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

Abstract

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 $\Gamma_i$ with the values on manifolds $\omega_{i}(\Gamma_i)$, where $\bigcup_i\overline{\Gamma_i}=\partial Q$ is a boundary of $Q$ and $\omega_i$ are $C^\infty$ 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.