Identifiability of diffusion coefficients for source terms of non-uniform sign

Markus Bachmayr; Van Kien Nguyen

doi:10.3934/ipi.2019045

Article Contents

2019, Volume 13, Issue 5: 1007-1021. Doi: 10.3934/ipi.2019045

This issue Previous Article On increasing stability of the continuation for elliptic equations of second order without (pseudo)convexity assumptions Next Article Lipschitz stability for the finite dimensional fractional Calderón problem with finite Cauchy data

Identifiability of diffusion coefficients for source terms of non-uniform sign

Markus Bachmayr^1, , and
Van Kien Nguyen^2,3,

1.
Institut für Mathematik, Johannes Gutenberg-Universität Mainz, Staudingerweg 9, 55128 Mainz, Germany
2.
Institute for Numerical Simulation, University of Bonn, Endenicher Allee 19B, 53115 Bonn, Germany
3.
Department of Mathematics, University of Transport and Communications, No.3 Cau Giay Street, Lang Thuong Ward, Dong Da District, Hanoi, Vietnam

^* Corresponding author
^* Corresponding author

Received: October 31, 2018

Revised: March 31, 2019

Published: July 2019

The authors acknowledge support by the Hausdorff Center of Mathematics, University of Bonn

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Abstract

The problem of recovering a diffusion coefficient $ a $ in a second-order elliptic partial differential equation from a corresponding solution $ u $ for a given right-hand side $ f $ is considered, with particular focus on the case where $ f $ is allowed to take both positive and negative values. Identifiability of $ a $ from $ u $ is shown under mild smoothness requirements on $ a $, $ f $, and on the spatial domain $ D $, assuming that either the gradient of $ u $ is nonzero almost everywhere, or that $ f $ as a distribution does not vanish on any open subset of $ D $. Further results of this type under essentially minimal regularity conditions are obtained for the case of $ D $ being an interval, including detailed information on the continuity properties of the mapping from $ u $ to $ a $.

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Mathematics Subject Classification: 35R30, 35J25.

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References

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