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Inverse Problems Read chapter Read first chapter

2017 | OriginalPaper | Chapter

3. Uniqueness and Stability in the Cauchy Problem

Author : Victor Isakov

Published in: Inverse Problems for Partial Differential Equations

Publisher: Springer International Publishing

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Abstract

In this chapter we formulate and in many cases prove results on the uniqueness and stability of solutions of the Cauchy problem for general partial differential equations. One of the basic tools is Carleman-type estimates. In Section 3.1 we describe the results for a simplest problem of this kind (the backward parabolic equation), where a choice of the weight function in Carleman estimates is obvious, and the method is equivalent to that of the logarithmic convexity. In Section 3.2 we formulate general conditional Carleman estimates and their simplifications for second-order equations, and we apply the results to the general Cauchy problem and give numerous counterexamples showing that the assumptions of positive results are quite sharp.

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previous chapter Ill-Posed Problems and Regularization
next chapter Elliptic Equations: Single Boundary Measurements
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Metadata
Title
Uniqueness and Stability in the Cauchy Problem
Author
Victor Isakov
Copyright Year
2017
Book
Inverse Problems for Partial Differential Equations

Print ISBN: 978-3-319-51657-8

Electronic ISBN: 978-3-319-51658-5

Copyright Year: 2017

DOI
https://doi.org/10.1007/978-3-319-51658-5_3

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