A set of Sobolev spaces and boundary-value problems for the curl and gradient-of-divergence operators

We will consider the scale of the Sobolev spaces H-m(G) vector fields in a bounded domain G of R-3 with a smooth boundary of Gamma. The gradient-ofdivergence and the rotor-of-rotor operators ( del div and rot(2)) and their powers are analogous to the scalar operator Delta(m) in R-3. They generate spaces A2(k) (G) and W-m (G) potential and vortex fields; where the numbers k,m > 0 are integers. It is proven that A(2)k (G) and W-m (G) are projections of Sobolev spaces H-2k (G) and H-m (G) in subspaces A and B in L-2(G). Their direct sums A(2K) (G) circle plus W-m (G) form a network of spaces. Its elements are classes C(2k,m) = A(2k) circle plus W-m. We consider at the properties of the spaces A(-m) and W-m and proved their compliance with the spaces A(m) and W-m. We also consider at the direct sums of A(k) (G) circle plus W-m (G) for any integer numbers k and m > 0. This completes the construction of the {C(..,..)}..,.. network. In addition, an orthonormal basis has been constructed in the space L-2(G). It consists of the orthogonal subspace A and B bases. Its elements are eigenfields of the operators. div and rot. The proof of their smoothness is an important stage in the theory developed. The model boundary value problems for the operators rot+lambda I del div+lambda I, their sum, and also for the Stokes operator have been investigated in the network {C(k,m)}(k,m). Solvability conditions are obtained for the model problems considered.