MiS Preprint Repository

Let $S\subset\mathbb{R}^2$ be a bounded domain with boundary of class $C^{\infty}$ and let $g_{ij} = \delta_{ij}$ denote the flat metric on $\mathbb{R}^2$. Let $u$ be a minimizer of the Willmore functional within a subclass (defined by prescribing boundary conditions on parts of $\partial S$) of all $W^{2,2}$ isometric immersions of the Riemannian manifold $(S, g)$ into $\mathbb{R}^3$.

In this article we study the regularity properties of such $u$. Our main result roughly states that minimizers $u$ are $C^{\infty}$ away from three kinds of line segments: Segments which intersect $\partial S$ tangentially, segments which bound regions on which $\nabla u$ is locally constant and segments for which $\nabla^2 u$ diverges near one endpoint. At segments of the third kind, we prove that $u$ is precisely $C^3$ (in the interior), and we obtain sharp estimates for the size of its derivatives.

Our main motivation to study this problem comes from nonlinear elasticity: On isometric immersions, the Willmore functional agrees with Kirchhoff's energy functional for thin elastic plates.