# Reflection of Willmore surfaces with free boundaries

## TL;DR

This paper investigates the properties of Willmore surfaces with free boundaries in three-dimensional space, deriving boundary conditions and proving regularity through reflection techniques.

## Contribution

It introduces new weak boundary conditions for Willmore surfaces with free boundaries and establishes their regularity using reflection methods.

## Key findings

- Derived weak forms of free boundary conditions for Willmore surfaces.
- Proved regularity of these surfaces via reflection techniques.
- Analyzed cases where the boundary meets a plane orthogonally or lies on a line.

## Abstract

We study immersed surfaces in $\mathbb{R}^3$ which are critical points of the Willmore functional under boundary constraints. The two cases considered are when the surface meets a plane orthogonally along the boundary, and when the boundary is contained in a line. In both cases we derive weak forms of the resulting free boundary conditions and prove regularity by reflection.