The methods for solving domain optimization problems depends on the case of study. There are methods that have been developed for the discretized problem, but not much is done in the infinite dimensional case. We analyze the theoretical aspects of the infinite dimensional case for a particular domain optimization problem where a portion of the boundary is parametrized, these results involve the existence of the solution to our problem and the calculation of the derivative of the shape functional. Shape optimization problems have a long history of mathematical study and a wide range of applications. In recent decades there has been an interest in solving these problems with partial differential equation (PDE) constraints. We consider a special class of PDE-constrained shape optimization problems where different boundary condition types (Dirichlet and Neumann) are imposed on the same boundary segment. We also consider the case where the interface between these different boundary condition types may also be parameter dependent. This study also includes special cases where the shape of the region where the PDE is imposed does not change, but the domain of the partial differential operator is parameter dependent, due to the change in boundary condition type. Our treatment centers on the infinite dimensional formulation of the optimization problem. We consider existence of solutions as well as the calculation of derivatives of the associated shape functionals via adjoint solutions. These derivative formulations serve as a starting point for practical numerical approximations. / Ph. D. / Optimization problems arise in a number of areas and are usually posed as finding values of design parameters that minimize a given cost function. Examples include finding the shape of a car or airplane wing to reduce drag and improve fuel economy which maintaining a desired level of performance. This is an example of a constrained optimization problem where the constraint is described by a physical model known as a partial differential equation (PDE). For shape optimization problems, we want to find the best shape to minimizes a certain cost function, and the cost depends on the shape through the solution to the PDE. The strategy for solving a shape optimization problem depends on the particular problem at hand. In many cases, one assumes that the solution of an optimization problem exists, so the development of methods to find or approximate possible solutions is the first step. In this dissertation, we study some theoretical aspects of the problem that can be used to guarantee the existence of an optimal (or locally optimal) solution to the problem. We focus our attention on a special class of PDE constraints where the cost function is calculated over a domain with an unknown portion that needs to be determined. We further consider a special case of boundary conditions for the PDE constraints known as mixed boundary conditions. In this work, we study the theoretical aspects to guarantee the existence of a solution, and then we provide formulations of the derivatives that permit algorithms to search for the shape of the domain that minimizes a given cost function. These formulations are important to develop efficient numerical approximations.

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