Uniqueness and stability in the Cauchy Problem

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Authors
Isakov, Victor
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Issue Date
2017
Type
Book chapter
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Isakov V. (2017) Chapter 3. Uniqueness and Stability in the Cauchy Problem. In: Inverse Problems for Partial Differential Equations. Applied Mathematical Sciences, vol 127. Springer, Cham, 47-103.
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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Publisher
Springer
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Series
Applied Mathematical Sciences;v.127
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ISSN
0066-5452
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