The Neumann problem for elliptic systems on a plane

Second order elliptic systems with constant leading coefficients are considered. It is shown that the Bitsadze definition of weakly connected elliptic systems is equivalent to the known Shapiro–Lopatinskiy condition with respect to the Dirichlet problem for weakly connected elliptic systems. An analogue of potentials of double layer for these systems is introduced in the frame of functional theoretic approach. With the help of these potentials all solutions are described in the Holder $C^\mu(D)$ and Hardy $h^p(D)$ classes as well as in the class $C(\overline D)$ of all continuous functions.