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Continuity in \(H^1\)-norms of surfaces in terms of the \(L^1\)-norms of their fundamental forms. (English) Zbl 1073.74009

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 \(k\to\infty\).

MSC:

74B05 Classical linear elasticity
74G40 Regularity of solutions of equilibrium problems in solid mechanics
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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
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