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MiS Preprint

4/2004

Regularity properties of isometric immersions

Stefan Müller and Mohammed Reza Pakzad

Abstract

We show that an isometric immersion $y$ from a two-dimensional domain $S$ with $C^{1, α}$ boundary to $R^{3}$ which belongs to the critical Sobolev space $W^{2, 2}$ is $C^{1}$ up to the boundary. More generally $C^{1}$ regularity up to the boundary holds for all scalar functions $V \in W^{2, 2} (S)$ which satisfy $det \nabla^{2} V = 0$ . If $S$ has only Lipschitz boundary we show such $V$ can be approximated in $W^{2, 2}$ by functions $V_{k} \in W^{1, \infty} \cap W^{2, 2}$ with $det \nabla^{2} V_{k} = 0$ .

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Received:: 28.01.04

Published:: 28.01.04

MSC Codes:: 53C42, 35B65, 74K20, 53A05

Keywords:: isometric immersion, regularity, rigidity, plates

Related publications

inJournal

2005 Repository Open Access

Stefan Müller and Mohammad Reza Pakzad

Regularity properties of isometric immersions

In: Mathematische Zeitschrift, 251 (2005) 2, pp. 313-331

BibTex DOI: 10.1007/s00209-005-0804-y