A right inverse of curl operator 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 div w=g0 and curl w= g in bounded star-shaped domains of R3. 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 g0 0, we readily obtain a bounded right inverse of 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 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.