Regularity results for flat minimizers of the Willmore functional
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Submission date: 04. Aug. 2008 (revised version: July 2009)
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Let be a bounded domain with boundary of class and let denote the flat metric on . Let u be a minimizer of the Willmore functional within a subclass (defined by prescribing boundary conditions on parts of ) of all isometric immersions of the Riemannian manifold (S, g) into . In this article we study the regularity properties of such u. Our main result roughly states that minimizers u are away from three kinds of line segments: Segments which intersect tangentially, segments which bound regions on which is locally constant and segments for which diverges near one endpoint. At segments of the third kind, we prove that u is precisely (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.