Inverse problems for elliptic equations in the space: I

We consider inverse problems of finding the right-hand side of a linear secondorder elliptic equation of general form. We study the first boundary value problem. Two ways of representation of the additional information (over-determination) are considered: specification of the trace of the solution of the boundary value problem inside the domain on some manifold of smaller dimension and specification of values of the normal derivative on a part of the boundary. The Fredholm alternative is proved for the considered problems. Stability estimates are given for the case of unique solvability. The analysis is performed in classes of continuous functions whose derivatives satisfy the Hölder condition.