## Inverse Problems for Partial Differential EquationsIn 8 years after publication of the ?rst version of this book, the rapidly progre- ing ?eld of inverse problems witnessed changes and new developments. Parts of the book were used at several universities, and many colleagues and students as well as myself observed several misprints and imprecisions. Some of the research problems from the ?rst edition have been solved. This edition serves the purposes of re?ecting these changes and making appropiate corrections. I hope that these additions and corrections resulted in not too many new errors and misprints. Chapters 1 and 2 contain only 2–3 pages of new material like in sections 1.5, 2.5. Chapter 3 is considerably expanded. In particular we give more convenient de?nition of pseudo-convexity for second order equations and included bou- ary terms in Carleman estimates (Theorem 3.2.1 ) and Counterexample 3.2.6. We give a new, shorter proof of Theorem 3.3.1 and new Theorems 3.3.7, 3.3.12, and Counterexample 3.3.9. We revised section 3.4, where a new short proof of exact observability inequality in given: proof of Theorem 3.4.1 and Theorems 3.4.3, 3.4.4, 3.4.8, 3.4.9 are new. Section 3.5 is new and it exposes recent progress on Carleman estimates, uniqueness and stability of the continuation for systems. In Chapter 4 we added to sections 4.5, 4.6 some new material on size evaluation of inclusionsandonsmallinclusions.Chapter5containsnewresultsonidenti?cation |

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### Contents

1 | |

Chapter 2 IllPosed Problems and Regularization | 23 |

Chapter 3 Uniqueness and Stability in the CauchyProblem | 47 |

Single BoundaryMeasurements | 104 |

Many BoundaryMeasurements | 149 |

Chapter 6 Scattering Problems and Stationary Waves | 211 |

Chapter 7 Integral Geometry and Tomography | 240 |

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algorithm analytic applied assumptions boundary condition boundary measurements boundary value problem bounded domain Carleman estimates Cauchy data Cauchy problem coefficients conclude consider constant continuous convergence convex Corollary defined depend derivatives Dirichlet data Dirichlet problem Dirichlet-to-Neumann map eigenvalues elliptic equations elliptic operator equa Exercise formula given gravimetry heat equation Helmholtz equation Hölder hyperbolic equation hyperbolic problem implies inequality integral equation inverse conductivity problem inverse problem Isakov lateral boundary data Lemma linear Lipschitz Lipschitz domain maximum principles method Neumann data nonlinear norm numerical observe obtain operator parabolic equation parabolic problem plane potential proof is complete proof of Theorem Radon transform regularization respect scattering Schrödinger Schrödinger equation Section smooth solves space stability estimate term Theorem Theorem 4.1 tion uniquely determines uniqueness results wave number zero