Euler-Lagrange Equation and Regularity for Flat Minimizers of the Willmore Functional

Let S subset of R(2) be a bounded domain with boundary of class C(infinity), and let g(ij) = delta(ij) denote the flat metric on R(2). Let u be a minimizer of the Willmore functional within a subclass (defined by prescribing boundary conditions on parts of partial derivative S) of all W(2,2) isometric immersions of the Riemannian manifold. (S, g) into 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 Sigma and C(infinity) away from a larger singular set Sigma boolean OR Sigma(0). We obtain a geometric characterization of these singular sets, and we derive the scaling of u and its derivatives near Sigma(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. (C) 2010 Wiley Periodicals, Inc.