## 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 | |

IllPosed Problems and Regularization | 20 |

Uniqueness and Stability in the Cauchy Problem | 41 |

Single Boundary Measurements | 89 |

Many Boundary Measurements | 127 |

Scattering Problems | 173 |

Integral Geometry and Tomography | 192 |

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algorithms analytic applied assume assumptions boundary condition boundary measurements boundary value problem Carleman estimates Cauchy data Cauchy problem coefficients conclude conformal mapping consider constant continuous convergence convex Corollary data g0 deﬁned deﬁnition depend derivatives Dirichlet data Dirichlet problem Dirichlet-to Dirichlet-to-Neumann map eigenvalues elliptic equations elliptic operator Exercise ﬁnd ﬁrst formula given gravimetry Green’s heat equation Helmholtz equation hyperbolic equation hyperbolic problem ill-posed problems implies inequality initial boundary value integral equation inverse conductivity problem inverse problem Isakov lateral boundary data Lemma linear Lipschitz maximum principle method Neumann data Neumann map nonlinear norm numerical observe obtain operator parabolic equation parabolic problem potential proof is complete proof of Theorem Prove uniqueness Radon transform regularization respect right side satisﬁes satisfying Section singular smooth solution solves space stability estimate Theorem Theorem 4.1 theory Uhlmann uniquely determines uniqueness results zero