Topological optimization and optimal transport

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This work explores the optimization of geometry and topology of shapes through discussions on shape representations, relaxation theory, and optimal transport, highlighting the mathematical tools involved. It delves into applications across various fields, including science, engineering, economics, social sciences, biology, physics, and image processing. The contents are divided into two parts. The first part addresses geometric issues in PDE problems related to the infinity Laplace operator, solutions to free boundary problems amid geometric uncertainties, and distributed and boundary control problems for the semidiscrete Cahn–Hilliard/Navier–Stokes system with nonsmooth Ginzburg–Landau energies. It also covers high-order topological expansions for 2D Helmholtz problems, a new phase field model for approximating interfacial energies in multiphase systems, and the optimization of eigenvalues and eigenmodes using the adjoint method, along with discrete varifolds and surface approximation. The second part focuses on weak Monge–Ampere solutions of the semi-discrete optimal transportation problem, optimal transportation theory with repulsive costs, and Wardrop equilibria. It examines the Lagrangian branched transport model and its equivalence with the Eulerian formulation, nonlinear evolution systems as perturbations of Wasserstein gradient flows, and pressureless Euler equations with a maximal density constraint. Finally, it di