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MiS Preprint
78/2007

Approximation of W2,2 isometric immersions by smooth isometric immersions

Peter Hornung

Abstract

Part I: This is the first of two papers in which we study W2,2 isometric immersions u from a flat domain SR2 into R3.

Here we study the geometry of the set on which u 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 W2,2 closure of the set of isometric immersions lying in W2,2(S;R3)C(S¯;R3) agrees with the set of all W2,2(S;R3) isometric immersions.

Part II: Let SR2 be a bounded Lipschitz domain and denote by Wiso2,2(S;R3) the set of mappings uW2,2(S;R3) which satisfy (u)T(u)=Id almost everywhere. Under an additional regularity condition on the boundary S (which is satisfied if S is piecewise continuously differentiable) we prove that the strong W2,2 closure of Wiso2,2(S;R3)C(S¯;R3) agrees with Wiso2,2(S;R3).

Download part II of the papers: PDF, 434 kB

Received:
03.09.07
Published:
03.09.07

Related publications

inJournal
2011 Repository Open Access
Peter Hornung

Approximation of flat W2,2 isometric immersions by smooth ones

In: Archive for rational mechanics and analysis, 199 (2011) 3, pp. 1015-1067