The article considers certain boundary-value problems for the Poisson system of equations in the 3D space the boundary conditions of which contain, apart from the function values, the values of gradient, divergence and curl. The statement of such problems is based on an identity containing unconditional link between the boundary values of any pair of functions and the values of vectors of their normal derivatives and curls. Such identity follows from well-known formula representing the Laplace operator in the curl-divergence form. Such conditions have a clear hydrodynamic sense and generate the corresponding subspaces of the Sobolev space. In addition, the obtained identity can be considered as the necessary condition for the theorem about the smoothness of generalized or weak solutions of the considered problems. The corresponding boundary-value problems are correctly solvable in the weak sense; that is, they have a unique weak solution. Some specific examples are given. For the proof, the author used the Galerkin method, which is based on the equality of the bilinear forms consistent with the standard formulation of the Laplace operator and its curl-divergence form. It is exactly the equality of these forms that determines different types of boundary conditions containing the basic operations of the field theory, namely, gradient, divergence and curl of the sought solution. It has been shown that the sequence of approximate solutions is compact in nature, and passage to the limit has been done. By using the proposed approach it becomes possible to formulate some nonstandard problems for the Stokes and Navier-Stokes systems of equations.

Some nonstandard boundary value problems are studied for the stationary Poisson system, Stokes system, and Navier–Stokes system. The problems under consideration are “intermediate” between the Dirichlet problem and Neumann problem. The well-posedness of these problems is proved.

Using decomposition results for Sobolev spaces of Clifford‐valued functions into direct sums of subspaces of monogenic and co‐monogenic functions variational problems will be studied.These variational problems are equivalent to PD‐models by the choice of special operators of conboundary differentiation. By a Galerkin scheme we construct the monogenic part as a weak solution of a non‐linear problem. The co‐monogenic potential is the solution of a weak Dirichlet problem. Copyright © 2002 John Wiley & Sons, Ltd.

My lecture is devoted to the following question. Let G ⊂ℂ_z^1 (ℂ_z^1 ≈ℝ_x,y^2 ,z = iy) be a bounded domain with smooth boundary, W_p^m ≡ W_p^m(G) , m = 0, ±1, …, the scale of Sobolev spaces, O_p^m = { f(z) ∈ W_p^m ,∂ _z̅ f(z) = 0(inD')} the scale of analytic Sobolev subspaces. It is easy to see that O_p^m is a closed subspace of W_p^m,i.e.O_p^m⊂ W_p^mandO̅_p^m = O_p^m.