Some Nonstandard Problems in the 3D Field Theory

Abstract

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.