Distances between classes in $W^{1,1}(\Omega;{\mathbb S}^1)$

TL;DR

This paper investigates the distances between classes of maps in Sobolev spaces with circle-valued functions, introducing a topological classification and establishing sharp bounds for these distances.

Contribution

It introduces a topological equivalence relation in $W^{1,1}(\

Findings

Established sharp bounds for distances between topological classes.

Extended analysis to $W^{1,p}$ spaces for $p>1$.

Provided a framework for understanding topological singularities in Sobolev maps.

Abstract

We introduce an equivalence relation on the space $W^{1,1}(\Omega;{\mathbb S}^1)$ which classifies maps according to their "topological singularities". We establish sharp bounds for the distances (in the usual sense and in the Hausdorff sense) between the equivalence classes. Similar questions are examined for the space $W^{1,p}(\Omega;{\mathbb S}^1)$ when $p>1$.