded open set in \(R^ 3\) with a connected complement and boundary in the class \(C^{2,\lambda}\), \(0<\lambda <1\), then for arbitrary harmonic in \({\mathcal D}\) function \(\phi\), such that \(\phi \in C^{1,\lambda}(\bar {\mathcal D})\), the considered problem has a solution.
scientific article; zbMATH DE number 4099777
scientific article