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MiS Preprint

Regularity results for flat minimizers of the Willmore functional

Peter Hornung


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.


Related publications

2008 Repository Open Access
Peter Hornung

Regularity results for flat minimizers of the Willmore functional