Zero-extension convergence and Sobolev spaces on changing domains

TL;DR

This paper develops a new framework for analyzing the convergence of Sobolev functions on changing domains without relying on a fixed reference, extending classical compactness results to this dynamic setting.

Contribution

It introduces a novel approach to weak and strong convergence for Sobolev functions on evolving domains, avoiding the need for a reference configuration.

Findings

Established new definitions of convergence on changing domains

Proved compactness theorems analogous to classical results

Demonstrated applications and compared with existing methods

Abstract

We extend the definition of weak and strong convergence to sequences of Sobolev-functions whose underlying domains themselves are converging. In contrast to previous works, we do so without ever assuming any sort of reference configuration. We then develop the respective theory and counterparts to classical compactness theorems from the fixed domain case. Finally, we illustrate the usefulness of these definitions with some examples from applications and compare them to other approaches.