Gradient flows in the shape optimization theory

The identification problem of an inclusion is considered in the paper. The inclusion is unknown subdomain of a given physical region. The available information on the inclusion is governed by measurements on the boundary of this region. In particular, the single measurement problem of impedance electrotomography and other inverse problems are included in our approach. The shape identification problem can be solved by the minimization of an objective function taking into account the measurement data. The best choice of such objective function is the Kohn–Vogelius energy functional. The standard regularization of the Kohn–Vogelius functional include the perimeter and Willmore curvature functional evaluated for an admissible inclusion boundary. In the two-dimensional case, a nonlocal existence theorem of strong solutions is proved for the gradient flow dynamical system generated for such a regularization of the Kohn–Vogelius functional.