Nonlinear estimates for hypersurfaces in terms of their fundamental forms
type: Item
instance of: scholarly article
title: Nonlinear estimates for hypersurfaces in terms of their fundamental forms
zbMATH Open document ID: 1388.53009
published in: Comptes Rendus. Mathématique. Académie des Sciences, Paris
publication date: 2017-11-23T00:00:00Z
review text: The paper deals with a sufficiently regular hypersurface immersed in the \((n+1)\)-dimensional Euclidean space. Such a hypersurface is determined up to a proper isometry of \(\mathbb R^{n+1}\) by its first and second fundamental forms. As a consequence, a sufficiently regular hypersurface with boundary, whose position and positively-oriented unit normal vectors are given on a non-empty portion of its boundary, is uniquely determined by its first and second fundamental forms. In this article other versions of these uniqueness results are established, by means of inequalities showing that an appropriate distance between two immersions from a domain \(\omega\) of \(\mathbb R^n\) into \(\mathbb R^{n+1}\) is bounded by the \(L^p\)-norm of the difference between matrix fields defined in terms of the first and second fundamental forms associated with each immersion. Some of the results of this article generalize or are based on previous results of \textit{P. G. Ciarlet} et al. [C. R., Math., Acad. Sci. Paris 355, No. 2, 226--231 (2017; Zbl 1361.53004)] and \textit{P. G. Ciarlet} and the second author [J. Math. Pures Appl. (9) 83, No. 1, 1--15 (2004; Zbl 1050.74030)]. The proofs of these theorems are only sketched and they are supposed to appear in a submitted paper of the authors.
reviewed by: Adela-Gabriela Mihai
Mathematics Subject Classification ID: 53A07
Mathematics Subject Classification ID: 46E35
Mathematics Subject Classification ID: 53C24
zbMATH DE Number: 6812090
zbMATH Keywords: sufficiently regular hypersurface
zbMATH Keywords: first and second fundamental forms
zbMATH Keywords: isometry
author: Maria Mălin
author: Cristinel Mardare
MaRDI profile type: Publication
full work available at URL: j.crma.2017.10.014
OpenAlex ID: W2767870359
cites work: Q4073917
cites work: New nonlinear estimates for surfaces in terms of their fundamental forms
cites work: On rigid and infinitesimal rigid displacements in shell theory.
cites work: Nonlinear Korn inequalities
cites work: Q3410224
cites work: A theorem on geometric rigidity and the derivation of nonlinear plate theory from three-dimensional elasticity
cites work: Nonlinear Korn inequalities on a hypersurface
cites work: Q3094428
DOI: 10.1016/J.CRMA.2017.10.014
description: scientific article; zbMATH DE number 6812090