  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.
