Global curvature for surfaces and area minimization under a thickness constraint

• Published in: Calculus of variations and partial differential equations 2006-04, Vol.25 (4), p.431-467
• Main Authors: Strzelecki, Paweł, Mosel, Heiko von der
• Format: Article
• Language: English
• Subjects: Calculus of variations; Convergence; Parameter estimation; Partial differential equations; Thickness measurement
• ISSN: 0944-2669; 1432-0835
• DOI: 10.1007/s00526-005-0334-9

Motivated by previous work on elastic rods with self-contact, involving the concept of the global radius of curvature for curves (as defined by Gonzalez and Maddocks), we define the global radius of curvature delta[X] for a wide class of continuous parametric surfaces X for which the tangent plane exists on a dense set of parameters. It turns out that in this class of surfaces a positive lower bound delta[X] greater than or equal to theta greater than 0 provides, naively speaking, the surface with a thickness of magnitude theta; it serves as an excluded volume constraint for X, prevents self-intersections, and implies that the image of X is an embedded C1-manifold with a Lipschitz continuous normal. We also obtain a convergence and a compactness result for such thick surfaces, and show one possible application to variational problems for embedded objects: the existence of ideal surfaces of fixed genus in each isotopy class. The proofs are based on a mixture of elementary topological, geometric and analytic arguments, combined with a notion of the reach of a set, introduced by Federer in 1959. [PUBLICATION ABSTRACT]