TY - JOUR

T1 - Euler-Lagrange Equation and Regularity for Flat Minimizers of the Willmore Functional

AU - Hornung, Peter

PY - 2011/3

Y1 - 2011/3

N2 - 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,W-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.

AB - 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,W-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.

UR - http://www.scopus.com/inward/record.url?scp=78650241496&partnerID=8YFLogxK

UR - http://dx.doi.org/10.1002/cpa.20342

U2 - 10.1002/cpa.20342

DO - 10.1002/cpa.20342

M3 - Article

VL - 64

SP - 367

EP - 441

JO - Communications on Pure and Applied Mathematics

JF - Communications on Pure and Applied Mathematics

SN - 0010-3640

IS - 3

ER -