Applications of Elliptic Carleman Inequalities to Cauchy and Inverse Problems

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Springer, Jun 3, 2016 - Mathematics - 81 pages

This book presents a unified approach to studying the stability of both elliptic Cauchy problems and selected inverse problems. Based on elementary Carleman inequalities, it establishes three-ball inequalities, which are the key to deriving logarithmic stability estimates for elliptic Cauchy problems and are also useful in proving stability estimates for certain elliptic inverse problems.

The book presents three inverse problems, the first of which consists in determining the surface impedance of an obstacle from the far field pattern. The second problem investigates the detection of corrosion by electric measurement, while the third concerns the determination of an attenuation coefficient from internal data, which is motivated by a problem encountered in biomedical imaging.

 

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Contents

1 Preliminaries
1
2 Uniqueness of Continuation and Cauchy Problems
7
3 Determining the Surface Impedance of an Obstacle from the Scattering Amplitude
38
4 Determining a Corrosion Coefficient from a Boundary Measurement and an Attenuation Coefficient from an Internal Measurement
63
Index
81
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