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

In this work, we investigate the system formed by the equations divw ->=g0$$ \operatorname{div}\kern0.3em \overrightarrow{w}={g}_0 $$ and curlw ->=g ->$$ \operatorname{curl}\kern0.3em \overrightarrow{w}=\overrightarrow{g} $$ in bounded star-shaped domains of Double-struck capital R3$$ {\mathrm{\mathbb{R}}}<^>3 $$. A Helmholtz-type decomposition theorem is established based on a general solution of the above-mentioned div-curl system. When g0 equivalent to 0$$ {g}_0\equiv 0 $$, we obtain a bounded right inverse of curl$$ \operatorname{curl}\kern0.1em $$ which is a divergence-free invariant. The restriction of this right inverse to the subspace of divergence-free vector fields with vanishing normal trace is the well-known Biot-Savart operator. This right inverse will be restricted 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 proposed.