Regularity results for flat minimizers of the

Let S ⊂ R2 be a bounded domain with boundary of class C ∞ and let gij = δij 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 ∂S) of all W 2,2 isometric immersions of the Rie-mannian manifold (S, g) into R3. In this article we study the regularity properties of such u. Our main result roughly states that minimizers u are C ∞ away from three kinds of line segments: Segments which in-tersect ∂S tangentially, segments which bound regions on which ∇u is locally constant and segments for which∇2u diverges near one endpoint. At segments of the third kind, we prove that u is precisely C3 (in the interior), and we obtain sharp estimates for the size of its derivatives. Our main motivation to study this problem comes from nonlinear elas-ticity: On isometric immersions, the Willmore functional agrees with Kirchhoff’s energy functional for thin elastic plates.