Non-standard Boundary Value Problems of the Field Theory on a Plane

Abstract

Two nonstandard boundary value problems in the theory of 2D vector fields for a system of Poisson equations are studied. The unique solvability of the problems is established, including an a priori assessment of solutions depending on the problem data. An example is given for the case of a rectangular area, demonstrating the non-standard boundary conditions of the problems set.

The proof of the main results is preceded by a criterion for the solvability of integral identities corresponding to the decomposition of vector fields into the sum of tangential and normal subspaces. Accordingly, the basic subspaces in which integral identities (and further boundary value problems) are considered are the kernels of normal and tangential trace operators in Sobolev space. The specified decomposition is uniquely determined by a tangential vector to the boundary of the domain in which the problems are considered. This is possible owing to the consideration of 2D vector fields and distinguishes the problems set from the 3D case. The criterion is proved using the Vishik-Lax-Milgram theorem.

It should be noted that the integral inequalities of the Friedrichs inequality type obtained in conducting the proof are, in our opinion, of certain independent interest. These inequalities have been obtained by proving a lemma showing the unconditional nature of the solvability of the integral identities under consideration.