Local solutions to a free boundary problem for the Willmore functional

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.