Optimization with PDE Constraints : ESF Networking Program OPTPDE This book on PDE Constrained Optimization contains contributions on the mathematical analysis and numerical solution of constrained optimal control and optimization problems where a partial differential equation (PDE) or a system of PDEs appears as an essential part of the constraints. The appropriate treatment of such problems requires a fundamental understanding of the subtle interplay between optimization in function spaces and numerical discretization techniques and relies on advanced methodologies from the theory of PDEs and numerical analysis as well as scientific computing. The contributions reflect the work of the European Science Foundation Networking Programme ’Optimization with PDEs’ (OPTPDE) Solution of 2D Contact Shape Optimization Problems Phase Field Methods for Binary Recovery Programming with Separable Ellipsoidal Constraints Adaptive Finite Elements for Optimally Controlled Elliptic Variational Inequalities Topology Design of Elastic Structures for a Contact Model Bisection Methods for Mesh Generation Differentiability of Energy Functionals for Unilateral Problems in Domains Two Sided Guaranteed Estimates of the Cost Functional for Optimal Control Problems with Elliptic State Equations Sensitivity Analysis of Work Functional for Compressible Navier Stokes Equations Exact Controllability to Trajectories for Navier Stokes Equations