Submitted Paper

Inserted: 12 nov 2024
Last Updated: 12 nov 2024

Year: 2024

Abstract:

Motivated by manifold-constrained homogenization problems, we construct an extension operator for Sobolev functions defined on a perforated domain and taking values in a compact, connected $C^2$-manifold without boundary. The proof combines a by now classical extension result for the unconstrained case with a retraction argument that heavily relies on the topological properties of the manifold. With the ultimate goal of providing necessary conditions for the existence of suitable extension operators for Sobolev maps between manifolds, we additionally investigate the relationship between this problem and the surjectivity of the trace operator for such functions.