ON SOLVABILITY OF NONLOCAL PROBLEMS FOR A 2ND-ORDER ELLIPTIC EQUATION

This article is an investigation of the solvability of nonlocal proble ms for an elliptic equation, in which the values of the solution on th e boundary of the domain Q under consideration are expressed in terms of its values at interior points and other points of the boundary. A n ew concept of solution (in the space of (n - 1)-dimensionally continuo us functions) is introduced, broader than concepts considered previous ly, and sufficient conditions are established for the problem to be Fr edholm with index zero. The connection between solvability of the prob lem in this formulation and in the classical formulation is studied. I n particular, there is a class of nonlocal problems (including some pr oblems studied previously) that are Fredholm with index zero in the fo rmulation introduced but not in the classical formulation (sometimes n ot even Fredholm). For a certain class of problems a theorem on unique solvability is proved.