Regularity properties of isometric immersions

Stefan Müller and Mohammed Reza Pakzad

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Submission date: 28. Jan. 2004
Pages: 24
published in: Mathematische Zeitschrift, 251 (2005) 2, p. 313-331 
DOI number (of the published article): 10.1007/s00209-005-0804-y
Bibtex
MSC-Numbers: 53C42, 35B65, 74K20, 53A05
Keywords and phrases: isometric immersion, regularity, rigidity, plates
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Abstract:
We show that an isometric immersion $y$ from a two-dimensional domain $S$ with $C^{1, \alpha}$ 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 $\det \nabla^2 V_k = 0$.