Global Uniqueness for a Two-Dimensional Inverse Boundary Value Problem

by: Adrian I Nachman

The Annals of Mathematics, Vol. 143, No. 1. (jan 1996), pp. 71-96.

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Abstract

We show that the coefficient $\γ(x)$ of the elliptic equation $\∇ \⋅ (\γ \∇ u) = 0$ in a two-dimensional domain is uniquely determined by the corresponding Dirichlet-to-Neumann map on the boundary, and give a reconstruction procedure. For the equation $\Σ \∂_i (\γ^ij \∂_j u) = 0$, two matrix-valued functions $\γ_1$ and $\γ_2$ yield the same Dirichlet-to-Neumann map if and only if there is a diffeomorphism of the domain which fixes the boundary and transforms $\γ_1$ into $\γ_2$.

@article{199601, abstract = {We show that the coefficient \$\\gamma(x)\$ of the elliptic equation \$\\nabla \\cdot (\\gamma \\nabla u) = 0\$ in a two-dimensional domain is uniquely determined by the corresponding Dirichlet-to-Neumann map on the boundary, and give a reconstruction procedure. For the equation \$\\Sigma \\partial\_i (\\gamma^{ij} \\partial\_j u) = 0\$, two matrix-valued functions \$\\gamma\_1\$ and \$\\gamma\_2\$ yield the same Dirichlet-to-Neumann map if and only if there is a diffeomorphism of the domain which fixes the boundary and transforms \$\\gamma\_1\$ into \$\\gamma\_2\$.}, author = {Nachman, Adrian I. }, citeulike-article-id = {2623459}, journal = {The Annals of Mathematics}, keywords = {bibtex-import}, month = {jan}, number = {1}, pages = {71--96}, posted-at = {2008-04-02 15:12:59}, priority = {2}, publisher = {Annals of Mathematics}, series = {2}, title = {Global Uniqueness for a Two-Dimensional Inverse Boundary Value Problem}, url = {http://links.jstor.org/sici?sici=0003-486X\%28199601\%292\%3A143\%3A1\%3C71\%3AGUFATI\%3E2.0.CO\%3B2-E}, volume = {143}, year = {1996} }

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TY - JOUR ID - 199601 TI - Global Uniqueness for a Two-Dimensional Inverse Boundary Value Problem SE - 2 JF - The Annals of Mathematics VL - 143 IS - 1 SP - 71 EP - 96 PB - Annals of Mathematics N2 - We show that the coefficient $\\gamma(x)$ of the elliptic equation $\\nabla \\cdot (\\gamma \\nabla u) = 0$ in a two-dimensional domain is uniquely determined by the corresponding Dirichlet-to-Neumann map on the boundary, and give a reconstruction procedure. For the equation $\\Sigma \\partial_i (\\gamma^{ij} \\partial_j u) = 0$, two matrix-valued functions $\\gamma_1$ and $\\gamma_2$ yield the same Dirichlet-to-Neumann map if and only if there is a diffeomorphism of the domain which fixes the boundary and transforms $\\gamma_1$ into $\\gamma_2$. KW - bibtex-import AU - Nachman, Adrian PY - 1996/jan// UR - http://links.jstor.org/sici?sici=0003-486X%28199601%292%3A143%3A1%3C71%3AGUFATI%3E2.0.CO%3B2-E ER -

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