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MiS Preprint
78/2007
Approximation of $W^{2,2}$ isometric immersions by smooth isometric immersions
Peter Hornung
Abstract
Part I: This is the first of two papers in which we study $W^{2,2}$ isometric immersions $u$ from a flat domain $S\subset R2$ into $R3$.
Here we study the geometry of the set on which $\nabla u$ 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 $W^{2,2}$ closure of the set of isometric immersions lying in $W^{2,2}(S; R3)\cap C^{\infty}(\bar{S};R3)$ agrees with the set of all $W^{2,2}(S; R3)$ isometric immersions.
Part II: Let $S\subset R2$ be a bounded Lipschitz domain and denote by $W^{2,2}_{\text{iso}}(S; R3)$ the set of mappings $u\in W^{2,2}(S;R3)$ which satisfy $(\nabla u)^T(\nabla u) = Id$ almost everywhere. Under an additional regularity condition on the boundary $\partial S$ (which is satisfied if $\partial S$ is piecewise continuously differentiable) we prove that the strong $W^{2,2}$ closure of $W^{2,2}_{\text{iso}}(S; R3)\cap C^{\infty}(\bar{S};R3)$ agrees with $W^{2,2}_{\text{iso}}(S; R3)$.
