Trace of Homogeneous Fractional Sobolev Spaces on Strip-like Domains

TL;DR

This paper characterizes the trace operator for homogeneous fractional Sobolev spaces on strip-like domains, introducing a new far screened Sobolev seminorm that captures non-local properties and provides a bounded right inverse.

Contribution

It introduces a new far screened Sobolev seminorm for fractional Sobolev spaces and establishes its role in the trace operator on strip-like domains, extending previous homogeneous Sobolev space results.

Findings

Defined intrinsic seminorms for the trace space.

Established the boundedness of the trace operator with the new seminorm.

Explored relationships between fractional and classical screened Sobolev spaces.

Abstract

In this paper, we discuss the trace operator for homogeneous fractional Sobolev spaces over infinite strip-like domains. We determine intrinsic seminorms on the trace space that allow for a bounded right inverse. The intrinsic seminorm includes two features previously used to describe the trace of homogeneous Sobolev spaces, a relation between the two disconnected components of the trace and the screened Sobolev seminorm. However, unlike its homogeneous Sobolev space equivalent, fractional Sobolev spaces require a far screened Sobolev seminorm that captures the non-local properties of fractional Sobolev spaces. We study some basic relationships between this new far screened Sobolev space with previously discussed screened Sobolev spaces.