Variations of Dirichlet-to-Neumann map and deformation boundary rigidity of simple 2-manifolds |
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Authors: | Vladimir Sharafutdinov |
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Affiliation: | (1) Sobolev Institute of Mathematics, 4 Koptjug Avenue, 630090 Novosibirsk, Russia |
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Abstract: | The Dirichlet-to-Neumann (DN) map Λg: C∞ (?M) → C∞(?M) on a compact Riemannian manifold (M, g) with boundary is defined by Λgh = ?u/?v¦in{t6M}, where u is the solution to the Dirichlet problem Δu = 0, u¦?M = h and v is the unit normal to the boundary. If gt = g + t? is a variation of the metric g by a symmetric tensor field ?, then Λg t = Λg + tΛ? + o(t). We study the question: How do tensor fields ? look like for which Λ? =0? A partial answer is obtained for a general manifold, and the complete answer is given in the two cases: For the Euclidean metric and in the 2D-case. The latter result is used for proving the deformation boundary rigidity of a simple 2-manifold. |
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Keywords: | KeywordHeading" >Math Subject Classifications 53A45 58J32 |
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