Ciarlet, Philippe G.; Gratie, Liliana; Mardare, Cristinel Continuity in \(H^1\)-norms of surfaces in terms of the \(L^1\)-norms of their fundamental forms. (English) Zbl 1073.74009 C. R., Math., Acad. Sci. Paris 341, No. 3, 201-206 (2005). Summary: The main purpose of this note is to show how a ‘nonlinear Korn’s inequality on a surface’ can be established. This inequality implies, in particular, the following interesting per se sequential continuity property for a sequence of surfaces. Let \(\omega\) be a domain in \(\mathbb{R}^2\), let \(\theta: \overline\omega\to\mathbb{R}^3\) be a smooth immersion, and let \(\theta^k: \overline \omega\to\mathbb{R}^3\), \(k\geq 1\) be mappings with the following properties: they belong to the space \({\mathbf H}^2(\omega)\); the vector fields normal to the surfaces \(\theta^k (\omega)\), \(k\geq 1\), are well-defined a.e. in \(\omega\) and they also belong to the space \({\mathbf H}^1(\omega)\); the principal radii of curvature of the surfaces \(\theta^k(\omega)\) stay uniformly away from zero; and finally, the three fundamental forms of the surfaces \(\theta^k (\omega)\) converge in \({\mathbf L}^1(\omega)\) toward the three fundamental forms of the surface \(\theta(\omega)\) as \(k\to\infty\). Then, up to proper isometries of \(\mathbb{R}^3\) the surfaces \(\theta^k(\omega)\) converge in \({\mathbf H}^1(\omega)\) toward the surface \(\theta (\omega)\) as \(k\to\infty\). Cited in 1 Document MSC: 74B05 Classical linear elasticity 74G40 Regularity of solutions of equilibrium problems in solid mechanics Keywords:nonlinear Korn’s inequality PDFBibTeX XMLCite \textit{P. G. Ciarlet} et al., C. R., Math., Acad. Sci. Paris 341, No. 3, 201--206 (2005; Zbl 1073.74009) Full Text: DOI References: [1] Adams, R. A., Sobolev Spaces (1975), Academic Press: Academic Press New York · Zbl 0314.46030 [2] Ciarlet, P. G., Continuity of a surface as a function of its two fundamental forms, J. Math. Pures Appl., 82, 253-274 (2002) · Zbl 1042.53003 [3] P.G. Ciarlet, L. Gratie, C. Mardare, A nonlinear Korn inequality on a surface, J. Math. Pures Appl., in press; P.G. Ciarlet, L. Gratie, C. Mardare, A nonlinear Korn inequality on a surface, J. Math. Pures Appl., in press · Zbl 1094.53001 [4] Ciarlet, P. G.; Laurent, F., Continuity of a deformation as a function of its Cauchy-Green tensor, Arch. Rational Mech. Anal., 167, 255-269 (2003) · Zbl 1030.74003 [5] Ciarlet, P. G.; Mardare, C., On rigid and infinitesimal rigid displacements in shell theory, J. Math. Pures Appl., 83, 1-15 (2004) · Zbl 1050.74030 [6] Ciarlet, P. G.; Mardare, C., Continuity of a deformation in \(H^1\) as a function of its Cauchy-Green tensor in \(L^1\), J. Nonlinear Sci., 14, 415-427 (2004) · Zbl 1084.53063 [7] Ciarlet, P. G.; Mardare, C., Recovery of a manifold with boundary and its continuity as a function of its metric tensor, J. Math. Pures Appl., 83, 811-843 (2004) · Zbl 1088.74014 [8] Ciarlet, P. G.; Mardare, C., Recovery of a surface with boundary and its continuity as a function of its two fundamental forms, Anal. Appl., 3, 99-117 (2005) · Zbl 1083.53007 [9] Friesecke, G.; James, R. D.; Müller, S., A theorem on geometric rigidity and the derivation of nonlinear plate theory from three dimensional elasticity, Comm. Pure Appl. Math., 55, 1461-1506 (2002) · Zbl 1021.74024 [10] Nečas, J., Les Méthodes Directes en Théorie des Equations Elliptiques (1967), Masson: Masson Paris · Zbl 1225.35003 [11] Szopos, M., On the recovery and continuity of a submanifold with boundary, Anal. Appl., 3, 119-143 (2005) · Zbl 1151.53346 [12] Reshetnyak, Y. G., Mappings of domains in \(R^n\) and their metric tensors, Siberian Math. J., 44, 332-345 (2003) [13] Whitney, H., Functions differentiable on boundaries of regions, Ann. of Math., 34, 482-485 (1934) · JFM 60.0217.03 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.