On Some Nonstandard Boundary Value Problems for 3D Vector Fields

We study two nonclassical boundary value problems for a system of Poisson equations in three-dimensional space whose boundary conditions contain the main first-order differential operations of field theory. The statements of the problems are based on the trace theorem for a linear combination of the vector of normal derivatives, the rotor, and the divergence. We prove two existence and uniqueness theorems for the weak solution of the problems under study.