Preprint 78/2007

Approximation of W2,2 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

Part I: This is the first of two papers in which we study formula26 isometric immersions u from a flat domain formula30 into R3. Here we study the geometry of the set on which formula34 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 formula26 closure of the set of isometric immersions lying in formula44 agrees with the set of all formula46 isometric immersions.

Part II: Let formula30 be a bounded Lipschitz domain and denote by formula50 the set of mappings formula52 which satisfy formula54 almost everywhere. Under an additional regularity condition on the boundary formula56 (which is satisfied if formula56 is piecewise continuously differentiable) we prove that the strong formula26 closure of formula62 agrees with formula50.

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

18.10.2019, 02:13