Let X be a smooth bordered surface in R-3 with a smooth boundary and (sigma)over cap s a smooth anisotropic conductivity on X. If the genus of X is given, then starting from the Dirichlet- to- Neumann operator Lambda(sigma) over cap on partial derivative X, we give an explicit procedure to find a unique Riemann surface Y (up to a biholomorphism), an isotropic conductivity s on Y and a quasiconformal diffeomorphism F : X -> Y which transforms (sigma) over cap into sigma.As a corollary, we obtain the following uniqueness result: if sigma(1) and sigma(2) are two smooth anisotropic conductivities on X with Lambda(sigma 1) = Lambda(sigma 2), then there exists a smooth diffeomorphism Phi : (X) over bar -> (X) over bar such that Phi vertical bar(partial derivative X) and Phi(*)sigma 1 = sigma 2.

Gel'fand-Calderón's inverse problem for anisotropic conductivities on bordered surfaces in ℝ3

Matteo Santacesaria
2012

Abstract

Let X be a smooth bordered surface in R-3 with a smooth boundary and (sigma)over cap s a smooth anisotropic conductivity on X. If the genus of X is given, then starting from the Dirichlet- to- Neumann operator Lambda(sigma) over cap on partial derivative X, we give an explicit procedure to find a unique Riemann surface Y (up to a biholomorphism), an isotropic conductivity s on Y and a quasiconformal diffeomorphism F : X -> Y which transforms (sigma) over cap into sigma.As a corollary, we obtain the following uniqueness result: if sigma(1) and sigma(2) are two smooth anisotropic conductivities on X with Lambda(sigma 1) = Lambda(sigma 2), then there exists a smooth diffeomorphism Phi : (X) over bar -> (X) over bar such that Phi vertical bar(partial derivative X) and Phi(*)sigma 1 = sigma 2.
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Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/11567/927672
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