Abstract
The authors discuss the existence and uniqueness up to isometries of En of immersions ϕ: Ω ⊂ Rn → En with prescribed metric tensor field (g ij ): Ω → S n> , and discuss the continuity of the mapping (g ij ) → ϕ defined in this fashion with respect to various topologies. In particular, the case where the function spaces have little regularity is considered. How, in some cases, the continuity of the mapping (g ij ) → ϕ can be obtained by means of nonlinear Korn inequalities is shown.
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Dedicated to Haïm Brezis on the occasion of his 70th birthday
This work was supported by a grant from the Research Grants Council of the Hong Kong Special Administration Region, China (Nos. 9041637, CiyuU100711).
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Ciarlet, P.G., Mardare, C. & Mardare, S. Recovery of immersions from their metric tensors and nonlinear Korn inequalities: A brief survey. Chin. Ann. Math. Ser. B 38, 253–280 (2017). https://doi.org/10.1007/s11401-016-1070-5
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DOI: https://doi.org/10.1007/s11401-016-1070-5