Reconstruction of the derivative of the conductivity at the boundary

Felipe Ponce-Vanegas

doi:10.3934/ipi.2020032

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2020, Volume 14, Issue 4: 701-718. Doi: 10.3934/ipi.2020032

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Reconstruction of the derivative of the conductivity at the boundary

Felipe Ponce-Vanegas

BCAM - Basque Center for Applied Mathematics, Mazarredo, 14 E48009 Bilbao, Basque Country – Spain

Received: September 2019

Revised: February 2020

Published: May 2020

The author is supported by the Basque Government and by the Spanish State Research Agency

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Abstract

We describe a method to reconstruct the conductivity and its normal derivative at the boundary from the knowledge of the potential and current measured at the boundary. The method of reconstruction works for isotropic conductivities with low regularity. This boundary determination for rough conductivities implies the uniqueness of the conductivity in the whole domain $ \Omega $ when it lies in $ W^{1+\frac{n-5}{2p}+, p}(\Omega) $, for dimensions $ n\ge 5 $ and for $ n\le p<\infty $.

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Mathematics Subject Classification: Primary: 35R30; Secondary: 42B37.

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Figure 1. Integration regions in equation (8)

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