Difference Potentials Method: Some Aspects of the Modified Calderon's Potentials Theory (parts I-III)

In this paper we elaborate some aspects of modified Calderon's potentials theory. These potentials are the basic continual object in the difference potentials method (DPM) due to V.Ryaben'kii. We transform classical Calderon's potentials as they arise in theory of PDE's to the form they were proposed in the general theory of DPM. Some problems of so called modified inner boundary conditions are discussed; we propose complete classification of such conditions with respect to the properties of independence and equivalence to non-modified inner conditions. Paragraph 3 is devoted to the problem of one-to-one parametrization of homogeneous differential equations solutions by means of potentials of special type - generalized multi-layer potentials. Finally a theory of generalized Poincare-Steklov operators is elaborated for a wide class of differential equations.