On the Laplacian vector fields theory in domains with rectifiable boundary

Given a domain Ω in ℝ³ with rectifiable boundary, we consider main integral, and some other, theorems for the theory of Laplacian (sometimes called solenoidal and irrotational, or harmonic) vector fields paying a special attention to the problem of decomposing a continuous vector field, with an additional condition, u on the boundary Γ of Ω ⊂ ℝ³ into a sum u = u⁺ + u^- were u ^± are boundary values of vector fields which are Laplacian in Ω and its complement respectively. Our proofs are based on the intimate relations between Laplacian vector fields theory and quaternionic analysis for the Moisil-Theodorescu operator.