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We consider a free boundary problem for the Willmore functional \(\mathcal{W}(f) = \frac{1}{4} \int _\Sigma H^2\,d\mu _f\). Given a smooth bounded domain \(\Omega \subset {\mathbb R}^3\), we construct Willmore disks which are critical in the class of surfaces meeting \(\partial \Omega \) at a right angle along their boundary and having small prescribed area. Using rescaling and the implicit function theorem, we first obtain constrained solutions with prescribed barycenter on \(\partial \Omega \). We then study the variation of that barycenter.

Local solutions to a free boundary problem for the Willmore functional