Zaremba's Problem for Elliptic Equations in the Neighbourhood of a Singular Point or at Infinity

The present paper deals with qualitative properties of the solutions of the mixed Zaremba's problem. In part I two aspect of the Zaremba's problem are studied. First the existence of Wiener's generalized solutions in limited domains is proven. The part of the boundary supporting Neumann's conditions is assumed to be locally smooth, whereas the other part of the alternating Schwarz's method, reducing the mixed problem in an arbitrary domain to a sequence of mixed problems in standard domains and corresponding Dirichlet's problem in domains adjoining adjoining the supports of Dirichlet's boundary conditions. Second, in the domains, satifying an isoperimertic inequality, the so called Growth Lemma, giving a quantitative estimate of the degree of “saddle” of the mixed problem solution, is proven. Being important in itself, this lemma be widely used in part 2 of the present paper.