Local solutions to a free boundary problem for the Willmore functional

TL;DR

This paper investigates a free boundary problem for the Willmore functional, constructing orthogonally meeting disks with prescribed small area and analyzing their barycenter variations.

Contribution

It introduces a method to construct constrained Willmore disks with prescribed area and barycenter in a free boundary setting.

Findings

Constructed orthogonal Willmore disks with small prescribed area.

Analyzed the barycenter variation of these disks.

Provided a framework for free boundary Willmore problems.

Abstract

We consider a free boundary problem for the Willmore functional. Given a smooth domain $\Omega$ in ${\mathbb R}^3$, we construct Willmore disks wich are critical in the class of surfaces meeting $\partial \Omega$ orthogonally along their boundary and having small prescribed area. Using rescaling we first obtain constrained solutions with prescribed two-dimensional barycenter, and then study the variation of the barycenter.