Search

MiS Preprint Repository

We have decided to discontinue the publication of preprints on our preprint server as of 1 March 2024. The publication culture within mathematics has changed so much due to the rise of repositories such as ArXiV (www.arxiv.org) that we are encouraging all institute members to make their preprints available there. An institute's repository in its previous form is, therefore, unnecessary. The preprints published to date will remain available here, but we will not add any new preprints here.

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 $W^{2,2}_{\text{iso}}(S; R3)$.

Download papers: Part I (PDF, 692 kB), Part II (PDF, 434 kB)

Received:
Sep 3, 2007
Published:
Sep 3, 2007

Related publications

inJournal
2011 Repository Open Access
Peter Hornung

Approximation of flat \(W^{2,2}\) isometric immersions by smooth ones

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