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Recovery of a manifold with boundary and its continuity as a function of its metric tensor. (English) Zbl 1088.74014
Summary: A basic theorem from differential geometry asserts that, if the Riemann curvature tensor associated with a field $${\mathbf C}$$ of class $${\mathcal C}^2$$ of positive-definite symmetric matrices of order $$n$$ vanishes in a connected and simply-connected open subset $$\Omega$$ of $$\mathbb{R}^n$$ then there exists an immersion $$\Theta\in{\mathcal C}^3(\Omega;\mathbb{R}^n)$$, uniquely determined up to isometries in $$\mathbb{R}^n$$, such that $${\mathbf C}$$ is the metric tensor field of the manifold $$\Theta(\Omega)$$, then isometrically immersed in $$\mathbb{R}^n$$. Let $$\dot \Theta$$ denote the equivalence class of $$\Theta$$ modulo isometries in $$\mathbb{R}^n$$ and let $${\mathcal F}:{\mathbf C}\to\dot\Theta$$ denote the mapping determined in this fashion.
The first objective of this paper is to show that, if $$\Omega$$ satisfies a certain “geodesic property” (in effect a mild regularity assumption on the boundary $$\partial\Omega$$ of $$\Omega)$$ and if the field $${\mathbf C}$$ and its partial derivatives of order $$\leq 2$$ have continuous extensions to $$\overline \Omega$$, the extension of the field $${\mathbf C}$$ remaining positive-definite on $$\overline\Omega$$, then the immersion $$\Theta$$ and its partial derivatives of order $$\leq 3$$ also have continuous extensions to $$\overline \Omega$$.
The second objective is to show that, under a slightly stronger regularity assumption on $$\partial\Omega$$, the above extension result combined with a fundamental theorem of Whitney leads to a stronger extension result: There exist a connected open subset $$\widetilde\Omega$$ of $$\mathbb{R}^n$$ containing $$\overline \Omega$$ and a field $$\widetilde{\mathbf C}$$ of positive-definite symmetric matrices of class $${\mathcal C}^2$$ on $$\widetilde\Omega$$ such that $$\widetilde{\mathbf C}$$ is an extension of $$\mathbb{C}$$ and the Riemann curvature tensor associated with $$\widetilde{\mathbf C}$$ still vanishes in $$\widetilde \Omega$$.
The third objective is to show that, if $$\Omega$$ satisfies the geodesic property and is bounded, the mapping $${\mathcal F}$$ can be extended to a mapping that is locally Lipschitz-continuous with respect to the topologies of the Banach spaces $${\mathcal C}^2 (\overline\Omega)$$ for the continuous extensions of the symmetric matrix fields $${\mathbf C}$$, and $${\mathcal C}^3(\overline\Omega)$$ for the continuous extensions of the immersions $$\Theta$$.

##### MSC:
 74B20 Nonlinear elasticity 74A05 Kinematics of deformation 53Z05 Applications of differential geometry to physics
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##### References:
 [1] Adams, R.A., Sobolev spaces, (1975), Academic Press New York · Zbl 0186.19101 [2] S. Anicic, H. Le Dret, A. Raoult, The infinitesimal rigid displacement lemma in Lipschitz coordinates and application to shells with minimal regularity, Math. Methods Appl. Sci., submitted for publication · Zbl 1156.35477 [3] Antman, S.S., Ordinary differential equations of nonlinear elasticity I: foundations of the theories of non-linearly elastic rods and shells, Arch. rational mech. anal., 61, 307-351, (1976) · Zbl 0354.73046 [4] Ball, J.M., Convexity conditions and existence theorems in nonlinear elasticity, Arch. rational mech. anal., 63, 337-403, (1977) · Zbl 0368.73040 [5] Ciarlet, P.G., Mathematical elasticity, vol. I, Three-dimensional elasticity, (1988), North-Holland Amsterdam [6] Ciarlet, P.G., Introduction to numerical linear algebra and optimization, (1989), Cambridge Univ. Press Cambridge [7] 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 [8] Ciarlet, P.G.; Larsonneur, F., On the recovery of a surface with prescribed first and second fundamental forms, J. math. pures appl., 81, 167-185, (2002) · Zbl 1044.53004 [9] 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 [10] Ciarlet, P.G.; Mardare, C., On rigid and infinitesimal rigid displacements in three-dimensional elasticity, Math. models methods appl. sci., 13, 1589-1598, (2003) · Zbl 1055.74006 [11] Ciarlet, P.G.; Mardare, C., On the recovery of a manifold with boundary in $$R\^{}\{n\}$$, C. R. acad. sci. Paris Sér. I math., 338, 333-340, (2004) · Zbl 1057.53013 [12] Ciarlet, P.G.; Mardare, C., Extension of a Riemannian metric with vanishing curvature, C. R. acad. sci. Paris Sér. I math., 338, 391-396, (2004) · Zbl 1066.53080 [13] P.G. Ciarlet, C. Mardare, Recovery of a surface with boundary and its continuity as a function of its two fundamental forms, in preparation · Zbl 1083.53007 [14] Dieudonné, J., Eléments d’analyse, tome 1 : fondements de l’analyse moderne, (1968), Gauthier-Villars Paris [15] 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 [16] Gurtin, M.E., An introduction to continuum mechanics, (1981), Academic Press New York · Zbl 0559.73001 [17] John, F., Rotation and strain, Comm. pure appl. math., 14, 391-413, (1961) · Zbl 0102.17404 [18] John, F., Bounds for deformations in terms of average strains, (), 129-144 [19] Kohn, R.V., New integral estimates for deformations in terms of their nonlinear strains, Arch. rational mech. anal., 78, 131-172, (1982) · Zbl 0491.73023 [20] Malliavin, P., Géométrie différentielle intrinsèque, (1972), Hermann Paris · Zbl 0282.53001 [21] Mardare, C., On the recovery of a manifold with prescribed metric tensor, Anal. appl., 1, 433-453, (2003) · Zbl 1060.53005 [22] S. Mardare, On isometric immersions of a Riemannian manifold with little regularity, Anal. Appl., submitted for publication [23] Nečas, J., LES Méthodes directes en théorie des equations elliptiques, (1967), Masson Paris · Zbl 1225.35003 [24] Reshetnyak, Y.G., Mappings of domains in Rn and their metric tensors, Siberian math. J., 44, 332-345, (2003) [25] Schatzman, M., Analyse numérique, (2001), Dunod Paris [26] Schwartz, L., Analyse II: calcul différentiel et equations différentielles, (1992), Hermann Paris [27] Whitney, H., Analytic extensions of differentiable functions defined in closed sets, Trans. amer. math. soc., 36, 63-89, (1934) · JFM 60.0217.01 [28] Yosida, K., Functional analysis, (1966), Springer-Verlag Berlin · Zbl 0217.16001
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