# On the Boundary Regularity of Phase-Fields for Willmore's Energy

## TL;DR

This paper investigates the boundary regularity of phase-fields related to Willmore's energy, showing potential boundary singularities and providing partial regularity results under certain boundary conditions.

## Contribution

It reveals boundary singularities in phase-fields with bounded Willmore energy and offers partial regularity results considering boundary conditions.

## Key findings

- Radon measures can be boundary singularities.
- Partial boundary regularity results depend on boundary conditions.
- Counterexamples show irregularities without boundary conditions.

## Abstract

We demonstrate that Radon measures which arise as the limit of the Modica-Mortola measures associated to phase-fields with uniformly bounded diffuse area and Willmore energy may be singular at the boundary of a domain and discuss implications for practical applications. We furthermore give partial regularity results for the phase-fields $u_\epsilon$ at the boundary in terms of boundary conditions and counterexamples without boundary conditions.