

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
Bibtex
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Abstract:
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.