Euler-lagrange equation and regularity for flat minimizers of the Willmore functional
✍ Scribed by Peter Hornung
- Publisher
- John Wiley and Sons
- Year
- 2010
- Tongue
- English
- Weight
- 764 KB
- Volume
- 64
- Category
- Article
- ISSN
- 0010-3640
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✦ Synopsis
Abstract
Let
\input amssym $S\subset{\Bbb R}^2$
be a bounded domain with boundary of class C^∞^, and let g~ij~ = δ~ij~ denote the flat metric on \input amssym ${\Bbb R}^2$. Let u be a minimizer of the Willmore functional within a subclass (defined by prescribing boundary conditions on parts of ∂S) of all W^2,2^ isometric immersions of the Riemannian manifold (S, g) into \input amssym ${\Bbb R}^3$. In this article we derive the Euler‐Lagrange equation and study the regularity properties for such u. Our main regularity result is that minimizers u are C^3^ away from a certain singular set Σ and C^∞^ away from a larger singular set Σ ∪ Σ~0~. We obtain a geometric characterization of these singular sets, and we derive the scaling of u and its derivatives near Σ~0~.
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. © 2010 Wiley Periodicals, Inc.