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

TL;DR

This paper investigates the solvability of nonlocal elliptic boundary-value problems in multidimensional domains, establishing conditions for Fredholm solvability and unique solutions using a priori estimates and regularizer construction.

Contribution

It introduces new methods to prove Fredholm and unique solvability for nonlocal elliptic problems with boundary conditions involving diffeomorphisms, expanding understanding of such problems.

Findings

Established a priori estimates for solutions.

Constructed a right regularizer to prove Fredholm property.

Proved unique solvability for problems with a parameter.

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.