Euler‐lagrange equation and regularity for flat minimizers of the Willmore functional
Péter Hornung · Communications on Pure and Applied Mathematics · 2010
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.