Abstract
We revisit and expand the intrinsic and parametric representations of hypersurfaces with application to the theory of thin and asymptotic shells. A central issue is the minimal smoothness of the midsurface to still make sense of asymptotic membrane shell and bending equations without ad hoc mechanical or mathematical assumptions. This is possible for a C 1,1-midsurface with or without boundary and without local maps, local bases, and Christoffel symbols via the purely intrinsic methods developed by Delfour and Zolésio starting with [14] in 1992. Anicic, Le Dret, and Raoult [1] introduced in 2004 a family of surfaces ω that are the image of a connected bounded open Lipschitzian domain in R 2 by a bi-Lipschitzian mapping with the assumption that the normal field is globally Lipschitzian. From this, they construct a tubular neighborhood of thickness 2h around the surface and show that for sufficiently small h the associated tubular neighborhood mapping is bi-Lipschitzian. We prove that such surfaces are C 1,1-surfaces with a bounded measurable second fundamental form. We show that the tubular neighborhood can be completely described by the algebraic distance function to ω and that it is generally not a Lipschitzian domain in R 3 by providing the example of a plate around a flat surface ω verifying all their assumptions. Therefore, the G 1-join of K-regular patches in the sense of Le Dret [20] generates a new K-regular patch that is a C 1,1-surface and the join is C 1,1. Finally, we generalize everything to hypersurfaces generated by a bi-Lipschitzian mapping defined on a domain with facets (e.g., for sphere, torus). We also give conditions for the decomposition of a C 1,1-hypersurface into C 1,1-patches.
This research has been supported by a discovery grant of the National Sciences and Engineering Research Council of Canada.
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Delfour, M.C. (2009). Representations, Composition, and Decomposition of C 1,1-hypersurfaces. In: Kunisch, K., Sprekels, J., Leugering, G., Tröltzsch, F. (eds) Optimal Control of Coupled Systems of Partial Differential Equations. International Series of Numerical Mathematics, vol 158. Birkhäuser, Basel. https://doi.org/10.1007/978-3-7643-8923-9_5
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DOI: https://doi.org/10.1007/978-3-7643-8923-9_5
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