Applications of Elliptic Carleman Inequalities to Cauchy and Inverse Problems (SpringerBriefs in Mathematics)

Annotation 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 Contents 6 Introduction 7 References 9 1 Preliminaries 10 1.1 Function Spaces 10 1.2 Geometric Assumptions 12 References 14 2 Uniqueness of Continuation and Cauchy Problems 15 2.1 Two-Dimensional Cauchy Problems 15 2.2 A Carleman Inequality for a Family of Operators 23 2.3 Three-Ball Inequalities and the Uniqueness of Continuation 27 2.4 Stability Estimates for Cauchy Problems 32 References 45 3 Determining the Surface Impedance of an Obstacle from the Scattering Amplitude 46 3.1 Carleman Estimate Depending Uniformly on Bounds on the Weight Function 46 3.2 Local Behavior at the Boundary of a Solution of an Obstacle Problem 52 3.3 Determining the Surface Impedance of an Obstacle 63 References 69 4 Determining a Corrosion Coefficient from a Boundary Measurement and an Attenuation Coefficient from an Internal Measurement 70 4.1 Local Behavior of Solutions at the Boundary 70 4.2 Determining the Corrosion Coefficient 74 4.3 An Inverse Medium Problem with Internal Data 76 References 87 Index 88 Front Matter....Pages i-ix Preliminaries....Pages 1-5 Uniqueness of Continuation and Cauchy Problems....Pages 7-37 Determining the Surface Impedance of an Obstacle from the Scattering Amplitude....Pages 39-62 Determining a Corrosion Coefficient from a Boundary Measurement and an Attenuation Coefficient from an Internal Measurement....Pages 63-80 Back Matter....Pages 81-81