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Carleman estimates for second order partial differential operators and applications. A unified approach. (English) Zbl 1445.35006

SpringerBriefs in Mathematics. Cham: Springer (ISBN 978-3-030-29529-5/pbk; 978-3-030-29530-1/ebook). xi, 127 p. (2019).
In the monograph, the Chinese mathematicians from the School of Mathematics, Sichuan University, Chengdu, give a brief and self-contained introduction to Carleman estimates for three typical partial differential equations of second order (elliptic, parabolic and hyperbolic equations) and their typical applications in control, unique continuation, and inverse problems. In the process, the authors consider null controllability for semilinear parabolic equations, exact controllability for semilinear hyperbolic equations and others. The main novelties of the book are as following. First, all estimates are derived from a fundamental identity for second-order partial differential operators. The difference is only the choice of weight functions. Second, only some rather weak smoothness and/or integrability conditions are needed for the coefficients appearing in the equations. Carleman estimates for various systems and problems have been rather actively studied in the recent years. The authors suggest their own view on such estimates. The book will be useful for specialists in the respective area of mathematical analysis as well as for specialists in the field of control theory.

MSC:

35-02 Research exposition (monographs, survey articles) pertaining to partial differential equations
35Kxx Parabolic equations and parabolic systems
35Lxx Hyperbolic equations and hyperbolic systems
35Jxx Elliptic equations and elliptic systems
35Q93 PDEs in connection with control and optimization
35B60 Continuation and prolongation of solutions to PDEs
35R30 Inverse problems for PDEs
93B05 Controllability
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