Sobolev Spaces and Elliptic Theory on Unbounded Domains in $\mathbb R^n$

TL;DR

This paper develops the theory of weighted Sobolev spaces on unbounded domains in Euclidean space and applies it to establish elliptic theory, including trace and extension results, similar to bounded cases.

Contribution

It introduces weighted Sobolev space theory on unbounded domains and extends elliptic operator results, including trace and extension theorems, to this setting.

Findings

Weighted Sobolev spaces are developed for unbounded domains.

Elliptic theory is extended to these weighted spaces.

Trace and extension results are established for unbounded domains.

Abstract

In this article, we develop the theory of weighted $L^2$ Sobolev spaces on unbounded domains in $\mathbb R^n$. As an application, we establish the elliptic theory for elliptic operators and prove trace and extension results analogous to the bounded, unweighted case.