*Published Paper*

**Inserted:** 2 may 2022

**Last Updated:** 6 may 2022

**Journal:** Arch. Rational. Mech. Anal.

**Volume:** 236

**Pages:** 1593-1676

**Year:** 2020

**Doi:** 10.1007/s00205-020-01499-2

**Abstract:**

We consider the problem of finding a surface $\Sigma \subset \mathbb{R}^m$ of least Willmore energy among all immersed surfaces having the same boundary, boundary Gauss map and area. Such a problem was considered by S. Germain and S.D. Poisson in the early XIX century as a model for equilibria of thin, clamped elastic plates. We present a solution in the case of boundary data of class $C^{1,1}$ and for when the boundary curve is simple and closed. The minimum is realised by an immersed disk, possibly with a finite number of branch points in its interior, which is of class $C^{1,\alpha}$ up to the boundary for some $0<\alpha <1$, and whose Gauss map extends to a map of class $C^{0,\alpha}$ up to the boundary.