A right inverse of curl operator which is divergence free invariant and some applications to generalized Vekua type problems

In this work, we investigate the system formed by the equations $\text{div }\vec{w}=g_0$ and $\text{curl }\vec{w}=\vec{g}$ in bounded star-shaped domains of ${\mathbb{R}}^3$. A Helmholtz-type decomposition theorem is established based on a general solution of the above-mentioned div-curl system which was previously derived in the literature. When $g_0\equiv 0$, we readily obtain a bounded right inverse of $\text{curl}$ which is a divergence-free invariant. The restriction of this operator to the subspace of divergence-free vector fields with vanishing normal trace is the well-known Biot–Savart operator. In turn, this right inverse of $\text{curl}$ will be modified to guarantee its compactness and satisfy suitable boundary-value problems. Applications to Beltrami fields, Vekua-type problems as well as Maxwell’s equations in inhomogeneous media are included.