On the stability of the index of unbounded nonlocal operators in Sobolev spaces

Abstract

Unbounded operators corresponding to nonlocal elliptic problems on a bounded region $G\subset\mathbb R^2$ are considered. The domain of these operators consists of functions from the Sobolev space $W_2^m(G)$ being generalized solutions of the corresponding $2m$-order elliptic equation with right-hand side from $L_2(G)$ and satisfying homogeneous nonlocal boundary conditions. It is known that such unbounded operators have the Fredholm property. It is proved in the paper that low-order terms in the differential equation do not affect the index of the operator. Conditions under which nonlocal perturbations on the boundary do not change the index are also formulated.