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

Abstract

In this work, we investigate the system formed by the equations and in bounded star‐shaped domains of . A Helmholtz‐type decomposition theorem is established based on a general solution of the above‐mentioned div‐curl system. When , we obtain a bounded right inverse of 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.