Introduction to Inverse Problems for Evolution Equations

Among several main formulations, the book treats inverse problems with single measurements by Carleman estimates and describes a method for proving the uniqueness and the stability for the first-order transport equations, parabolic equations, and hyperbolic equations. The book gives self-contained derivations of Carleman estimates, which do not rely on any general theory and simplified application method of the Carleman estimates to the inverse problems. Thus, the target audience is graduate students of faculty of sciences and interested graduate students from engineering fields. The book tries to limit the preliminary knowledge to a standard 4-year undergraduate course. In order to master the methodology, we restrict equations within first-order transport equations, parabolic equations and hyperbolic equations, but it is expected to apply the same methods for other evolution equations such as fluid dynamics.

Accessible accounts: needed knowledge is calculus at undergraduate level Self-containedness Applications to inverse problems: unified approach staring with the derivation of Carleman estimates

Autorentext

Masahiro Yamamoto , born in 1958 in Tokyo, Japan, completed his Ph.D. at The University of Tokyo in 1988 and has held various academic positions there, including Associate Professor from 1990 to 2010 and Professor from 2010 to 2024. He is currently a Project Professor at The University of Tokyo and a Professor at Zonguldak Bülent Ecevit University in Türkiye. He has engaged in multiple visiting professorships over the past decade at esteemed institutions such as Sapienza Università di Roma and has a notable history with various French universities and was an Alexander von Humboldt Fellow at Technische Universität München from 1992 to 1993.

Inhalt

• 1. How inverse problems appear and what should we study about inverse problems?.- 2. Application of the multiplier method to an inverse wave source problem.- 3. Inverse problems by Carleman estimates for first-order transport equations.- 4. Inverse problems by Carleman estimates for parabolic equations.- 5. Inverse problems by Carleman estimates for hyperbolic equations.- 6. Conditional stability and convergence rates of the Tikhonov regularization.- 7. Other estimates of Carleman type.- 8. Derivation of Carleman estimates for elliptic equations.