Properties of solutions of a class of planar elliptic operators with degeneracies

In this paper we investigate properties of solutions of first and second order elliptic equations that degenerate along a simple closed curve in $\mathbb{R}^2$. These equations are generated by a $\mathbb{C}$-valued vector field $L$. To the vector field $L$, we associate the second order operator $\mathbb{P}=\mathrm{Re}\left[L\overline{L}+p L \right]$, where $p$ is a $\mathbb{C}$-valued function. We establish a one-to-one correspondence between the solutions of the equation $\mathbb{P}u=0$ and those of an associated first order equation of type $Lw=Aw+B\overline{w}$.