A Resolution of the Poisson Problem for Elastic Plates

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.