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

TL;DR

This paper investigates the stability of the index of unbounded nonlocal elliptic operators in Sobolev spaces, showing that low-order terms and certain boundary perturbations do not alter the index, thus ensuring its robustness.

Contribution

It establishes conditions under which the index of nonlocal elliptic operators remains unchanged despite low-order and boundary perturbations.

Findings

Low-order terms do not affect the operator's index.

Nonlocal boundary perturbations under certain conditions preserve the index.

The operator retains the Fredholm property under these perturbations.

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.