Nonlocal elliptic problems with nonlinear argument transformations near the points of conjugation

Abstract

We consider elliptic equations of order $2m$ in a domain $G\subset\mathbb R^n$ with nonlocal conditions that connect the values of the unknown function and its derivatives on $(n-1)$-dimensional submanifolds $\Upsilon_i$ (where $\bigcup_i\Upsilon_i=\partial G$) with the values on $\omega_{is}(\overline\Upsilon_i)\subset\overline G$. Nonlocal elliptic problems in dihedral angles arise as model problems near the conjugation points $g\in\overline\Upsilon_i\cap\Upsilon_j\ne\varnothing$, $i\ne j$. We study the case where the transformations $\omega_{is}$ correspond to nonlinear transformations in the model problems. It is proved that the operator of the problem remains Fredholm and its index does not change as we pass from linear argument transformations to nonlinear ones.