MiS Preprint Repository

Let $S\subset {\mathbb{R}}^2$ be a bounded domain with boundary of class $C^{\mathrm{\infty }}$ 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 $C^{\mathrm{\infty }}$ away from three kinds of line segments: Segments which intersect $\partial S$ tangentially, segments which bound regions on which $\mathrm{\nabla }u$ is locally constant and segments for which ${\mathrm{\nabla }}^{2}u$ 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.