Gradient Flow in Shape Optimization
Монография
This book investigates gradient flow dynamical systems in the context of shape optimization problems, with a focus on regularization of the shape functional. Two representative model problems are selected for this purpose. The first is the single-measurement identification problem, which arises in electrical impedance tomography, particularly in geophysical applications. The second is the Bernoulli free boundary problem, known for its wide range of applications. Throughout the monograph, these problems serve as illustrative models for fundamental methods
and approaches in shape optimization theory. Variational formulations are introduced for both problems. In the case of the inverse problem, the formulation is based on minimizing the integral Kohn–Vogelius functional . A regularization strategy involving the one-dimensional Willmore functional is discussed. Basic concepts from the differential geometry of curves in the Euclidean plane are recalled, and explicit
formulas for the gradients of the Kohn–Vogelius and Willmore functionals are derived. The chapter outlines the main results of the monograph, including expressions for the Hessians of both functionals. The application of the Nash–Moser method is described to establish the solvability of the gradient flow equations, and linearized equations are formulated in normal coordinates. Finally, potential generalizations of the presented results are briefly discussed.
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