Non-local elliptic problems with non-linear argument transformations near the points of conjugation

P. L. Gurevich

Izvestiya. Mathematics , Tome 67 (2003) no. 6, pp. 1149-1186

Résumé

We consider elliptic equations of order $2m$ in a domain $G\subset\mathbb R^n$ with non-local conditions that connect the values of the unknown function and its derivatives on $(n-1)$-dimensional submanifolds $\overline\Upsilon_i$ (where $\bigcup_i\overline\Upsilon_i=\partial G$ with the values on $\omega_{is}(\overline\Upsilon_i)\subset\overline G$. Non-local elliptic problems in dihedral angles arise as model problems near the conjugation points $g\in\overline\Upsilon_i\cap \overline\Upsilon_j\ne\varnothing$, $i\ne j$. We study the case when the transformations $\omega_{is}$ correspond to non-linear 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 non-linear ones.