On the Fredholm and unique solvability of nonlocal elliptic problems in multidimensional domains

We consider elliptic equations of order $2m$ in a bounded domain $Q\subset {\mathbb{R}}^n$ with nonlocal boundary-value conditions connecting the values of a solution and its derivatives on $(n-1)$-dimensional smooth manifolds ${\mathrm{\Gamma }}_i$ with the values on manifolds ${\omega }_i({\mathrm{\Gamma }}_i)$, where ${\bigcup }_i\overline{{\mathrm{\Gamma }}_i}=\mathrm{\partial }Q$ is a boundary of $Q$ and ${\omega }_i$ are $C^{\mathrm{\infty }}$ diffeomorphisms. By proving a priori estimates for solutions and constructing a right regularizer, we show the Fredholm solvability in weighted space. For nonlocal elliptic problems with a parameter, we prove the unique solvability.