Global solvability of the Laplace equation in weighted Sobolev spaces

TL;DR

This paper investigates the solvability of the Laplace equation in unbounded domains within weighted Sobolev spaces, establishing conditions under which weak solutions are also strong solutions, with implications for non-local boundary problems.

Contribution

It provides new results on the weak and strong solvability of non-local Laplace boundary value problems in weighted Sobolev spaces with Muckenhoupt weights, addressing a gap in elliptic PDE theory.

Findings

Weak solutions in $W_{ u}^{2,p}$ are also strong solutions.

Solutions satisfy corresponding boundary conditions.

Special techniques are developed for non-standard elliptic problems.

Abstract

We consider a non-local boundary value problem for the Laplace equation in unbounded studding the weak and strong solvability of that problem in the framework of the weighted Sobolev space $W^{1,p}_\nu$, with a Muckenhoupt weight. We proved that if any weak solution belongs to the space $W_{\nu}^{2,p}$, then it is also a strong solution and satisfies the corespding boundary conditions. It should be noted that such problems do not fit into the general theory of elliptic equations and require a special technique.