On One Tangential Boundary Value Problem of the Field Theory on a Plane

Abstract

One tangential boundary value problem of the field theory on a plane for the system of Poisson equations is studied. The peculiarity of the problem is that the boundary condition posed on the desired solution in nonlocal in nature. A weak solution of the problem formulated is to be found. In view of the fact that boundary condition posed on the problem formulated is nonlocal in nature, an appropriate space of functions is introduced, among which the solution of the problem is sought. Since the problem is invariant with regard to the additive vector constant, the search for a solution narrows down to the space of functions having an average integral value equal to zero in the specified domain. The weak solution of the problem formulated is defined, and it is shown that it is well-posed in nature. The main result of the work is the theorem of the existence and uniqueness of a weak solution of the problem. The proof is carried out in a few stages, and an example for the case of a rectangular domain is given.