Local solvability of rotationally invariant differential forms

Abstract

We will deal with local solvability of classes of linear partial differential operators. More precisely, we will deal with solvability of classes of first order differential equations in context of differential forms. Let \Omega=A(z)dz+B(z)d\bar{z} be a smooth differential 1-form defined in a neighborhood of the origin in \mathbb{R}^2. We say that \Omega is rotationally invariant if \Omega\wedge R^*_\alpha\Omega=0 for all rotation of angle \alpha, R_\alpha, of \mathbb{R}^2. Let \Omega be a rotationally invariant smooth differential 1-form, singular at (0,0) and elliptic for all (x,y)\in\mathbb{R}^2\setminus{(0,0)}. We are interested in studying equations in the formdu\wedge\Omega=\eta\wedge\Omega,where \eta is a smooth differential 1-form defined in a neighborhood of the origin of \mathbb{R}^2. The relation between the order of the vanishing of the 2-forms\Omega\wedge\overline{\Omega} and \Omega\wedge(\bar{z}dz+zd\bar{z}) has influence in the solvability. (AU)