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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 $C^{1, \alpha}$ boundary to $\mathbb{R}^3$ 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 $\det \nabla^2 V_k = 0$.
