Mappings of Domains in Rn and Their Metric Tensors |
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Authors: | Reshetnyak Yu G |
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Institution: | (1) Sobolev Institute of Mathematics, Novosibirsk |
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Abstract: | We consider quasi-isometric mappings of domains in multidimensional Euclidean spaces. We establish that a mapping depends continuously in the sense of the topology of Sobolev classes on its metric tensor to within isometry of the space. In the space of metric tensors we take the topology determined by means of almost everywhere convergence. We show that if the metric tensor of a mapping is continuous then the length of the image of a rectifiable curve is determined by the same formula as in the case of mappings with continuous derivatives. (Continuity of the metric tensor of a mapping does not imply continuity of its derivatives.) |
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Keywords: | quasi-isometric mapping metric tensor locally weak convergence of Jacobians semicontinuity of functionals of calculus of variations |
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