Preprint 52/2008

Regularity results for flat minimizers of the Willmore functional

Peter Hornung

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Submission date: 04. Aug. 2008 (revised version: July 2009)
Pages: 50
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Let formula15 be a bounded domain with boundary of class formula17 and let formula19 denote the flat metric on formula21. Let u be a minimizer of the Willmore functional within a subclass (defined by prescribing boundary conditions on parts of formula25) of all formula27 isometric immersions of the Riemannian manifold (S, g) into formula31. In this article we study the regularity properties of such u. Our main result roughly states that minimizers u are formula17 away from three kinds of line segments: Segments which intersect formula25 tangentially, segments which bound regions on which formula41 is locally constant and segments for which formula43 diverges near one endpoint. At segments of the third kind, we prove that u is precisely formula47 (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.

07.06.2018, 02:11