Approximation of W^2,2 isometric immersions by smooth isometric immersions

Peter Hornung

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Submission date: 03. Sep. 2007 (revised version: September 2010)
Pages: 48
published in: Archive for rational mechanics and analysis, 199 (2011) 3, p. 1015-1067 
DOI number (of the published article): 10.1007/s00205-010-0374-y
Bibtex

Abstract:
Part I: This is the first of two papers in which we study isometric immersions u from a flat domain into R3. Here we study the geometry of the set on which is locally constant and the properties of local line of curvature parametrizations for nonconvex S. A main result is that u(S) can be approximated by flat surfaces consisting of finitely many planar regions and finitely many developable regions. In a companion paper we will use this to prove that, for a large class of domains S, the strong closure of the set of isometric immersions lying in agrees with the set of all isometric immersions.

Part II: Let be a bounded Lipschitz domain and denote by the set of mappings which satisfy almost everywhere. Under an additional regularity condition on the boundary (which is satisfied if is piecewise continuously differentiable) we prove that the strong closure of agrees with .

Download papers: Part I (PDF, 692 kB), Part II (PDF, 434 kB)