Mathematical Sciences: Some Inverse Problems for Partial Differential Equations

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This project studies uniqueness and stability of solutions of well-known but ill-posed inverse problems. It considers the inverse problem of potential theory that is fundamental in geophysics, the inverse conductivity problem with applications in electrical tomography and in crack detection and a non-hyperbolic Cauchy problem for hyperbolic equations as well as related heat conduction parameter identification, earthquakes prediction and acoustic and seismic prospecting. The work will involve some classical (potential theory, Carleman type and energy estimates, orthogonality relations) and more recent methods ( the logarithmic convexity and the Calderon-Sylvester-Uhlmann's method). The aim is to solve fundamental questions, provide applications and derive new mathematical tools.