On the Lavrentiev gap for manifold-valued maps

TL;DR

This paper examines when smooth maps densely approximate manifold-valued maps in variational problems, highlighting conditions for the Lavrentiev gap phenomenon on compact manifolds.

Contribution

It provides new insights into the conditions under which the Lavrentiev gap occurs or is avoided for manifold-valued maps.

Findings

Identification of conditions leading to the Lavrentiev gap

Examples of manifold settings where density fails

Criteria for the validity of modular density

Abstract

We investigate the validity and the failure of modular density of smooth maps on compact manifolds.