On One Normal Boundary Value Problem of the Field Theory

Abstract

One normal problem of the field theory on a plane for a system of Poisson equations is studied. This problem is one of implementations of the general approach to formulation of boundary value problems of the field theory, which is based on studying the kernels of trace operators and kernels of trace functionals, which are taken as the main space of desired solutions.

A peculiarity of this problem is that one of the boundary conditions imposed on the desired solution is nonlocal in nature. The considered boundary condition nonlocality is due to the fact that the base space in which the solution of the problem is sought is the kernel of the regular trace functional in the Sobolev space. The need to supplement the problem on the one-dimensional cokernel of the above-mentioned functional, which arises in this case, determines the second boundary condition, which is local (pointwise) in nature. That is why, in the framework of the duality of the Sobolev space and its conjugate space, the solution to the problem is the pair (u, с), where с is a constant that compensates for the one-dimensional “defect” of the problem.

It should be noted that the theory of boundary value problems for vector fields on a plane differs essentially from the theory of boundary value problems for three-dimensional vector fields. One of the reasons for the difficulties in formulating three-dimensional boundary value problems is that a number of important surfaces cannot be made parallel to each other, which rules out the possibility to consider “tangential” boundary value problems, that is, problems with boundary tangential fields. In this respect, the theory of boundary value problems of plane vector fields differs from the three-dimensional theory for the better. On the other hand, the fact that a formula that would interrelating the Laplace operator with first-order field theory operations is not available on a plane is an obstacle to identifying new non-standard boundary value problems for plane fields. Revealing of this interrelation is a separate issue, the solution of which leads to the concept of the trace of first-order field theory operators as a singular functional above the space of traces of functions from the first-order Sobolev space. Thus, boundary value problems for plane fields are a separate section of the general theory of field boundary value problems.

One of the non-standard problems in the theory of plane vector fields, which contains non-local boundary conditions, is considered. A theorem for existence and uniqueness of a weak solution of the problem posed is formulated and proved. The proof of the theorem includes a few steps:

• using the Galerkin method, the existence of a solution to the equation is established;
• the passage to the limit is made, and the vector function u(x) is found;
• the number α is determined, and the uniqueness of the function u and the number α is established.

The main result of the work is that the problem posed is correct in nature.